TREATISE ON A NALYSlS Volume 111
This is Volume 10-111 in PURE AND APPLIED MATHEMATICS A series of Monographs and Textbooks EILENBERG AND HYMAN BASS Editors: SAMUEL A list of recent titles in this series appears at the end of this volume.
Volume 10 TREATISE ON ANALYSIS 10-1. Chapters I-XI (Foundations of Modern Analysis, enlarged and corrected printing, 1969) 10-11. Chapters XII-XV, 1970 10-111. Chapters XVI-XVII, 1972 1 0-IV. Chapters XVIII-XX, 1974
TREATISE O N
A NA LYS IS J. DIEUDONNE Universitt de Nice Facuitt des Sciences Parc Valrose, Nice, France
Volume 111
Translated by
1. G. Macdonald University of Manchester Manchester, England
ACADEMIC PRESS
1972
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AMS(M0S) 1970 Subject Classifications: 58-01,58A05, 58A10,58A30 PRINTED IN 82
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“Treatise on Analysis,” Volume 111 First published in the French Language under the title “Elements d’Analyse,”,tome 3 and copyrighted in I970 by Gauthier-Villars,Editeur, Paris, France.
SCHEMATIC PLAN OF THE WORK I. Elements of
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CONTENTS
Notation..
................................
Chapter XVI
DIFFERENTIAL MANIFOLDS
.....................
1. Charts, atlases, manifolds. 2. Examples of differential manifolds. Diffeomorphisms. 3. Differentiable mappings. 4. Differentiable partitions of unity. 5. Tangent spaces, tangent linear mappings, rank. 6. Products of manifolds. 7. Immersions, submersions, subimmersions. 8. Submanifolds. 9. Lie groups. 10. Orbit spaces and homogeneous spaces. 11. Examples: unitary groups, Stiefel manifolds, Grassmannians, projective spaces. 12. Fibrations. 13. Definition of fibrations by means of charts. 14. Principal fiber bundles. 15. Vector bundles. 16. Operations on vector bundles. 17. Exact sequences, subbundles, and quotient bundles. 18. Canonical morphisms of vector bundles. 19. Inverse image of a vector bundle. 20. Differential forms. 21. Orientable manifolds and orientations. 22. Change of variables in multiple integrals. Lebesgue measures. 23. Sard's theorem. 24. Integral of a differential n-form over an oriented pure manifold of dimension n. 25. Embedding and approximation theorems. Tubular neighborhoods. 26. Differentiable homotopies and isotopies. 27. The fundamental group of a connected manifold. 28. Covering spaces and the fundamental group. 29. The universal covering of a differential manifold. 30. Covering spaces of a Lie group.
ix
1
Chapter XVll
DIFFERENTIAL CALCULUS ON A DIFFERENTIAL MANIFOLD I. DISTRIBUTIONS AND DIFFERENTIAL OPERATORS. . . . . . . 230 1. The spaces &(')(U) (U open in R"). 2. Spaces of C" (resp. C') sections of vector vii
viii
CONTENTS
bundles. 3. Currents and distributions. 4. Local definition of a current. Support of a current. 5. Currents on an oriented manifold. Distributions on R". 6. Real distributions. Positive distributions. 7. Distributions with compact support. Point-distributions. 8. The weak topology on spaces of distributions. 9. Example: finite parts of divergent integrals. 10. Tensor products of distributions. 11. Convolution of distributions on a Lie group. 12. Regularization of distributions. 13. Differential operators and fields of point-distributions. 14. Vector fields as differential operators. 15. The exterior differential of a differential pform. 16. Connections in a vector bundle. 17. Differential operators associated with a connection. 18. Connections on a differential manifold. 19. The covariant exterior differential. 20. Curvature and torsion of a connection. Appendix
.......................
347
.................................
378
...................................
381
MULTILINEAR ALGEBRA
8. Modules. Free modules. 9. Duality for free modules. 10. Tensor product of free modules. 11. Tensors. 12. Symmetric and antisymmetric tensors. 13. The exterior algebra. 14. Duality in the exterior algebra. 15. Interior products. 16. Nondegenerate alternating bilinear forms. Symplectic groups. 17. The symmetric algebra. 18. Derivations and antiderivations of graded algebras. 19. Lie algebras.
References Index
NOTATlO N
In the following definitions, the first number is the number of the chapter and the second number is the number of the section within that chapter.
dimx(X) (resp. dim,, ,(X)) dim(X), dim,(X) CIV
44 TXW)
%, x 0, 9
2,
Tx(f)
rkxf
dimension of a differential (resp. complexanalytic) manifold X at a point x : 16.1 dimension of a pure differential (resp. complex-analytic) manifold X: 16.1 restriction of a chart c to an open set V: 16.1 image of a chart c under a homeomorphism u : 16.2 tangent space at a point x of a differential manifold X: 16.5 mapping of Tx(X) onto R" induced by a chart c = (U, cp, n) at the point x : 16.5 canonical mapping of Tx(E) onto E, where E is a finite-dimensional affine space: 16.5 tangent linear mapping to f at the point x: 16.5 rank off at the point x : 16.5
x
NOTATION
JX
x x,Y
differential at the point x of the mapping f (resp. f) of X into a vector space (resp. into R): 16.5 Hessian off at the point x: 16.5 differential manifold underlying a complex-analytic manifold X: 16.5 mapping of T,(XIR) onto R'", where X is a complex-analytic manifold of dimension n: 16.5 R-linear automorphism of the tangent space at x to a complex-analytic manifold, defined by multiplication by i: 16.5 space of antilinear forms on the tangent space at x to a complex-analytic manifold X: 16.5 C-linear and C-antilinear parts of dxf, where f is a complex function on a complex-analytic manifold : 16.5 gradient off at the point x: 16.5, Problem 7 sets of jets of order k from X to Y: 16.5, Problem 9 jet of order k of the mapping f at the point x: 16.5, Problem 9 fiber-product of two manifolds X, Y over Z: 16.8, Problem 10 general linear groups in n variables over R, C, H: 16.9 actions of elements s, t of a Lie group G on a tangent vector h, at a point x E G: 16.9 special linear groups: 16.9 s an element of a Lie group G, x a point of a manifold X on which G acts, h, a tangent vector to G at s, k, a tangent vector to X at x: 16.10 spaces of quadratic (resp. hermitian) forms over a real (resp. complex or quaternionic) vector space E : 16.11 orthogonal groups: 16.11 rotation groups: 16.1 1 unitary groups: 16.1 1
NOTATION
x x,x X xGF
Oh4
b(B;R) (resp. b(B; C)), b(B) Mor(E, E')
E' 8 E" s' 0 S"
E' @ E"
xi
special unitary group: 16.11 Stiefel manifolds : 16.1 1 Grassmannians: 16.11 projective spaces: 16.11 rotation group: 16.1 1, Problem 4 symplectic group : 16.11, Problem 6 inverse image of a fiber bundle: 16.12 inverse image of a section: 16.12 set of C"-sections of a bundle X over a submanifold B' of the base: 16.12 fiber-product of two bundles over B: 16.12 bundle associated with a principal bundle X (with group G), with fiber-type F: 16.14 element of X x F corresponding to X E X and Y E F: 16.14 x a point of X, y a point of F, h, a tangent vector at x to X, k, a tangent vector at y to F: 16.14 zero section of a vector bundle E: 16.15 zero element of a fiber Eb of a vector bundle E: 16.15 rank of a vector bundle E at a point b of the base manifold: 16.15 tangent bundle of a differential manifold M: 16.15 canonical projection of the tangent bundle T(M): 16.15 algebra of real- (resp. complex-) valued C"-functions on B: 16.15 vector space of morphisms of E into E', where E and E' are vector bundles over the same base B: 16.15 vector space of C"-sections of a vector bundle E over B: 16.15 direct sum (Whitney sum) of two vector bundles E , E" over B: 16.16 direct sum of a section s' of E' and a section s" of E": 16.16 tensor product of two vector bundles E , E" over B: 16.16
xii
NOTATION
s' €3 sn
mE, E@"'
K E
tensor product of a section s' of E' and a section s" of E": 16.16 direct sum (tensor product) of m copies of a vector bundle E: 16.16 mth exterior power of a vector bundle E: 16.16
exterior product of m sections of E: 16.16 0
E@O,A E, I Hom(E', E")
mth exterior power of a homomorphism of vector bundles: 16.16 trivial line-bundle: 16.16 vector bundle of homomorphisms of a vector bundle E' into a vector bundle E" : 16.16
a
dual of a vector bundle E: 16.16 E a section of E, s* a section of E*: 16.16 transpose of a homomorphism u of vector bundles : 16.16 homomorphism of Hom(E, E") into Hom(F', F") corresponding to two vector bundle homomorphisms u': F' + E', u": E" -+ F": 16.16 canonical vector bundle over Gn,JR), (resp. GnJC)): 16.16, Problem 1 bundle of symmetric (antisymmetric) tensors of order m over a vector bundle E: 16.17 space of symmetric (antisymmetric) tensors of order m over the fiber Eb: 16.17 contraction of the contravariant index j and the covariant index k: 16.18 trace of an endomorphism u of a vector bundle : 16.18 antisymmetrization operator: 16.18
SAt
exterior product of a section of A E
Hom(u', u")
P
4
AE
i;,E* i(s)z*, i,
- Z*
and a section of A E : 16.18 exterior algebra bundle of a vector bundle E: 16.18 mth exterior power of the dual (= dual of the mth exterior power) of E: 16.18 interior product of a section s of E and
NOTATION
xiii
P
a section z* of AE*: 16.18 complex vector bundle obtained by extension of scalars from a real vector bundle E : 16.18 inverse image of a vector bundle E: 16.19 tensor bundle of type (p, q) over a manifold M: 16.20 vector space of C" tensor fields of type (p, q) on M : 16.20 vector space of C" real differential p forms on M: 16.20 differential of a function f of class C': 16.20 canonical differential l-form on the cotangent bundle T(X)* : 16.20 inverse image under f of a p-form a and of a covariant tensor field Z: 16.20 exterior products of vector-valued differential forms relative to a bilinear mapping B: 16.20 u a differential n-form on an oriented manifold X : 16.21 f a submersion of X onto Y (oriented manifolds of dimensions n and m, respectively), 5, an m-covector at the point y =f ( x ) , u, (resp. u) an n-covector at the point x (resp. an n-form): 16.21
-X
integral of an n-form v over an oriented manifold X of dimension n: 16.24 X an oriented manifold: 16.24
Y an oriented submanifold of dimensionp of the manifold X, c~ a p-form on X: 16.24 surface measure of the sphere Sn-': 16.24 integral along the fibers of a differential p-form a : 16.24, Problem 1 1 connected sum of two connected manifolds of the same dimension: 16.26, Problem 15 fundamental group: 16.27
xiv
NOTATION
g P ( X ; K), 9F)(X; K)
9(X; K), @')(X; K), 9(X), 9 q X )
&=,
nth homotopy group (n 2 2): 16.30, Problem 3 nth relative homotopy group: 16.30, Problem 4 space of complex-valued C"-functions on the open set U: 17.1 space of complex-valued C'-functions on the open set U: 17.1 space of C'-sections of a vector bundle E over an open set U: 17.2 space of C" (C? complex-valued differential p-forms on X: 17.3 the subspace of gP(X)(d':)(X)) consisting of p-forms with support contained in the compact set K: 17.3 the union of the g P ( X ;K) (9:)(X; K)) for all compact subsets K of X: 17.3 particular cases of 9 J X ; K), g:'(X; K), 9,(X), 9F)(X) for p = 0: 17.3 Dirac p-current defined by the tangent p-vector z,: 17.3 T a pcurrent, w a differential q-form (q S p ) : 17.3 T apcurrent, Y a vector field: 17.3 T a current, u a proper mapping: 17.3 space of p-currents (p-currents of order S r ) on X: 17.3 space of distributions (distributions of order S r ) on X: 17.3 T a p-current, u a differential p-form with compact support: 17.3 T a distribution, f a function: 17.3 support of a current T: 17.4 inverse image of a current under a local diffeomorphism IL : 17.4 space of locally integrable (n - p)-forms on X: 17.5 p-current defined by an (n - p)-form /?on an oriented manifold: 17.5 n-current defined by a scalar function f: 17.5
NOTATION
xv
n-current defined by a distribution T, relative to an n-form uo: 17.5 distribution defirsed by a function f (relative to an n-form u,,): 17.5 derivative of order v (v = multi-index) of a distribution T E 9'(U), where U is an open set in R": 17.5 Heaviside's function: 17.5 singular support of a distribution T: 17.5 inverse image of a current under a submersion: 17.5, Problem 8 value of the trace measure tr on a function f: 17.5, Problem 10 space of real-valued C' differential p forms on X: 17.6 space of distributions with compact support: 17.7 image of a compactly supported current T of order S r - 1 under a c'mapping I[: 17.7 T a p-current with compact support, a a differentialp-form: 17.8 Laplacian : 17.9 finite part of rc: 17.9 finite part of 2,: 17.9 distribution on R: 17.9
8
d'Alembertian: 17.9 distribution on R": 17.9
P.V. - ,P.V.(g(x))
Cauchy principal value: 17.9, Problem 1
i-
tensor product of two distributions: 17.10 value of S @ T atf: 17.10 convolution of n distributions Tj on a Lie group : 17.1 1 T a distribution on a Lie group: 17.11 algebra of differential operators on X: 17.13 P a differential operator, T a distribution: 17.13 Lie derivative relative to a vector field X:
Diff(X)
'P * T
xvi
NOTATION
dcl
CT
da
&a, d”a
C
rel,
vu do
rr r r
I
t
‘24 t
U-1
‘X
17.14 Lie bracket of two vector fields X, Y: 17.14 exterior differential of a p-form a : 17.15 boundary of ap-current T: 17.15 exterior differential of a vector-valued p-form a: 17.15 C-linear and C-antilinear parts of du on a complex manifold : 17.15 linear connection in a vector bundle : 17.16 horizontal lifting of a vector field relative to a connection C: 17.16 covariant derivative of a mapping G of a manifold N into a vector bundle E, in the direction of a tangent vector h, to N, relative to a linear connection in E: 17.17 covariant differential of a tensor field U on M, relative to a linear connection on M: 17.18 covariant exterior differential (relative to a linear connection on E) of a differential p-form with values in E: 17.19 curvature morphism relative to the mapping f: 17.20 curvature morphism (or curvature) of a connection in a vector bundle E: 17.20 curvature tensor of a connection in E: 17.20 torsion morphism of a connection on M: 17.20 torsion tensor of a connection on M: 17.20 x an element of a module, x* an element of its dual: A.9.1 canonical mapping of a module E into its bidual: A.9.1 transpose of a linear mapping: A.9.3 contragradient of an isomorphism of one module onto another: A.9.3 transpose of a matrix X: A.9.4
NOTATION
E@",T"(E), T",E)
s * z, a . z UAV
ib
xvii
tensor product of linear mappings: A.10 tensor product of finitely-generated free A-modules: A.10.3 nth tensor power of a finitely-generated freemodule: A.ll.l module of tensors of type (p, 4):A.11.1 contraction of the contravariant index i and the covariant index j: A.11.3 trace of an endomorphism: A.11.3 transform of a contravariant tensor z by a permutation 0: A.12.1 symmetrization and antisymmetrization of a tensor: A.12.2 exterior product of two antisymmetris tensors: A.13.2 mth exterior power of a finitely-generated free module: A.13.3 m
basis elements of A E: A.13.3
AE
mth exterior power of a linear mapping: A.13.4 exterior algebra of a finitely-generated free A-module E: A.13.5 interior product of a q-vector and a (p q)-form: A.15.1 interior product by a vector x : A.15.4 symplectic groups: A.16.4 symmetric product of two symmetric tensors: A.17.1 nth symmetric power of a finitelygenerated free A-module E: A.17 symmetric algebra of a finitely-generated free A-module E: A.17 basis element of a symmetric power: A. 17 inner derivation in an associative algebra: A.18.2 inner derivation in a Lie algebra: A.19.4 algebra of derivations of a Lie algebra: A.19.4
+
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CHAPTER XVI
DIFFERENTIAL MANIFOLDS
One of the dominant themes of modern mathematics may be described as “analysis on differential manifolds.” Of course, the word “ analysis” is to be understood here in its widest sense; in it are inextricably blended a range and variety of concepts whose latent fertility would be stunted by confining them within the old and artificial divisions of algebra, geometry, and analysis in the classical sense of these words. The traditional domain of differential geometry, namely the study of curves and surfaces in three-dimensional space, was soon realized to be inadequate, particularly under the influence of mechanics. A solid body depends on six parameters, and a system of n pointsdependson 3nparameters, in general connected by certain relations. It is therefore natural to represent such systems by points in a space RN (where N is any positive integer) restricted to lie on a “submanifold ” of this space, defined by certain equations; and it is important to have available on such a “manifold” algorithms generalizing those of differential and integral calculus on open sets in RN, which were developed in Chapters VIII, X, and XI11 (see [37], Introduction). This and the two following chapters are devoted to the development of such algorithms. The classical viewpoint in the differential study of surfaces consists in regarding them as embedded in the ambient space R3. This point of view can be generalized, and in fact there is no loss of generality in considering only differential manifolds embedded in RN (Section 16.25, Problem 2). In certain problems, such an embedding can be a useful device (Sections 16.25 and 16.26). Nevertheless, this conception is patently artificial (for example, a sphere remains the same surface whether we embed it in R3 or in R4).Moreover, the necessity of embedding every manifold in a space RN would be an intolerable constraint in relation to many operations which arise naturally on manifolds (for example, the formation of “ orbit manifolds ” (16.10.3)). 1
2
XVI
DIFFERENTIAL MANIFOLDS
The first of the two fundamental notions which are developed in this Chapter is therefore the concept (which goes back to Gauss) of a differential manifold as an intrinsic object, independent of any adventitious embedding in a Euclidean space. The essential idea is that focaffya differential manifold of dimension n can be identified with an open subset of R” by means of a chart (16.1), and the structure of the manifold resides in the manner in which the charts are patched together. The property, for two mappings of R into R”, of being tangent at a point to E R (8.1) is invariant under a differentiable change of variables in R or in R”.This fact makes it possible to define, for two differentiable mappings f, g of R into a differential manifold M of dimension n, the relation “f and g are tangent at a point to E R ” (which implies that f ( t o ) = g(to) = x, E M). The equivalence classes for this relation (with to and xo fixed) are no longer called “derivatives ” but tangent vectors to M at thepoint xo. They form in a natural way a real vector space T,,(M) of dimension n, called the tangent vectorspace to M at thepoint xo (16.5). The fundamental difference, compared with analysis in vector spaces, is that the space TJM) varies with x (whereas all the values of the derivative of a function with values in R” are considered as belonging to the same vector space). This variation of the tangent space may seem familiar, from the example of surfaces in R3;unfortunately, geometric intuition here is misleading, because it suggests that the “tangent plane” also is embedded in R3. To see that one cannot arrive in this way at a correct conception of tangent vectors, it is enough to remark that a tangent vector at a point x of a surface S depends firstly on the two parameters which determine x, and then for each x on two more parameters which fix the vector in the tangent plane TJS). The tangent vectors to S must therefore be considered as forming a,four-dimensional manifold, which clearly cannot be embedded in R3.The notion which is appropriate here, and which enables us to “ p u l l ” the tangent vectors out of the ambient space, is the second fundamental idea in this chapter, that of aJiber bundle. In its various forms it dominates nowadays not only differential geometry, but all of topology (see [43] and [49]). This chapter is very long, and the greater part of it consists essentially of transcriptions:either in order to give intrinsic expression to properties (usually of a focal nature) of mappings of one differential manifold into another, by reducing to the case of open sets in R“ by means of charts; or in order to transpose to the context of vector bundles the elementary notions and results of linear and multilinear algebra. Apart from the theorem on the existence of orbit-manifolds (16.10.3) and its consequences, there is no substantial theorem before Section 16.21. The only aid to digestion of this accumulation of definitions and trivialities that I have been able to devise is to make the chapter yet longer, by inserting in the text and in the problems as many and
1 CHARTS, ATLASES, MANIFOLDS
3
various examples as possible, in order to show the full richness of the ideas introduced. The, most important of these examples are connected with the notion of Lie 'group (which will be studied in greater depth in Chapters XIX and XXI)and the closely related notion of homogeneous spaces. It is a fact of experience that the most important manifolds in applications are homogeneous space (for example, the upper half-plane Y z > 0 is a homogeneous space of the unimodular group SL(2, R)). From the fact that such a space can be put in the form G/H, one has algorithms available arising from the group structure of G; the precept of "lifting everything up to the group" has shown its validity in all studies of homogeneous spaces, and the reader will have many opportunities to see it in action. From Section 16.21 gnwards, we can at last begin to study some elementary global questions on differential manifolds: orientation (16.21), integration on a manifold (16.22-1 6.24), elementary properties of approximation and homotopy (16.25 and 16.26), and finally the theory of covering spaces, limited to differential manifolds (16.27-16.30).
1. CHARTS, ATLASES, MANIFOLDS
Let X be a topological space. A chart of X is a triplet c = (U, cp, n), where U is an open set in X, n is an integer 20,and cp is a homeomorphism of U onto an open set in R". The integer n is called the dimension of the chart c, and the open set U is its domain of definition. If V is an open set contained in U, it is clear that the restriction cplV is a homeomorphism of V onto an open set in R", and therefore V = (V, cp I V, n ) is a chart, called the resfriction of c to V. Consider two charts c = ( U , q , n ) and c ' = ( U , cp',n') of X with the same domain of definition U. They are said to be compatible if the two homeomorphisms
CI
: cp(U) cp'(u), cp 0 q'-' : cp'(U) + q(U) cp'
O
'P-l
+
(called the transition homeomorphisms) are indefinitely differentiable (8.12). This implies that, for each x E cp(U), the derivative D(q' 0 cp-')(x) is a bijective linear mapping of R" onto R"', hence n = n'. (16.1.1) For two charts (U, cp, n ) and (U, cp', n ) with the same domain of definition to be compatible, it is necessary and sufJicient that cp' 0 c p - l should be an indefinitely differentiable bijection of cp(U) onto cp'(U) whose derivative D(q' c p - I ) has rank n at every point of cp(U). 0
4
XVI
DIFFERENTIAL MANIFOLDS
The condition is clearly necessary, and the sufficiency follows from (10.2.5).
Two arbitrary charts c = (U, cp, n) and c' = (U', cp', n') of X are said to be compatible if either U n U' = 0 or the restrictions (U n U , cp I (U n U'), n) and (U n U', cp' I (U n U'), n') of c and c' to U n U' are compatible. If c and c' are compatible, then for each pair of open sets V c U and V' c U , the restrictions c ( V and c'lV' are compatible. An atlas of X is a set of charts of X, each pair of which are compatible and whose domains of definition cover X.Two atlases a,b of X are said to be compatible if %u b is an atlas of X,or equivalently if each chart in 2l is compatible with each chart in b.
On the set of atlases of X, the relation R : "aand b are compatible " is an equivalence relation.
(16.1.2)
Reflexivity and symmetry are obvious; we have to prove that the relation is transitive. ket a,,a,,'$1, be three atlases of X such that 9Il and a, are compatible, and 912 and a, compatible. We shall show that if c, = (U,, ql, n,) and c3 = (U,, cp,, n,) are charts belonging to 211 and a,, respectively, then they are compatible. We may assume that U1n U, # 0, or there is nothing to prove. If f,,f3 are the restrictions of cpl, cp3 to U, n U, , it is clear that f3 0f;' is a bijection of cpl(Ul n U,) onto (~3(U1n U,). Moreover, for each x E U1 n U 3 , there, is a chart c, = (U,, cp, ,n,) in '$ such I,that XEU,'.If g , , g , , 9 , are the restrictions of cpl, cp,, 9, to U1n U, n U,, the hypothesis that 911 and a, (resp. a, and a,) are compatible implies that n, = n, (resp. n2 = n,), and (denoting the common value of n,, n, , n3 by n) that g , g;' (resp. 9, g i ' ) is indefinitely differentiable with derivative of rank I? at every point of 'pl(Ul n U2n U,) (resp. cp2(Ul n U, n U,)). It follows that g 3 o g;' = (9, o g ; , ) o (9, g;') is indefinitely differentiable with derivative of rank n at every point of cpl(U, n U, n U,) ((8.12.10) and (8.2.1)). This shows that f3 0 f;' is indefinitely differentiable and has derivative of rank n at every point of cp,(U, n U,). Hence the result, by (16.1.1). 0
0
0
(16.1.3) It follows from (16.1.2) that the union of all the atlases of a given equivalence class is the largest atlas in this class. Such an atlas is said to be saturated. A chart which is compatible with all the charts in an atlas 91 belongs to the saturated atlas of the equivalence class of 2f.
A differentialmanifold is by definition a separable metrizable topological space X on which is given an equivalence class of atlases (with respect to the
1 CHARTS, ATLASES, MANIFOLDS
5
relation R) or, equivalently, a saturated atlas. The topological space X is called the underlying topological space of the differential manifold defined be a saturated atlas on X. If X is a differential manifold, an atlas of X is any atlas in the equivalence class defining X, and a chart of X is any chart belonging to one of these atlases (or, equivalently, any chart belonging to the saturated atlas of X). If c = (U, cp, n ) is a chart on X, we say that c is a chart ofX at a for each point a E U. The real-valued functions 'pi = pri cp :U + R ( I 5 i 2 n ) are called coordinates in U (for the chart c). For each point a E U, we say that ( c p i ) l s i s , is a system of local coordinates at a, and the numbers cpi(a)are the local&>rdinates of a for the chart c. Since all translations in R" are indefinitely differentiable, there always exists a chart (U, cp, n) of X at a point a E X such that cp(a) = 0. An atlas on a separable metrizable space X defines a structure of differential manifold on X, namely that defined by the equivalence class of a. 0
(16.1.4) (i) A diferential manifold is locally compact and locally connected, and every point has a neighborhood homeomorphic to a complete metric space. The set of (open) connected components of a diflerential manifold is at most denumerable. (ii) For each open covering (V,)ap, of a diflerential manifold X, there exists a locally finite denumerable open covering (U,) which is finer than (V,) and consists of relatively compact connected sets which are domains of definition of charts on X.
Assertion (i) is an immediate consequence of the definitions (cf. (3.18.1), (3.19.1), and (3.20.16)) and of the fact that a differential manifold is separable. Since clearly there exists an open covering which is finer than (V,) and consists of connected domains of definition of charts on X, assertion (ii) follows from (12.6.1) and from the fact that the restriction of a chart on X to an open set contained in its domain of definition is again a chart on X. When the differential manifold X is compact, the number of connected components is finite (because they form an open covering of X) and in (16.1.4, (ii)) we may take the covering (U,) to befinite. (16.1.5) Let X be a differential manifold, x a point of X. Then for all the charts (U, cp, n) on X such that x E U, the integer n is the same; it is called the dimension of X at the point x and is written dim,(X). Since dim,(X) = dim,(X) for all y E U, the function x I+ dim,(X) is a continuous mapping of X into the discrete space N; hence it is constant on each connected component
6
XVI
DIFFERENTIAL MANIFOLDS
of X (3.19.7). When x H dim,(X) is constant over X, the manifold X is said to bepure. If X is pure and not empty, the common value of the numbers dim,(X) is called the dimension of X, written dim(X). A pure differential manifold of dimension 1 (resp. 2) is often called a curve (resp. surface); but these terms can lead to confusion, since they have several commonly accepted meanings.
Remarks (16.1.6) (i) A separable metrizable space X is said to be a topological manifold if these exists a family of charts on X whose domains of definition cover X. The topological space underlying a differential manifold is therefore a topological manifold, but examples are known of compact topological manifolds which are not the underlying topological spaces of any differential manifold. (ii) If in the definition of compatible charts given above we replace “indefinitely differentiable ” by “analytic (9.3), we have the notion of real-analytically compatible charts. A real-analytic atlas on a topological space X is a set of charts each pair of which are real-analytically compatible and whose domains of definition cover X. The notion of “real-analytic compatibility” for two such atlases is defined as above, and (16.1.2) extends immediately to this new definition, by virtue of (9.3.2). A real-analytic manifold is then a separable metrizable space endowed with an equivalence class of real-analytic atlases. Since a real-analytic atlas is an atlas and since analytically compatible atlases are compatible, it follows that the analytic atlases of a real-analytic manifold X define on X a structure of differentiable manifold, called the differential manifold underlying the analytic manifold X. (iii) We can also replace R“ by C”in all the definitions: in this way we define a complex-analyticatlas and a complex-analyticmanifold. A complexanalytic atlas of a complex-analytic manifold X is also a real-analytic atlas, hence defines on X a structure of a real-analytic manifold, called the realanalytic manifold underlying the complex-analytic manifold X. is a system of (complex) local coordinates at a point a of a If (& complex-analytic manifold X, the 2n real-valued functions 9 q ~ and j 9 q ~ (1 sj 6 n) form a system of local coordinates at the point a for the realanalytic manifold underlying X. If at a point X E X the dimension of the complex-analytic manifold X (written dim,, ,(X)) is n, then the dimension at x of the underlying real-analytic manifold is 2n. If X is a pure manifold, the common value of the numbers dim,,,(X) is written dim,(X) and called the (complex) dimension of X. ”
j
7
2 EXAMPLES OF DIFFERENTIAL MANIFOLDS: DIFFEOMORPHISMS
A Riemann surface is a pure complex-analytic manifold of dimension 1 (so that the underlying real analytic manifold has dimension 2). In this book we shall not study the general theory of analytic manifolds; we shall merely mention, from time to time, the existence of analytic structures which arise naturally on certain differential manifolds (see [7], [17], [181, [191, WI, [411, W I , W1).
PROBLEMS
+
1. Let X be the subspace of R3consisting of the points ( x l , x 2 , x3) satisfying x: = x: x: (a cone of revolution). Show that X is not a topological manifold. (Consider the connected components of V - {x}, where V is an open neighborhood of a point x e X.) 2.
The differential manifolds defined in the text are also called "C"-manifolds," and topological manifolds (16.1.6) are also called " Co-manifolds." Define in the same way, for each integer r 1, C'-manifolds, by replacing in the definition of differential manifolds the phrase "indefinitely differentiable" by " r times continuously differentiable." Examine the validity for C'-manifolds of the properties proved in the text.
2. EXAMPLES O F DIFFERENTIAL MANIFOLDS: DIFFEOMORPHISMS
(16.2.1) On any (at most denumerable) discrete space X there is a unique structure of a pure differential manifold of dimension 0. If (x,,) is the sequence of points of X, the charts are the triplets ({x,,}, cp,, 0) where cpn is the unique mapping of {x,} onto Ro = {O}. (16.2.2) Let E be a real vector space of finite dimension n, endowed with the unique Hausdorff topology compatible with its vector space structure (12.13.2). Let cp: E R" be a bijective linear mapping. The triplet c = (E, cp, n) is a chart and 2l= {c} is an atlas. Moreover, if cp' is another bijective linear mapping of E onto R", then cp' 0 cp-' is a linear bijection of R" onto itself, hence is indefinitely differentiable (indeed analytic). The equivalence class of the atlas 2l is therefore independent of the choice of the linear bijection cp. In future, whenever we consider a finite-dimensional vector space E as a differential or analytic manifold, it is always the structure (called canonical) defined by this equivalence class that is to be understood, unless the contrary is expressly stated. E is a pure manifold of dimension n. --f
(16.2.3) Let (ei)ojisn be the canonical basis of the space R"+l, and let us identify R" with the hyperplane spanned by el, ..., e n . Let (XI y) be the
8
XVI DIFFERENTIAL MANIFOLDS
usual scalar product on R"+l, such that (eil ej) = 6, (Kronecker delta), and llxll the corresponding norm (6.2). The sphere whose equation is llxll = 1 relative to this norm is denoted by S,, and is called the "Euclidean unit sphere of dimension n," considered as a subspace of R"". We shall define on S, a structure of a pure real-analytic manifold of dimension n. To do this, we associate with each point x # e, of S,, with coordinates t i (0 5 i 6 n) the point y where the line through e, and x meets the hyperplane R".A simple calculation gives
iX=
(16.2.3.1)
IlYl12 - 1 e,
2
+
llYll2+1
IIYII2+lY'
and consequently these formulas define a homeomorphism 'pi :
S,,- {e,} R"
called sterographic projection with pole e, . In the same way we define the stereographic projection q 2 : S,,-{-e,}+R"
with pole -e, , such that for x # -e, (16.2.3.2)
q 2 ( x )= (1
+ to)-'(x - toeo).
We have thus defined two charts = (Sn
- {eo}, 9 1 , n),
~2
= (Sn
- {-eo>,
92
9
n)*
Let us show that they are analytically compatible. For each y # 0 in R" we have, by(16.2.3.1)and(16.2.3.2), (16.2.3.3)
and since Vl((Sn - {ed)
(Sn - {-ed)> = V,((Sn - {ed) = R" - {0},
this proves our assertion, since lly1)2=
n
(q')' i=1
(Sn - {-',I))
is a polynomial. Since
S, - {eo>and S, - { - e,} cover S,, , we have defined an analytic atlas 2l= {cl, c2} on S,, . In future, whenever we speak of S, as a manifold (analytic
2 EXAMPLES OF DIFFERENTIAL MANIFOLDS: DIFFEOMORPHISMS
9
or differential), it is always the structure defined by the equivalence class of the atlas % that is meant (cf. (16.2.7)), unless the contrary is expressly stated. The differential manifold S, so defined is compact, since it is a bounded closed subset of R"" ((3.1 5.1), (3.17.3), (3.17.6), and (3.20.16)) and connected if n 2 1 by virtue of (3.19.2). We remark that, since a nonempty open set in R"cannot be compact if n 2 1, an atlas of a nondiscrete compact manifold contains at least two charts (cf. Problem 5). (16.2.4) Let X be a differential manifold, % an atlas of X, and Y an open set in X. We have seen (16.1) that the restrictions to Y of any two charts c, c' belonging to % are compatible. Hence, as c runs through %, the restrictions clY form an atlas on Y, called the restriction of % to Y, and written %IY. Moreover, the equivalence class of %lY depends only on that of %, and therefore defines on Y a structure of a differential manifold depending only on that of X. This structure on Y is said to be induced by that on X. Here again, whenever we consider an open subset Y of a differential manifold as a differential manifold, it is always the induced structure that is meant. If Y is an open set in R", then (Y, ,1 ,n) is a chart on Y, called the canonical chart. (16.2.5) Let X be a separable metrizable space and an open covering of X.Suppose we are given on each X, a structure of a differential manifold in such a way that for each pair (01, jl) the structures of differential manifold induced on the open set Xu n X, by those on X, and X, (16.2.4) are the same. It then follows immediately from the definitions that if %, is an atlas of Xuand % the union of the aU, then % is an atlas of X, whose equivalence class depends only on those of the aU. The structure of differential manifold on X defined by is said to be obtained by patching together the differential manifolds X,;it is clear that it induces on each Xu the given structure of differential manifold. (16.2.6) Let X be a differential manifold, X' a topological space, u : X + X' a homeomorphism of X onto X . For each chart c=(U,cp,n) of X, the triplet (u(U), cp u - ' , n) is a chart on X , which we denote by u(c). If c and c' = (U', cp', n') are compatible, then so are u(c) and u(c'), because (cp' o u - ' ) (cp u-')-l = cp' cp-'. As c runs through the saturated atlas of charts on X, the u(c) from a saturated atlas of X',defining a structure of differential manifold, which is said to be obtained by transporting the structure on X by means of the homeomorphism u. If X, Y are two differential manifolds, a mapping u :X + Y is a direomorphism (or an isomorphism of diflerential manifolds) if u is a homeomorphism and if the structure of differential manifold on Y is the Same as 0
0
0
0
10 XVI
DIFFERENTIAL MANIFOLDS
that obtained by transporting the structure on X by means of u. Two differential manifolds X, Y are said to be diyeomorphic if there exists a diffeomorphism of X onto Y. Remarks
Consider the real line R, endowed with its canonical structure of differential manifold (16.2.2), and let u be the real-valued function such that u(t) = t for t 5 0, and u(t) = 2t for t >= 0. It is clear that u is a homeomorphism of R onto itself (4.2.2), and we may therefore endow R with the structure of differential manifold defined by the single chart u(c), where c = (R,I,, 1) is the single chart defining the canonical structure. Since u is not differentiable at the point t = 0, the charts c and u(c) are not compatible. If XIand X2are the differential manifolds defined on the underlying space R by c and u(c), respectively, then u is a diffeomorphism of XI onto X,, and we have therefore defined on R two distinct (but isomorphic) structures of differential manifold. In other words, the identity mapping 1, is not a diffeomorphism of XI onto X, . It can be shown that, for certain values of n 7, there exist on the topological space S, several nonisomorphic structures of differential manifold, having the same underlying topology. It can also be shown that the only connected differential manifolds of dimension 1 are (up to diffeomorphism) R and S, (Problem 6).
(16.2.7)
Let X be a differential manifold, X' a set, u: X + X' a bijection of X onto X . We can begin by transporting the topology of X to X' by means of u, by defining the open sets in X' to be the images under u of the open sets in X. Since u then becomes a homeomorphism of X onto X' we can transport to X' (again by means of u) the structure of differential manifold on X,as explained in (16.2.6). (16.2.8)
PROBLEMS 1. Show that the space T"= R"/Z" (12.11) is endowed with a structure of real-analytic manifold for which there exists an atlas of n 1 charts whose images are translationsin R" of the open cube I", where 1 = 10, 1[ (cf. (16.10.6)).
+
2. (a) Let K be a compact subset of R" and B a closed ball whose interior contains K. Show that for each E > 0 there exists a homeomorphismf o f R" onto itself, such that f ( x ) = x for all x 6 B and such that the diameter off(K) is 6 E. (We may assume that 0 is the center of B; takefto be of the formf(x) = xtp( Ilxll), where tp is a suitably chosen real-valued function.)
2 EXAMPLES OF DIFFERENTIAL MANIFOLDS: DIFFEOMORPHISMS
11
(b) Let W be an open set in Rn and a a point of W. Show that, for each open ball B with center a contained in W, there exists a homeomorphism of W onto an open neighborhood of a contained in B, which coincides with the identity on a neighborhood of a (same method).
3. Let X be a metrizable space and A a compact subset of X such that there exists a fundamental system ( v k ) of relatively compact open neighborhoods of A which are all homeomorphic to R". In these conditions, the space X/A (Section 12.5, Problem 10) is homeomorphic to X; moreover, for each relatively compact neighborhood U of A, there exists a homeomorphism h of X/A onto X such that, if w :X + X/A is the canonical mapping, h w coincides with lx on X - U . (We may restrict ourselves to the case where U = V1,and Vk+' C v k . Using Problem 2(a) and (3.16.5), show that there exists a sequence ( g k ) of homeomorphisms of X onto itself with the following properties: (i) g1 = l X ;(ii) g k + l agrees with g k on a neighborhood of X - v k ; (iii) the diameter of g k ( v k ) is s l / k . Deduce that the sequence ( g k ) converges uniformly to a continuous mapping g of X into itself such that g(A) is a single point, and show that g factorizes into h w ; h is the required mapping.) 0
0
4. Let A be a compact subset of R", and suppose that there exists a homeomorphism h of Rn/A onto an open set W in R" such that, if r : R"?-. R"/A is the canonical mapping, we have h(w(A))= (a}. Show that there exists a fundamental system ( v k ) of relatively compact open neighborhoods of A which are homeomorphic to R". (Using Problem 2(b), show that there exists a fundamental sequence ( u k ) of relatively compact open neighborhoods of a in R",and for each k a homeomorphism A of W onto u k , which coincides with 1 R n on a neighborhood of a. Take v k = n - - ' ( h - ' ( u k ) ) , and show that there exists a homeomorphism g k of R" onto V kwhich coincides with 1,. on a neighborhood of A and is such that h o w o gk = A o h w . ) 0
5. Let M be a compact connected metrizable space which has an open covering consisring of two subspaces X, Y each homeomorphic to R". Then M is homeomorphic to S. (Morton Brown's theorem). (If A = M - Y c X, observe that M/A is homeomorphic to S., hence X/A is homeomorphic to an open set in R"; then apply Problems 3 and 4.) 6. Let X be a connected differential manifold of dimension 1. Then there exists an atlas ((Uk,v k ,l)), where the vk(Uk)are open intervals in R, such that ( u k ) is an at most denumerable locally finite covering of X by relatively compact open sets, and such that v k extends to a homeomorphism of an open neighborhood of onto an open interval in R. but that neither of (a) Suppose that u h n u k # Show that there are just two possibilities:
u h
, Ur is contained in the other.
(a) ph(Uhn U,) is an interval, one of whose endpoints is also an endpoint of v h ( u h ) , and vk(Uhn U,) is an interval, one of whose endpoints is also an endpoint of vk(uk);
(8)
vh(Uhn u k ) is the union of two disjoint intervals, each of which has an endpoint which is also an endpoint of v h ( u h ) , and likewise for P k ( u h n u k ) and P k ( u k ) . In this case (/?),show that every other U, is contained in u h U u k and that X is diffeornorphic to S1.
12
XVI DIFFERENTIAL MANIFOLDS
(b) Deduce from (a) that X is diffeomorphic to either R or S1.(Assume that X is not Mwmorphic to S1,that each uk+1intersects U1u . u u k and that neither of these two sets is contained in the other. Using (a) and induction, construct a diffeomorphism fc of Ul u u U, onto an open set in R,such thatft+lextendsx.) +
3. DIFFERENTIABLE MAPPINGS
Let X, Y be two differential manifolds. For each integer p 2 0, a mapping f :X + Y is said to be p times continuously differentiable (resp. indefinitely dzferentiable) iff is continuous on X and satisfies the following condition: for each pair of charts (U, cp, n) and (V, JI, m) of X and Y, respectively, such that f(U) c V, the mapping F=JI0(flU)ocpp-' : cp(u)-+$(v) (which is called the local expression off for the charts under consideration) is p times continuously differentiable (resp. indefinitely differentiable) (8.12). (Forp = 0 we make the convention that the derivative of order 0 of F is F itself.) (16.3.1) For a mapping f :X -+ Y to be p times continuously diflerentiable (resp. indefinitely differentiable) it is necessary and suflcient that for each xo E X there exists a chart (U, cp, n) of X, a chart (V, $, m) of Y and a mapping F :cp(U) + $(v) which is p times continuously diflerentiable (resp. indefinitely diflerentiable), such that xo E U, f(xo) E V and such that f I U = JI-' o F 0 cp.
The condition is clearly necessary. Conversely, suppose that it is satisfied. Clearly it implies that f is continuous on X. Let ( U , cp', n') and (V', $', m') be charts of X and Y, respectively, such thatf(U')c V'. We have to show that $' (f1 U') 0 q'-' satisfies the appropriate differentiability condition. For each xo E U', let (U, cp, n), (V, JI, m), and F be two charts and a mapping satisfying the conditions of the proposition. Then first of all we have n = n' and m = m', because xo E U n U and&) E V n V'. Replacing U and U' by U n U', and V and V' by V n V', we may assume that U = U' and V = V'. However, then we have 0
$' o(f
lu)
o
q'-' =($'
o $-I)
o
F o (cp o cp'-')
and the result follows from the definition of compatible charts (16.1) and from (8.12.10). In the notation of (16.3.1), if z = (I;i)lsisn F off is of the form F(z) = F(C1,. .., = (F'(C', . . ,
r)
E
cp(U), the local expression
. r),..., FYI;', ..., r))
DIFFERENTIABLE MAPPINGS
3
13
sj
where the Fj (1 p m) are scalar functions defined on cp(U); to say that F is p times continuously differentiable (resp. indefinitely differentiable) means that the Fj have this property (8.12.6). If (cp'), ($j) are coordinates in U, V, respectively (16.1), we have (16.3.1.1)
+j(f(x))= Fj(cpl(x), ...,cp,(x))
(1 S j 5 m)
sj s
for all x E U. The Fj (1 m) are said to constitute the local expression off for the given charts. A p times continuously differentiable (resp. indefinitely differentiable) mapping is also called a mapping of class C p (resp. a mapping of class C", or a morphism of differential manifolds). A mapping of class C p (resp. C") into R is also called (if there is no risk of confusion) a function of class C p (resp. of class C") defined on X. It is clear that a mapping of class CP (where p is an integer or co) is also of class Cqfor all q < p.
The sum andproduct of two functions of class C p on X are functions of class CP. I f f is a function of class CP such that f ( x )# 0 for all x E X , then Ilfis a function of class C p .
(16.3.2)
This follows from (8.12.9), (8.12.10), and (8.12.1 1). (16.3.3) (i) Let X, Y, Z be three differential manifolds and f:X -,Y , g :Y -,Z two mappings. I f f and g are of class C p( p an integer or a),then so is g of. (ii) For a mapping f : X + Y to be a diffeomorphism of X onto Y it is necessary and sufficient that f be bijective and that f and f - be of class C".
(i) Let x E X and let (U, a, a) and (V,j?, b) be charts on X and Y, respectively, such that x E U, f ( x ) E V and f I U = j?-' o f i 0 a, where fi is of class Cp. Likewise, let (V', j?', 6') and (W, y. c) be charts on Y and Z, respectively, such that f ( x ) E V', g ( f ( x ) )E W and g1 V' = y-' 0 g1 0 b', where g1 is of class C p . Replacing V and V' by V n V', and U by f -'(V n V'), we may assume that V' = V, from which it follows that b' = b. We have then
u =Y-l
( 9 0f)l
O
(Sl
O
(8' j?-9oh) O
O
a,
and the result now follows from the definition of compatible charts and from (8.12.10). (ii) The necessity of the condition is an immediate consequence of the definitions. To prove the sufficiency it is enough to show, for each chart c = (U, cp, n) on X, if we put f l =f I U, that f(c) = (f(U), cp 0 f T1,n) is a chart on Y. Since f is a homeomorphism, it is clear first of all that f ( c ) is a
14
XVI
DIFFERENTIAL MANIFOLDS
chart of the topoIogicalspace Y, and it is enough to show that it is compatible with every chart c' = (V, $, rn) of the manifold Y (16.1). We may assume that V =f(U), and then it follows from the definition of morphisms and from the hypotheses that $ (cp of;')-' = $ o f i cp-' and (cp of;') $-I are indefinitely differentiable. Hence the result. 0
0
0
Examples (i) When X and Y are open subsets of finite-dimensional real vector spaces, the definition of a mapping of class C p (p an integer or m) agrees with that of (8.12), by virtue of (16.2.2). If X is any differential manifold and (U, cp, n) is any chart on X,then cp is a diffeomorphism of U onto the open set cp(U) in R". Conversely, every diffeomorphism cp of an open set U in X onto an open set cp(U) in R"defines a chart (U, cp, n) on X. If Y is an open subset of a differential manifold X,the canonical injection of Y into X i s a mapping of class C". (ii) The mapping (16.3.4)
(16.3.4.1)
f : X W
2x 1-
llXllZ
is a diffeomorphism ofthe open ball B: llxll < 1 in R" (Ilxll being the Euclidean norm (16.2.3)) onto R".The inverse diffeomorphism is (16.3.4.2)
The mapping (16.3.4.3)
g :X H -
X
lIxIlZ
is a diffeomorphism of the exterior llxll > I of B onto the complement of (0)in B. The cornpositionfo g is therefore a diffeomorphism of the exterior of B onto R" - (0). (iii) For the two manifolds XI, X, defined in (16.2.7), both of which have
R as underlying space, the identity map 1, is not of class C ' , whether considered as a mapping from X, to X, or from X, to X,. If u is the real-valued function t~ t 3 (which is a homeomorphism of R onto itself) and if we endow R with the structure of differential manifold defined by the single chart u(c), we obtain a differential manifold X, again having R as underlying
3
DIFFERENTIABLE MAPPINGS
15
space and distinct from both Xi and X2.This time, the mapping l,, considered as a mapping from X, to X, , is of class C" but is not a diffeomorphism, since the inverse mapping is not even of class C'.
Remark (16.3.5) If in the definition of a mapping of class C" we replace differential manifolds by real-analytic (resp. complex-analytic) manifolds, and indefinitely differentiable mappings of open sets of R" (resp. C")into R"'(resp. Cm) by analytic mappings (9.3), we arrive at the definition of an analytic mapping of one real-analytic (resp. complex-analytic) manifold into another. In the complex case, such mappings are also called holomorphic. We leave to the reader the task of formulating for such mappings the analogues of the propositions of this section.
PROBLEMS
Let X be a pure differential manifold (resp. a real-analytic manifold, r a p . a complexanalytic manifold) of dimension n. For each open set U c X, let F(U) be the set of C"-mappings of U into R (resp. real-analytic mappings of U into R, resp. complexanalytic mappings of U into C). Show that the sets F ( U ) have the following properties: (a) For each open set V C U, the restrictions to V of the functions f~ F ( U ) belong to FW). (b) For each open set U c X and each covering (U.) of U by open sets contained in U, if a function fdefined on U is such that f l U. E F(U.) for each a,then fe F(U). (c) For each point x E X, there exists a homeomorphism u of an open neighborhood U of x onto an open set in R"(resp. R",resp. C") such that, for each open set V c U, FW)is the set of all functions of the form g 0 u where g runs through the set of C mmappings of u(U) into R (resp. real-analytic mappings of u(U) into R, resp. complexanalytic mappings of u(U) into C). Conversely, let X be a separable metrizable space and suppose we are given, for each open set U in X, a set F ( U ) with the above properties. Show that there exists a unique structure of differential manifold (resp. real-analytic manifold, r a p . complex-analytic manifold) on X for which F ( U ) is the set of C"-mappings of U into R (resp. realanalytic mappings of U into R, resp. complex-analytic mappings of u into C) for each open set U in x. (Observe that, if u = (u', . ,uJ), the functions UJ belong to F(U).)
..
In C, considered as a complex-analytic manifold, every nonempty simply connected open set other than C is isomorphic to the unit disk 1 z 1 < 1, and the latter is not isomorphic to C. Hence there are two classes of simply connected nonempty open sets with respect to the relation of isomorphism (Section 10.3, Problem 4). Deduce that in the plane R2 any two simply connected nonempty open sets are diffeomorphic. Give an example of two nonisomorphic complex-analytic manifolds having the same underlying structure of differential manifold.
16
XVI
DIFFERENTIAL MANIFOLDS
3. (a) Let X,Y be two connected real-analytic(resp.complex-analytic)manifoldsandf,g two analytic mappings of X into Y.Show that if there exists a nonempty open set U c X on which f and g agree, then f= g (9.4.2). (b) Let X be a connected complex analytic manifold and let f be a holomorphic complex-valued function on X,not identically zero. Show that the set of points x E X such thatf(x) # 0 is a connected dense open set. (If a, b are two points of X,show that there exists a sequence ( c ~ of )points ~ of ~ X~such ~ that ~ co = a, c. = b and such that ,n - 1, the points ct and ct+l both belong to the domain of definifor each i = 0, 1, tion U of a chart (U, 'p, n), where cp
...
4. Give an example of two distinct structures of real-analytic manifold on R (resp. two
distinct structures of complex-analytic manifold on C)which are isomorphic and have as underlying structure of differential manifold the canonical structure (16.2.2).
5. Let X be a compact metrizable space and let B be a Banach subalgebra of Vc(x) which is a Dirichlet algebra (Section 15.3, Problem 9(c)). Let xo be a character of B and let p be the unique representative measure of xo. Let P c X(B) be the Gleason part of xo (Section 15.3, Problem 18). For each character h E P, the unique representative measure of A can be written #A . p, where #A and l/G are bounded in measure (relative to p) (Section 15.3, Problem 19). For each function f~ Z ' ( p ) (Section 15.3, Problem 15)
putf(A)
=/
f i A dp. Iff, g E Z z ( p ) ,we havef(h)g(h) = /fg#.
dp (Section 15.3,Problem
13(f)). If alsofg E 9 ' ( ~then ) , fg E Z ' ( p ) (Section 15.3, Problem 15). (a) Suppose that P does not consist of the single point xo, and let x1 E P be. distinct from xo . The set C c X z ( p ) of functions f such that f(xl) = 0 is of the form q Z ' ( p ) where q E C is such that 141 = 1 on X (Section 15.3, Problem 15(c)). Show that < 1 for all A E P. (If not, q would be almost everywhere equal to a constant, which would contradict the relation q(xl) = 0.) (b) Put Us f =q-'(f-f(x1)) f o r f e X z ( p ) .Then U is a continuous linear mapping of .W2(p) into itself. Show that there exists a constant p such that /I U"II 5 p for all integers n 2 1. (Calculate the norm of U considered as an operator on Yz(pl), where p, = * p is the representative measure of x i , and show that this norm is < I .) Deduce that, for each f~ Z z ( p )and each integer n 2 1, we have
I (u" . f ) ( N I 5 . Nz(f)Nz(#J for all h E P. Deduce that the function
is holomorphic in the disk I z I < 1 and that
f0)=P(dh)) for all h E P. (Observe that f=
"-1
k=O
(Uk.f)(xl)qk
+ (U".flq".)
(c) I f f , g, and fg are all in Z ' ( p ) , show that for all integers n 2 1 we have
4
DIFFERENTIABLE PARTITIONS OF UNITY
17
(Multiply together the expressions f o r f a n d g in terms of the CJk ' f a n d U k . g used in (b).) Deduce that fb = (fg)'. (d) Show that the mapping h ~ q ( / is \ )a bijection of P o n t o the unit disk IzI < I . (Use (a) t o show that q ( h l )= q(h2)implies hl = h 2 . By virtue of (c), for each zo in the unit disk, f w f ( z 0 ) is a character ho E X(B). To show that ho E P, observe that if z, , z2 are two points of the disk I z I < 1, there exists a constant c < 2 such that I F(z,) - F(z2)I 5 c for every holomorphic function F such that I F(r) I 5 1 at all points of the disk. Finally, by passing t o the limit in P ( p )show that we havef(ho) =f(z0) for allfE P Z ( p ) and , in particular for f = 4.) (e) Let v be the inverse of the mapping h~ q(h). Show that v is continuous on the disk IzI < 1. (Argue by contradiction: observe, by using the compactness of X(B), that if q~ were not continuous at a point zo there would exist a character x E X(B) distinct from v(zo) and such that&) =f(p(zo)) for all f~ B, by remarking that f o v is continuous.) The set P is therefore endowed with a structure of complex-analytic manifold.
4. DIFFERENTIABLE PARTITIONS O F U N I T Y
The following proposition is a sharpening of (12.6.3) for differential manifolds: (16.4.1) If (A,) is an at most denumerable locally finite open covering of a diferential manifold X , there exists a partition of unity (f,,) on X subordinate to (A,,) and consisting offunctions of class C".
We shall apply the remark (12.6.5) to the set % of functions of class C" on X; since the properties (2) and (3) in this remark follow from (16.3.2), we have only to establish property ( I ) . The proof of this consists of several steps. (16.4.1.1)
For each integer n 2 0, we have
lim t-"e'
=
+a.
I-++"
For it follows from the power-series expansion of e' that et 2 t"+'/(n+ I)! for t 2 0. (16.4.1.2) (16.4.1.3)
Thefunction h :R + R defined by h(t) =
is indefinitely differentiable.
0 exp(-tZ)
f o r t 5 0, for t > 0
18
XVI
DIFFERENTIAL MANIFOLDS
For it is easily shown (8.8) by induction on n that for t > 0 we have D"h(t) = P,(t-') exp( - t 2 )
lim
where P, is a polynomial; hence
t-'D"h(t) = 0 by (16.4.1.1). This
r-0, f > O
proves (16.4.1.2) by induction on n.
(16.4.1.4) Let I be the interval [ - I ,
+
I ] in R. There exists a function g of class C" on R" which is > O in the interior of K = I", zero on the exterior of K, and such that
S-S
g(tl,
. . . , t,)
dt,
. . . dt,
= 1.
Put h,(t) = h( 1 + t)h( 1 - t ) , where h is the function defined by (16.4.1.3). Then we may take g ( t l , . . . , t,) = ch,(t,) . . . ho(t,) with a suitable constant c. ( 16.4. I .5) .End ofthe proof
Let M be a compact subset of X, and N a closed subset of X such that M n N = /zr. For each x E M , there exists a chart ( U , , cp,, n,) such that x E U,, U, n N = /zr, cp,(U,) 2 I" and cp,(x) = 0. The real-valued function f, which is equal tog cp, on U, and 0 on the complement of U, is of class C" and is >O on V, = cp-'(i"), which is an open neighborhood of x . We can cover M by a finite number of such neighborhoods V X i ; the function is of class C", vanishes everywhere on N and is >O every0
x,fxi I
where on M. If
c(
=
inf c , f x i ( x ) ,we have a > 0 (3.17.10), and the function
Ii f x satisfies i condition ( I ) xsM
f = d-'
i
of (12.6.5).
(16.4.2) Let X be a differential manifold, K a compact subset of X, and (Ak)lsksm a j n i t e covering of K by open subsets of X. Then there exist m functions fk of class C" on X with values in the interval [0, I ] such that Supp(f,) c A, for I 5 k 5 m , I . f k ( x ) = 1 for all x E K and x f k ( x ) 5 I for
all x E X.
k
k
This follows from the preceding result and from (12.6.5) and (12.6.4). (16.4.3) Let X be a differential manifold, F a closed subset of X, and g a mapping of F into R. Suppose that,,for each x E F, there exists an open neighborhood V, of x in X and a function,f, of class C' ( r an integer or co) on Vx which is equal to g on V, n F. Then for each open neighborhood U of F there exists a,functionf of class C' which is zero on the complement of U and equal to g on F.
+
4 DIFFERENTIABLE PARTITIONS OF UNITY
19
For each X E CF, let V, be an open neighborhood of x which does not intersect F. For each x E F, on the other hand, we may assume that V, c U (by replacing V, by V, n U). Let (A,,) be a denumerable open covering of X which is locally finite and finer than the covering (V.J,.x(12.6.1), For each n, choose an x such that A,, c V,. If x # F, let f,, denote the zero function on A,,, and if x E F, letf. denote the restriction off, to A,,. If (h,,) is a C" partition of unity on X subordinate to the covering (A,) (16.4.1),then the function gnwhich is equal to h,,f,, on A,, and is zero on the complement of A, is of class c' on X, and the functionf = g,, satisfies the required conn ditions.
(16.4.4) For brevity we shall say that a function g with the property stated in (16.4.3) is of class C' on F (although in general F is not a differential manifold); equivalently, g is the restriction to F of a function of class C' on X (cf. Problem 6).
PROBLEMS
Let KO,K1be disjoint closed subsets of the sphere S.. Show that there exists a C "function f o n R"+'- (0) which is equal to 0 on KOand to 1 onK, ,satisfiesf(tx) = f ( x ) for all real numbers t > 0 and is such that for each multi-index a,Ilxlll"Wf(x) remains bounded as x + O (Ilxll denotes the Euclidean norm). For each C"-function g on Rn+' such that D"g(0)= 0 for each multi-index a,the function gfextends to a C"-function on R"+l. Let (xk)kk be a sequence of distinct points of R" tending to 0. For each k let akbe a multi-index such that I akI + m, and let ( c k , v ) v Nn be a multiple sequence of numbers such that ck,v = 0 for 1 v1 5 I I and v # a,. Show that there exists a c "-function f on R"with the following properties: (a) DVf(xt) = ct," for all v E N"; (b) D'f(0) = 0 for all Y. (Use the method of Problem 4 of Section 8.14 to construct by induction a sequence of C"-functionsfk, whose supports are pairwise disjoint and do not contain 0, such that (i) D"ji(xk)= ck,v for each multi-index v, and (ii) llD"hiI 5 T kfor all v such that (YI < l a k ( . Then takef=xfk.)
+
k
Let X be a differential manifold. Show that for each x E X there exists a chart (U, v, n) at the point x such that 9 is the restriction to U of a C"-mapping of X into R". Let F be a closed subset of R"and U its complement. Let h, k, 1) be three numbers in the interval 10, I[. Show that there exists a denumerable covering of U by open Euclidean balls B(Q, r l ) with the following properties:
20
X V I DIFFERENTIAL MANIFOLDS
(a) the balls B(ui, kri) cover U; (b) rl = hd(ul, F) for each i; (c) there exists an integer N(h, k, 9) depending only on h, k, 7 such that for each x E U the closed ball with center x and radius qd(x, F) meets at most N closed balls B'(uI,rd. (Let e be a real number >O. For each m E Z, let F,,, be the set of points x E U such that d(x, F) = (1 E ) ~ and , let T, be an at most denumerable subset of F, consisting of points whose mutual distances are 2 e(1 e)'", and such that the open balls with centers at these points and radii equal to &(I 6)"' cover F, . Let (u,) be the sequence T,, arranged in any order. Show that if we take consisting of the points of
+
u
+ +
INEZ
ri = hd(ui, F), the required conditions are satisfied provided that E < &hk.If x E U and 8 = d(x, F), observe that there exists rn f Z such that (1 + E)"' 8 < (1 + E)"+', and deduce that d(x, T,) 5 2 4 1 + e)'". Then show that there exist two constants c > 0 and C > 0, depending only on h and 7, and such that (i) if B'(ui, r,) meets the ball with center x and radius 78,then d(x, ul)5 C8, and (ii) if j # i is another index with the same property, then d(ui, u,) 2 c8.)
5. With the notation of Problem 4, put B1= B(ui, r,). Show that there exists a Cmpartition of unity (U,)subordinate to the covering (B,), and for each a E N"a constant C. such that IID"ur(x)ll5 Cu(d(x,F))-Iuifor all x E R",i and m. 6. Let F be a closed subset of R".Generalizing the definition of a mapping of class C', a mapping f : F -+R" is said to be of class C' (resp. C ") if for each multi-index a E N" such that I a I r (resp. for each multi-index a) there exists a mapping f.:F -+ R" with fo = A such that the following conditions are satisfied: if for each integer s 5 r (resp. each integer s => 0) we write
where x E F, z E F, and I a I 5 s, then for each xo E F, each E > 0, and each pair (a, s) with 1 a ( 9 s, there exists p > 0 such that [ I R J x , z)ll e(Ix - z((J-l'l for all x, z E F such that IIx - xo II < p and I I z - xo II < p. These conditions imply that thef. are continuous on F. When F = RR,this definition is equivalent to the previous definition of functions of class C (resp. C "). (a) Show that if the mapping f:F 4 R" is of class C' then f can be extended to a mapping h :R"-+ R" of class C' on R" and of class C" on U = CF, and hence the definition above agrees with (1 6 4.4). (The Taylor polynomial of order s 5 r off at the point z E F is the polynomial in xl, ...,x"
With the notation of Problem 5 , show that the function h defined by
where b, E F is such that d(ui, b,) = &ai, F), satisfies the required conditions. For this purpose, show that if 1 a I r , then D"h(x)- D"T:. f ( x ) -+ 0 as x 4 u E Fr(F), where
4
DIFFERENTIABLE PARTITIONS OF UNITY
21
xo E F is such that d(x, xo) = d(x, F). Using Leibniz's formula, this reduces to majorizing the norm IlD6T;,f(x) - DBT:,f(x)ll for fi 5 a, by using the results of Problems 4 and 5.) (b) Show that iffis of class C" there exists an extension h offto R"which is of class C (Whitney's extension theorem). As a consequence, the definition above agrees with that of (16.4.4). (For each integer r, show that there exists a number d, such that the relations z E F, x E R",IJx- zll 5 d,, s 5 r, and I a I 5 s imply IID"T:f(x) - D"T:f(x)ll I_Ilx - ZIP-'". If V, is the neighborhood of F consisting of the points x such that d(x, F) 5 id,, let rl denote the largest r such that a, E V, (we can always suppose that the sequence (d,) tends to 0). Then define f( x ) if X E F , h(x) = u,(x)z:f(x) if x E v,
{
the points b, being defined as in (a). Show that h has the required properties by arguing as in (a).) With the notation of Problems 4 and 5, suppose that F is compact. Show that for each p > 0 one can define a function up 2 0 of class C? on R",such that u,(x) = 1 whenever d(x, F) 5 p , u,(x) = 0 whenever d(x, F) 2 2p, and such that for every functionf: Rn+R of class C' which vanishes on F together with all its partial derivatives D9f of order 1 a I 5 r, the functions u , f , and their partial derivatives D"(u,f) of order I a I 5 r tend uniformly to 0 on R" as p +0. (Take u, to be a sum of certain of the functions u, defined in Problem 5.) Let F be a closed subset of R" and let f b e a real-valued function of class C' on F, and
g a real-valued function of class C' on the open set &F.For each z E R" let P, be the polynomial in x ' , ...,x" which is equal to T: fif z E F, and is equal to T:g if z $ F, in
the notation of Problem 6. Show that there exists a function h of class C' on R"such that P, = T: h for all z E R"if and only if the coefficients of P, are continuous functions of z. (Reduce to the case r = 1 by induction, then to the case n = 1 by using (8.9). Then we have P,(x) = a(z) ( x - a)b(z) where a and b are continuous functions of z E R. Reduce to the case b = 0 and then show that the function a(z) has zero derivative at each z E R.)
+
(a) Let f be a real-valued function 20 of class C2 on a neighborhood of 0 in RN. Suppose that f a n d its derivatives of order (2 all vanish at 0, and that there exist positive real numbers c, M such that IDID I f ( x )I 5 M for all pairs of indices (i,j) and all x E RN such that IxJl 5 2c (1 N). Show that if 1x11
+
sj
1x21
+...+
lXNl
sc,
we have (1 )
I D,f(x)I 5 2 M f M
(1 S j I_ N). 5 c implies that I D J f ( x ) l (= Mc for all j . Then J by supposing that at some point x satisfying I x'l 5 c the
(Observe first that the relationx 1 xJl argue by contradiction,
I
inequality (1) is false for some index j , withf(x) > 0; use Taylor's formula to conclude 91 2c (1 < j 5 N).) thatf(y) < 0 for some pointy such that 1 (b) Let f b e a real-valued function 20 of class C2 on an open set U in R",such that
<
22
XVI
DIFFERENTIAL MANIFOLDS
all the derivatives of order 4 2 of fvanish at the zeros off. Show that f 1 I 2 is of class C' on U (use (a)). c2)u(x), where u is (c) For each E > 0, let f, be the function defined byf,(x) = (x' a function of class C " on R such that u(x) = 1 for I x I 5 4 and u(x) = 0 for I x I >= 2. Let (a") be a sequence of strictly positive numbers such that the series
+
s=1
+ 2(al + a2+ ... + a. +. ..)
converges, and let (&) be another sequence of strictly positive numbers which tend to 0 sufficiently rapidly so that P./a:+O for each integer k > 0. Finally let (E.) be a sequence of strictly positive numbers tending to 0. Put s, = 1
+ 2(a1+ ... + an-1)+ an,
Show that the function g(x) = ~ g . ( x ) is 2 0 and of class C" on R, and that all its n
derivatives vanish at the zeros of g. For each p E 10, A[, show that it is possible to g is not of class C ' ; also that it is possible choose the sequence (E.) so that the function ' to choose the sequence (E.) so that the function g1'2(which is of class C1 by virtue of (b) above) is not of class C2. 10. Let E be a finite-dimensional real vector space and let M, ( 1 sj
spaces of E. Show that the following conditions are equivalent:
5 r ) be vector sub-
(a) If mj=codimMj, then for each subset H of [ l , r ] in N the codimension of (I M,is m,. JEH
J€H
(b) The sum of the annihilators M,O C E* of the M, in the dual E* of E is direct. (c) There exists a direct sum decomposition of E of the form P @ N1@ ... @ N, such that each M j is the direct sum of P and the N kwith k # j .
(d) I f P =
n
15jCr
M j , t h e n c o d i m P = x m,. I =I
(To show that (a) implies (c), observe that if P =
n
lSJSI
MI and
Qj
=
n Mk, then
k f l
P has codimension m, in Q j .) A family of vector subspaces MIsatisfying these conditions is said to be ingeneral position in E. Let V be the union of the M, and letfbe a real-valued function on V such that the restriction offto MI is of class C' for 1 5 j 5 r . Show thatfis the restriction to V of a function of class C' on E. (Proceed by induction on r.) 11. Give an example of a C"-mapping f: R+R2 such that f(R) is the square sup( Ix 1 I, 1x2 I ) = 1.
5. T A N G E N T SPACES, T A N G E N T LINEAR MAPPINGS, RANK
(16.5.1) Let X, Y be two differential manifolds, x a point of X. Let f,,fi be two C'-functions, each defined on an open neighborhood of x, with values in Y. The functionsf,,fi are said to be tangent at the point x if
5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, RANK
23
f i ( x ) =fz(x) and if the following condition is satisfied: If (U, cp, n) is a chart of X at x such that U is contained in the domains of definition of f l and f i
and if (V, $, m) is a chart of Y at the pointfl(x) such thatfi(U) and f z ( U ) are contained in V, then the functions o (fi I U) cp-' and $ o (fzI U) o cp-' (that is to say, the local expressions of fl and f i ) are tangent at the point cp(x) (8.1), i.e., they have the same derivative at this point. If this condition is satisfied for one choice of the charts (U, cp, n) and (V, $, m),then it is satisfied for any other pair of charts (U',cp', n) and (V', $', rn) satisfying the same conditions. For we may assume without loss of generality that U = U' and V = V', and then we have 0
$'o(fi~U)ocp'-'=(II/'~II/-')~($~(f;~u)~cp-').(cp'"cp-')-' for i = 1, 2, and the assertion therefore follows from (8.2.1). Furthermore, it follows immediately from this definition that the relation ''fland f 2 are tangent at the point x " is an equivalence relation.
(16.5.1.1) Consider in particular the real line R, a differential manifold X, a point x E X, and the relation " f l and f 2 are tangent at the point 0" between two functions f l and f z of class C', defined on an open neighborhood of 0 in R, with values in X and such that f l ( 0 ) =f2(0)= x. The equivalence classes for this relation are called the tangent vectors to X a t the point x , and the set of them is denoted by TJX). Let c = (U, cp, n ) be a chart on X at the point x . Then the definition just given shows that we obtain a bijection 0, :TJX) -+ R" (also denoted by 0,. ). by mapping the equivalence class of a mapping f :V -+ X (where V is an open neighborhood of 0 in R) of class C' and such that f ( 0 ) = x, to the vector (D(q f))(O). The inverse of this bijection maps a vector h E R" to the tangent vector, belonging to T,(X), which is the equivalence class of the mapping t w p - ' ( p ( x )+ (h), where 5 belongs to a sufficiently small neighborhood of 0 in R. If c' = (U, cp', n ) is another chart of X at x (we may assume that c' and c have the same domain of definition), then the mapping 0,. 0 0;' is the bijective linear mapping 0
(1 6.5.1.2)
0,, 0 0:': hM(D(cp'0 cp-')(cp(x))* h.
It follows that we can define a structure of a real vector space of dimension n on T,(X) by transporting by means of 0,' the vector space structure of R", that is to say by defining 0,-'(h)
+ 0,-'(h')
= B,-'(h
+ h')
and
1 * 0,-'(h)
= O,-'(Lh)
24
XVI DIFFERENTIAL MANIFOLDS
for 1 E R;moreover, this vector space structure is independent of the choice of the chart c because the mapping (16.5.1.2) is linear. The set T,(X), endowed with this vector space structure, is called the tangent vector space, or simply the tangent space, to the differential manifold X at the point x. If (ei)’ is the canonical basis of R”,the tangent vectors ec;:(ei) (1 5 i 5 n) form a basis of the tangent space T,(X). This basis is said to be associated with the chart c. The reader should beware of confusing the notions of tangent vector and tangent space defined here with the elementary notions of “tangent vector” or “tangent plane” defined for ordinary “surfaces” in R3.The relationship between these notions will be made clear in (16.8.6). For each tangent vector h, E T,(X),
is called the local expression of h, , relative to the chart c.
Example (16.5.2) Let E be a real vector space of dimension n, endowed with its canonical structure of differential manifold (16.2.2). For each linear bijection cp: E + R” and each x E E, the triplet c(cp, x ) = (E, cp, n) is a chart on E, hence defines a linear bijection O,,,, x) :T,(E) + R”,and therefore by composition a linear bijection (16.5.2.1)
2,
= cp-1
eCc(,,,) : TJE)
-,E
which is independent of the linear bijection cp, by virtue of (16.5.1) and the relation D(q’ cp-’)(cp(x)) = cp‘ 0 cp-’ for two linear bijections cp, cp’ of E onto R”(8.1.3). The bijection 2, is called canonical. 0
(16.5.3) Now let X, Y be two differential manifolds, f:X -+ Y a mapping of class C1, x a point of X, and y =f(x). Let c = (U, cp, n) and c‘ = (V, $, rn) be charts of X and Y at x , y , respectively, such thatf(u) c V, and consider U)0 cp-’ off relative to c and c’. This local the local expression F = $ o expression is a C’-mapping of cp(U) into $(V), and its derivative F’(cp(x)) (8.1) is therefore a linear mapping of Rninto R”.We shall show that the linear mapping
(fl
(16.5.3.1)
T , c ~ )= e,
1
~‘(cp(x)) e, : T,(x)
-+
T,(Y)
is independent of the choice of charts c, c‘ at x, y. For if we replace c and c’ by two other charts c1 = (Ul, cpl,n) and c; = (Vl, $‘,rn) at x and y ,
5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, RANK
25
respectively, we may assume that U = U, and V = V,, by replacing U and U, by U n U,, and V and V, by V n V1;then the local expression o f f relative to the charts c, and c; is ($, o$-')oF0(cpl o c p - ' ) - ' , and the assertion follows from (8.2.1) and (16.5.1.2). The mapping T,( f ) is called the tangent linear mapping to f at the point x. For a point z = (ci) E R", put F(z) = (F1(cl,..., p), ...,Fm(cl, .., p)). Since the matrix of F'(cp(x)) with respect to the canonical bases is the Jacobian matrix (Dj F'(cp'(x), . . .,cp"(x))) of type (m, n) (8.10), this Jacobian matrix is (8;'(ej)),,,,,. also thematrixofT,(f)relative to the bases(8;'(ei))l.i..and The mapping F'(cp(x)), or its matrix relative to these bases, is called the local expression of T x ( f )relative to the charts c and c'. To say that f and g are tangent at a point X E X therefore means that f(x) = g(x) and TXU) = Tx(g). The rank of the linear mapping Tx(f) is called the rank o ff at the point x and is denoted by rkxCf). The mapping xwrk,(f) of X into the discrete space N c R is lower semicontinuous on X ((10.3) and (12.7)). We have
.
rkx(f
1 5 inf(dim,(X),
dimf(*)(V).
(16.5.4) Let X, Y, Z be three differential manifolds, and f :X + Y, g : Y -t Z two mappings of class C' . For each x E X,we have (16.5.4.1)
Tx(g o
f
1 = Tf(X,(S)
O
TAf 1-
This follows immediately from the definitions and from (8.2.1). Let X beadirereenrialmanifold,Yadflerentialmanifold, andf: X + Y a mapping of class C'. For f to be locally constant on X it is necessary and suflcient that T,( f ) = 0 for all x E X (or equivalently, that rk,(f) = 0 for all x E X). (16.5.5)
It is clear that the condition is necessary. Conversely, since each point
x E X has a connected neighborhood contained in the domain of definition of a chart at x, the relation TJf) = 0 for all x implies that f is locally con-
stant (8.6.1).
If X is connected, the condition Tx(f) = 0 for all x E X therefore forces f to be constant on X, because if xo E X the set of points X E X such that f(x) =f(xo) is both open and closed (3.15.1). Let X, Y be two differentialmanifoldr,f : X + Y a mapping of class C' (r an integer >O, or co),x a point of X. Then the following conditions are equivalent : (16.5.6)
26
XVI
DIFFERENTIAL MANIFOLDS
(a) T,(f) is a bijective linear mapping; (b) rk,Cf) = dim,(X) = dimf(,)(Y); (c) There exists an open neighborhood U of x in X such that f lU is a homeomorphism of U onto an open neighborhood V of f(x), and the inverse homeomorphism is of class C'. The equivalence of (a) and (b) is linear algebra (A.4.18). For the equivalence of (a) and (c) we reduce immediately, by using charts, to the situation where X = R" and Y = R"',and then the result follows from (10.2.5). When the conditions of (16.5.6) are satisfied with r = oc), the mapping f is said to be a local direomorphism at x, or &ale at x, and X is said to be &taleover Y at thepoint x (relative to f). If X is an open subset of Y, endowed with the induced structure of differential manifold (16.2.4), then the canonical injection of X into Y is Ctale.
Remark
A bijective local diffeomorphism is clearly a diffeomorphism, but a mapping f:X -,Y can be a local diffeomorphism at each point of X without being injective, even if X is connected. An example is the analytic mapping ZHZ' of C - {0} onto itself (cf. (16.12.4)).
(16.5.6.1)
(16.5.7) Now let X be a differential manifold, E a finite-dimensional real vector space, f :X E a mapping of class C', and x a point of X. Then the linear mapping (cf. (1 6.5.2))
is an element of Hom(T,(X), E), called the dzrerential off at the point x, and denoted by d, f. In the particular case where Xis also a finite-dimensionalreal T,(f) Z; is prevector space G, it is immediate that the mapping z,,, cisely the derivative Df(x) defined in (8.1) (an element of Hom(G, E)). Hence in this case we have, if h, E T,(X), 0
(16.5.7.2)
0
d, f * h, = Df(x) * z,(h,).
If u is any linear mapping of E into another finite-dimensional real vector space F, it follows immediately from the above definition that (16.5.7.3)
d,(u o f) = u o d,f.
5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, RANK
In particular, if we take a basis (bj),
27
of E, so that
where the f j are real-valued functions of class C' on X, then by taking for u in (16.5.7.3) the coordinate functions on E we obtain rn
d, f * h, =
(16.5.7.4)
C (d,f j , h,)bj
j= 1
for h, E T,(X). This is also written in the abbreviated form (16.5.7.5)
(instead of C ( d , f j ) @ b,, which is the correct form when T,(X)* @ E i
is identified with Hom(T,(X), E)). The differentials d, f j belong to the dual T,(X)* of T,(X). The elements of this dual space are called tangent covectors to X at x (or simply covectors at x). (16.5.8) Let c = (U, rp, n) be a chart on X at x . Then the bijection 0, introduced earlier is given by (16.5.8.1)
0, = d,rp,
for if we take the chart c' = (R",I,,, n ) on R",the definition (16.5.3.1) shows that T,(rp) = 0; o 0,, and our assertion follows from (1 6.5.2.1) and the definition of the differential (16.5.7.1). This shows that the covectors d,rp' form the basis dual to the basis (OF1(ei)) of T,(X). This dual basis is likewise said to be associated with the chart c. From this result and the definition (16.5.7.1) we see that if f is a C'mapping of X into a finite-dimensional real vector space E, and if F=(flU)Orp-': R"+E,
then
for h, E T,(X), where Di F(rp(x)), the partial derivative of F at the point q ( x ) E R",is identified with a vector in E (8.4). If we identify Hom(T,(X), E) canonically with E @I (T,(X))*, then the formula (16.5.8.2) takes the form
28
XVI DIFFERENTIAL MANIFOLDS
c DiF(CpW) 8 dx n
(16.5.8.3)
dx f =
'pi,
I=1
and in particular, when E = R (so that the Di F(cp(x)) are scalars) (16.5.8.4)
dxf =
iD, F ( 4 W 4
i=1
'pi.
These are the localexpressions of d, f and dxf relative to the chart c. Finally, consider a mapping n :Y + X of class C'. For each C'-mapping f :X + E, where as above E is a finite-dimensionalreal vector space, we have for each y E Y
and in particular, when E = R,
by the definition of the transpose of a linear mapping.
.
(16.5.9) Let X be a differential manifold, let f I , ..,f , , be n functions of class C" dejined on an open neighborhood V of a point x E X,and let f denote the of V into R".Then thefollowing conditions are equivalent: mapping (f')l
(a) There exists an open neighborhood U c V of x such that (U, f I U, n) is a chart on X at the point x ; (b) The dzrerentials d,f' (1 5 i S n) form a basis of (Tx(X))*. For if (W, cp, n) is a chart on X at x , and we put F'
d, f =
c Dj F'(cp(x)) d, cp',
=f'
0
cp-',
we have
n
j=1
and condition (b) signifies that the Jacobian matrix (DjF'(cp(x))) is invertible. The result therefore follows from (16.5.6). (16.5.1 0 ) Let f be a real-valuedfunction of class C' on a differential manifold X . I j f attains a relative minimum (resp. a relative maximum) at a point xo E X, that is to say iff@) 2 f ( x o )(resp. f ( x ) S f(xo))for allpoints x insomeneighborhood of xo , then d,J= 0.
We reduce immediately to the case X = R", and then it is enough to prove that the partial derivatives Dif ( x o ) are all zero, and so we reduce to
5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, RANK
29
the case n = 1. However, thenf'(x,) is the limit at the point 0 of the function h w c f ( x 0 + h) -f(xo))/h, which is defined for all sufficiently small h # 0, and is 20 for h > 0 and $0 for h c 0. Hence the result, by (3.15.4). (16.5.11) The converse of the proposition (16.5.10) is false, as is already shown by the example of the function t I-+ t3 at the point t = 0. At the points x E X such that d, f = 0 we say that f is stationary, or that x is a criticalpoint off; the numberf(x) is called a critical value off. To see whether, at such a point, f has a relative minimum or maximum or neither, we introduce (assuming thatfis of class C2) a quadratic form on the vector space T,(X), as follows. Consider a C2-mapping u :V + X,where V is a neighborhood of 0 in R and u(0) = x. We shall show that the hypothesis that f is stationary at the point x implies that, for the real-valued function v =f 0 u of class C2, the value v"(0) depends only on the tangent vector h, which is the class of the function u. To see this, let c = (U, cp, n) be a chart of X at the point x, and let F = f o cp-l be the corresponding local expression off; we may write v = F w, where w = cp o u is a C2-mapping of V into R". Then we have, by (8.1.4) and (8.12.1), 0
d ( t ) = DF(w(t)) * w'(t),
-
v"(t)= D*F(w(t)) (w'(t), w'(t))
+ DF(w(t))- w"(t);
but by hypothesis DF(cp(x)) = 0, so that
the tranIf c1 = (U, cp,, n) is another chart on X at x and $ = cp 0 cp; sition homeomorphism, and if we put F, =f 0 c,p'; w, = cp, 0 u, then we have F, = F $, so that for all y E U and t E R" 0
DFl(cp,O) * t
-
= DF(cpdy)) . (D$(cp,(Y)) t).
Differentiating again, putting y we shall obtain
= x,
and remembering that DF(cp(x))= 0,
D2Fl(cpl(~))* (5, t) = D2F(cp(x))* (D$(cpl(X))* s, D$(cp,(xN * t).
-
Since on the other hand w = J/ w,, we have B,(h,) = D+(cp,(x)) OC,(h,) and this shows that v"(0) depends only on h,. The formula (16.5.11.1) shows moreover that there is a symmetric bilinear form on T,(x), called the Hessian off at thepoint x and denoted by Hess,(f), such that 0
v"(0) = Hess,(f)
- (h,,
hx).
30
XVI DIFFERENTIAL MANIFOLDS
The symmetric bilinear form D2F(cp(x))on R" is the local expression of the Hessian o f f at the critical point x relative to the chart c; its matrix with respect to the canonical basis of R" is therefore the symmetric matrix (DiDjF(cp(x))),called the Hessian matrix of F at the point q ( x ) (8.12.3). We have now the following suflcient criterion for a C2-function to have a relative minimum or maximum at a point of X : (16.5.12) Let f be a function of class C2 on a diferential manifold X. If at a point x E X we have d, f = 0 and fi Hess,( f ) is positive dejinite (resp. negative dejinitive), then f attains a relative minimum (resp. relative maximum), at the point x.
We reduce immediately to the case X = R". Suppose that the Hessian is positive definite. Then as h runs over the sphere Sn-l, the continuous function h H D 2 f ( x )* ( h , h) is always > O ; hence its greatest lower bound a is > O (3.17.10). Since the function (y, h ) H D 2 f i y ) * ( hh) , is continuous on X x Sn-l,there exists p > 0 such that D2f(y) ( h , h) 2 +a for all y such that Ily - xIJ p and all h E Sn-l. Now Taylor's formula (8.14.2) gives, for 5 E R,
-=
and the result follows.
Remark Let X be a complex-analytic manifold and let XI, denote the underlying differential manifold (16.1.6). As at the beginning of this section we can define the notion of holomorphic mappings f,,f 2 of X into a complex-analytic manifold Y which are tangent at a point. In particular, the tangent vectors to X at a point x will be the equivalence classes of holomorphic functions defined on a neighborhood of 0 in C, with values in X. A chart c = (U, cp, n ) on X at x defines a bijection 8, :T,(X) -+ C" as before, and we deduce that T,(X) is endowed intrinsically with the structure of a complex vector space of dimension n. However, since c is also a chart of X I , , there is also a bijection OclSi T,(X,,) + R2",and therefore, by identifying canonically C" with R2", a bijection 8;; o O,:T,(X) -+ T,(XIR) which is R-linear and does not depend on the choice of the chart c. Hence, by means of this canonical bijection, we may identifv T,(XIR)with the real vector
(16.5.13)
5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, R A N K
31
space obtained by restricting the scalars to R in T,(X). Multiplication by i= is an R-automorphism J, : h,Hih, of the real vector space T,(X,,) such that J: = - I , , where I , is the identity automorphism. The notion of a diyerential is defined just as above for holomorphic mappings of X into a complex vector space E of finite dimension. The elements of the dual T,(X)* of the complex vector space T,(X) are called covectors at x. The dual T,(X)* = Homc(Tx(X), C) can be embedded canonically in HomR(Tx(XlR), C) = T,(XIR)* 8 iTx(XIR)* = (Tx(XlR)*)(q. To be precise, we have by transposition an automorphism ‘J, of T,(XIR)*, which extends canonically to a C-automorphism (also denoted by ‘J,) of (Tx(x\R)*)(C) :
‘J;(h:@<)=(‘J;
h,*)@<
for 5 E C. The C-endomorphisms
p:
= +(Ix
- i ‘J,),
p;
= +(I,
+ i ‘J,)
of (TX(XlR)*)(,-),are projectors on this space, such that p; + p; = I,, and their respective images are Tx(X)* (the space of C-linear forms on T,(X)) and T,(X)* (the space of C-antilinear forms on Tx(X), or equivalently of complex conjugates of C-linear forms), so that we have (T,(XI R)*)(c) = T.JX)* 8 TAX)*If c = (W, q,n) is a chart of the complex-analytic manifold X at the point x , then the forms d,q’ form a basis of T,(X)* over C, and their complex conjugates d,q’ a basis of T,(X)* over C. If now f is a C’-mapping of the diferential manifold XI, into C,so that d, f E T,(XIR)* 8 iT,(X,R)*, we put
If we consider c = (U, cp, n ) as a chart on X I , , the corresponding local coordinates are Wq’ and Yq’ (16.1.6). If F(<’,q ’ , . . ., t”, q”) is the local expression off relative to this chart, then by (16.5.7.4) we have d,cp’ = d,(Wq’)
and consequently
+ i d,(Yq’),
= d,(Wqj) -
i dx(Yqi)
32
XVI
DIFFERENTIAL MANIFOLDS
(the derivatives of F being taken at the point cp(x)). It follows that f is h o b morphic if and only if d;f = 0 for all x E X (9.10.2). In the same way we define d: f and d; f, where f is a C’-mapping of X I Rinto a finite-dimensional complex vector space.
PROBLEMS
1. Letfbe a real-valued function of class C2 on a differential manifold X. (a) lffattains a relative minimum at a point x E X,show that the symmetric bilinear form Hess,(f) is positive (definite or semidefinite). (b) For the functionsf,((, 7)= 5’ 74 and f2((, 7)= f 2 - g4, defined on the plane R2, the point (0,O)is a critical point at which the Hessian is positive but not positive definite. For fl ,show that this point is a relative minimum, but not for f2 .
+
2. Let f b e a real-valued function of class C2 on a differential manifold X,and let x be a critical point off. For each pair of tangent vectors h,, k, at x, there exists a C “-mapping w of a neighborhood V c R2 of (0,O) into X such that w(0,O) = x, To(w) el = h,, To(w) .e2 = k,. If F = f o w, show that DID2F(0) = Hess,cf) . (h,, k,).
-
3. Let f b e a real-valued function of class C2 on a pure differential manifold of dimension n. At a critical point x off, the Morse index offat x is defined to be the number of coefficients< 0 in any reduction of Hess,(f) to diagonal form (or equivalently the maximum dimension of a subspace of T,(X) on which the Hessian is negative definite). Suppose that X is an open set in R”.Let K be a compact subset of X,and suppose that at every critical point x offbelonging to K the Morse index offis L k . Show that there exists E > 0 such that, if g is any C2-function on X satisfying the conditions for all i, j and all z E K, then at each critical point of g belonging to K the Morse index of g is z k . (For each z E X put
and let
u;k) 2 u3z) 2 ..* s u;(z)
be the sequence of (real) eigenvalues of the matrix (DlD,f(~))1SI,j6n, each counted according to its multiplicity. Observe that the hypothesis on f implies that the number m&) = sup(c,(z), -uJ(z)) is strictly positive at each point z E K ; then show that the function z++mf(z)is continuous on X, by using (9.17.4) or Problem 8 of Section 11.5; finally compare m, and mf.) 4.
Let f be a function of class C2 on a pure differential manifold X of dimension n. A critical point x offis said to be nondegenerate if the symmetric bilinear form Hess,(f) is nondegenerate.
5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, RANK
33
(a) If x is a nondegenerate critical point o f i show that there exists a local coordinate system at x for which the local expression of f i s F({l,
...,e)=f(~)-({')'-.*.-({~)>'+({"')>'+*..+ (1")>'
(cf. Section 8.14, Problem 7). (b) Deduce from (a) that the nondegenerate critical points offare isolated. 5. Let G be a finite group of diffeomorphisms of a differential manifold X, and let XG be the set of points of X which are fixed by G.
(a) If x E XG,show that there exists a chart on X at x such that the local expressions of the dfleomorphisms s E G are finear mappings. (Reduce to the case where X is a neighborhood of 0 in R".I f f i s a positive definite quadratic form on R",consider the function g(x) = f(s .x) and use Section 8.14, Problem 7.) (Cf. Section 19.1, Probaee
lem 6.) (b) Deduce that XG is a closed submanifold of X (16.8.3). (c) Suppose that X is connected. If s E G is such that there exists a point xo E XGsuch that the tangent linear mapping T+) is the identity, show that s is the identity mapping. (Use (a) to show that the set of points x E X such that s(x) = x is both open and closed.) 6. If we fix an orjgin in a finite-dimensional real affine space E, the canonical topology (12.13.2) of the vector space so obtained does not depend on the choice of origin, and is
called the canonical topology on E.The dimension of a convex set in E is the dimension of the affine-linear variety generated by the set. A conuex body in E is by definition a closed convex set in E, of dimension equal to the dimension of E; equivalently, it is a closed convex set in E whose interior is not empty (Section 12.14, Problem ll(d)). A conuex polyhedron in E is the intersection of a finite number of closed half-spaces. Hence the intersection of two convex polyhedra is a convex polyhedron. Show that the frontier of a convex polyhedron P of dimension n is the union of a finite number of convex polyhedra of dimension n - 1 which are intersections of P with hyperplanes of support (Section 5.8, Problem 3) of P. These are well determined by this condition and are called the faces of P.
7.
Let E be a real affine space of dimension n and let f:E +R be a C2-function, bounded below.Suppose that for each x E E the symmetric bilinear form (h,k ) w D 2 f ( x ).(h, k) is positive definite. Show that f is strictly convex, and that for a > inf f(x) the x€E
set A. = { x E E :f ( x ) 5 a} is a closed strictly convex set of dimension n, whose frontier is the set Fa = {x E E :f(x) = a}. Through each point x E Fathere passes a unique hyperplane of support, whose equation is for all u E E. If a > inff(x),
then grad f(x) # O
for all
XE
F.. Put g(x) =
X.2E
(gradf(x))/[lgradf(x)II and show that g is a homeomorphism of F. onto Sn-l and that
XVI DIFFERENTIAL MANIFOLDS
34
both g and the inverse homeomorphism ho are of class C'. For each z = t u in R", with t 2 0 and I/u11 = 1, let H(z) = (z I ho(u)), which is a C1-function on R" - (0).We have
H(z) =
SUP
(Y lz)
yEAa
(the f i c t i o n of support of Aa). The function H is convex and positively homogeneous. 8. Let A be a compact convex body in R", having 0 as an interior point.
(a) Prove that for each E > O there exists a convex polyhedron P such that A c P c (1 &)A.(Separate each point of the frontier of (1 &)Afrom A by a hyperplane (Section 12.15, Problem 4(d)).) (b) Let P be a compact convex polyhedron of dimension n in R", having 0 as an interior point. We may suppose that P is defined by m inequalities gJ(x) 5 1, where each gJ is a nonzero linear form on R". Let N > 0 and put
+
+
f(x) =
I
exp(N(g,(x) - 1)).
J=l
Show that the real-analytic function f satisfies the conditions of Problem 7, and that the convex set B = {x E Rn :f(x) 5 l} satisfies P c B c (1 N-' log m) P. (c) Deduce from (a) and (b) that for each E > 0 there exists a real-analytic function f on R" satisfying the conditions of Problem 7 and such that if B = { x E R":f(x) =< l}, we have A c B c (1 &)A.
+
+
9. Let X, Y be two differential manifolds andf, g two mappings of class C'(r 2 1) defined on an open neighborhood of a point x E X, with values in Y. If k is an integer such that 0 5 k 5 r, the functions f and g have contact of order 2 k at the point x e X if f(x) = g(x) and if, for each chart (U, 'p,.n) on X at x and each chart (V,I/J, m) on Y at the point f(x) =g(x), the local expressions F, G off, g are such that IIF(t)- G(t)ll/liz - tJJk tends to 0 as t E R"tends toz = ~ ( x )or,equivalently, ; if DpF(z) = DpG(z) for 1 5 p 5 k. If this condition is satisfied for one pair of charts, then it is satisfied for all pairs. Iff and g have contact of order z k for all k, they are said to have contact of infinite order at x. The relation "fand g have contact of order hk at the point x" is an equivalence relation
between Ck-mappingsdefined on a neighborhood of x with values in Y. An equivalence class for this relation is called a jet of order k from X to Y , with source x and target y (the common value of the mappings in the equivalence class). The equivalence class of f i s denoted by J:( f ) and is called the jet of order k o f f a t the point x . The set of jets of order k with source x and target y is written J:(X, Y ) y ;the set of jets of order k with source x (resp. with target y ) is written J:(X, Y ) (resp. Jk(X, Y ) y )The . union of the sets J:(X, Y), for all x E X and all y E Y is written Jk(X,Y). If Y = R, we write Pt(X) in place of J:@, Y), and P:(f) in place of J:( f). The set P:Q has a natural R-algebra structure, and we have
Then P:(X), = m is the unique maximal ideal of this algebra; we have I I I ~ +=~ 0, and m/m2is canonically isomorphic to the vector space T,(X)* of covectors at the point x. The set J$(R", R")', of jets of order k from R" to R" with source and target at the origins of these spaces is denoted by L:. ". This set carries a natural structure of a real vector space of dimension m (1
(" ")
- m,
and the jets of the monomials xt-rx" . e,
5j 5 m, 0 < Ia 1 5 k ) form a canonical basis. Every set of jets J:(X, Y),
is in one-
6
PRODUCTS OF MANIFOLDS
35
one correspondencewith L:. ,,,by means of charts at the points x, y (where dim,(X) = n and dim,(Y) = m), but if k 1 2 the vector space structure on J:(X, Y), obtained by transporting that of Lk. ,, depends on the choice of charts. When k = 1, J&R, X), is the tangent space Tx(X).
6. PRODUCTS OF MANIFOLDS
All the definitions and all the results of the next three sections (16.6)(16.8) (with the single exception of (16.8.9)) can be transposed to the contexts of real- or complex-analytic manifolds, simply by replacing C"-mappings by analytic mappings in the statements and the proofs. We shall therefore make use of them for real- and complex-analytic manifolds without further comment. Let X,, X, be two topological spaces. If c, =
w,, n,) 4p1,
and
c2 = (U,, (P2 9 n2)
are charts of X,, X,, respectively, the triple (U, x U, , (P, x 'pZ,n , + n,) is a chart of X, x X, (3.20.15 and 12.5); it is denoted by c1 x c,. If c;, c; are two other charts on X,, X, , respectively, and if ci and c; are compatible for i = 1,2, then c, x c2 and c; x c; are compatible, by (8.12.6). If a,is an atlas of X, and a,is an atlas of X2 , the set 2I of charts cl x c2 , where c, E 211 and c2 E a,,is therefore an atlas of X, x X2, and is denoted (by abuse of notation) by 2I, x %, . Moreover, if 21iand 21; are compatible atlases of X i (i = 1, 2), then the atlases a,x 21, and x 2ti are cornpatible. If X, and X, are differential manifolds, the product space X = X, x X2 is separable and metrizable (3.20.16), and the atlases x a,, where %I (resp. a,)runs through the equivalence class of atlases defining the structure of differential manifold on X, (resp. X,), are all equivalent. Hence their equivalence class defines on X a structure of differential manifold which depends only on the structures of XI and X, . The space X endowed with this structure is called the product of the differential manifolds X, and X, . It should be noted that even if 21, and %, are saturated atlases, %XIx ' i l l , will in general not be saturated. Whenever we consider XI x X, as a differential manifold, it is always the product structure as defined above that is meant, unless the contrary is expressly stated. Example (16.6.1) If El, E, are two finite-dimensional real vector spaces, each endowed with its canonical structure of differential manifold, it follows from the definitions (16.2.2) that the product manifold El x E, is the product vector space endowed with its canonical structure of differential manifold.
36
X V I DIFFERENTIAL MANIFOLDS
Let X I , X , be two direrentialmanifolds,X = X I x X, theirproduct. The projections prI :X + XI, pr, : X + X, are morphisms (16.3). For each point ( x , , xz) E X , the mapping (16.6.2)
(T(x,,xz)(Prl),T(XI,X2)(PrZ)) :T(,I, is an isomorphism of vector spaces.
x2)(x1 x X,)
-,T,,(X,)
x T,,(X,)
In view of the definition of the product manifold structure, we reduce immediately to the situation where XI and X, are open sets in R"' and R"' respectively. The first assertion is then a trivial consequence of (8.12.10), and the second follows from (8.1.5) applied to a C'-mapping of a neighborhood of 0 in R,with values in R"' x R"'. We shall identify canonically T(x,,xz)(X:x X,) with the product T,,(X,) x T,,(X,) by means of the isomorphism defined in (16.6.2). The canonical injection T,,(X,) -,T(,,, J X 1 x X,) resulting from this identification is just the tangent linear mapping at the point x , to the injection y , H (yl,xz), which is a morphism of X, into X. Likewise for the canonical injection T,,(X,) -,T,,,, ,JX1 x X,). It is clear that
(16.6.4) Let Y , X , , X , be three digerential manifolds and fl :Y -,X,, fz : Y -,X , two mappings. Then the mapping f = (5,fz): Y -,X I x X, is of class C' ( r an integer >O, or 00) if and only if f, and f, are of class c'. Moreover, for all y E Y , we have
T,((.h
9
fz))= (Ty(fi1, Ty(f2))
with the identification (16.6.2).
Once again we reduce to the case in which X I and X2 are open sets in R"' and R"', and the result then follows from (8.12.6). (16.6.5) Let X I , X , , Y,, Y, be diferential manijbids, and A : Yi + X i (i = 1,2) mappings of class C'. Thenfi x f, : Y, x Y, + X, x X 2 is a mapping of class C', and we have
T(y,,Y J f i
x f 2 ) = T,,(fl) x
TY'(fZ),
rk(Yl,YZ)(flx fz) = rky1(f1)+ rky*(f2). The second formula is a trivial consequence of the first, and that follows from (16.6.4) and (16.6.2), since fi x f, = (fl prl,f 2 prJ. 0
0
7 IMMERSIONS, SUBMERSIONS, SUBIMMERSIONS
37
(16.6.6) Let X,, X2, Z be three diyerential manifolds, f X, x X, + Z a mapping of class C' (ran integer >O, or a),and (a,, a,) apoint of X1 x X, . Let f(a,, * ) (resp. f( , a,))denote the partial mapping X,H f(a,, x,) (resp. x1w f ( x l , a,)). Then we have
-
T(4,,u2)(f)= T4,(f( *
9
ad)
0
PI
+ TaZ(f(a1, . ))
o
P2
9
where PI
= T(u1,42)(~r1) : T(ai,uz)(X~x X2)+Ta,(XA
P2
= T(41, 42)(Pr2) : T(41,4z)(x1
x2)
T4,(x,)
are the canonicalprojections (with the identijcation (16.6.2)). Once more, the proof reduces to the case where X,, X, ,Z are open sets in R"',R"', and R'", and then it follows from (8.9.1). In particular, if Z = E is a finite-dimensional real vector space, we have
(16.6.8)
With the hypotheses and notation of (16.6.6), suppose that T42f(a1,
*
) : TaZ(X2)
-+
T,(Z)
(where c =f(al, 4 )
is bijective. Then there exists an open neighborhood U , of a, in X, and an open neighborhood U , of a, in X2 with the following properties: for each x, E U , there exists a unique point u(x,) E U , such that f ( x , , u(xI)) = c, and u is a C'-mapping of U, into U, . Furthermore, we have
("implicit function theorem "). We reduce to the case where X,, Xz ,and Z are open sets in R"',R"', and R", respectively, and then the theorem is a particular case of (10.2.3).
7. IMMERSIONS, SUBMERSIONS, SUBIMMERSIONS
(16.7.1) Let X, Y be two differential manifolds, f : X + Y a mapping of class C", and x a point of X. The mapping f is said to be a subimmersion at the point x if there exists a neighborhood U of x in X such that the function x ' ~ r k , . ( f ) is constant on U. The mapping f is said to be an immersion
38
XVI
DIFFERENTIAL MANIFOLDS
(resp. a submersion) at x if the linear mapping T,cf) is injective (resp. surjective). By virtue of the lower semicontinuity of the rank of f (16.5), this implies that rk,.Cf) = dim,,(X) (resp. rk,,(f) = dimf(,.,(Y)) for all x' in some neighborhood of x, and hence f is a subimmersion at the point x. A mapping f : X Y of class C" is both an immersion and a submersion at the point x if and only iff is Ctale at x (16.5.6). It is clear that the set U of points of X at which f is a subimmersion (resp. a submersion, resp. an immersion, resp. Ctale) is open in X. The mapping f is said to be an immersion (resp. a submersion, a subimmersion, Ctale) if U = X. For example, the projections of a product manifold XI x X, onto its factors X I , X, are submersions (16.6.2). -+
(16.7.2) I f f : X + Y and g : Y Z are both submersions (resp. both immersions), then g f : X Z is a submersion (resp. an immersion). -+
0
-+
This follows immediately from the definitions (16.7.1) and from (16.5.4). We remark that the composition of two subimmersions is not necessarily a subimmersion (Section 16.8, Problem l(b)). (16.7.3) If fi : XI -+ Y, and f , : X , + Y , are both submersions (resp. immersions, resp. subimmersions), then flx f i : XI x X, -+ Y, x Y, is a submersion (resp. an immersion, resp. a subimmersion).
This follows from the definitions (1 6.7.1) and from (1 6.6.5). (16.7.4) Let f : X --+ Y be a mapping of class C". In order that f should be a subimmersion of rank r at a point x E X, it is necessary and sufficient that there should exist a chart ( U , cp, n) of X , a chart ( V , II/, m ) of Y and a C"-mapping F :q(U) $(V) such that x E U , f ( x ) E V, q ( x ) = 0, $ ( f ( x ) ) = 0, f I U = $-' F cp, and such that the local expression F o f f is the restriction to q ( U ) of the mapping -+
0
0
of R" into R"
This is an immediate consequence of the rank theorem (10.3.1). (16.7.5) r f f : X -+ Y is a submersion, the image under f of any open set U in X is open in Y .
8 SUBMANIFOLDS
39
If x E U, it follows from (16.7.4) that there exists an open neighborhood W c U o f x such that f ( W ) is open in Y; now apply axiom (0,)of topological spaces (12.1). (16.7.6) It should be remarked that if f : X -+ Y is an immersion, f ( X ) is not necessarily closed, noreven locally compact, even iff is injective (16.9.9.3). On the other hand, a C"-mapping f : X -+ Y can be injective (resp. surjective) without being an immersion (resp. a submersion), as is shown by the example of the bijective mapping l e t 3 of R onto R (cf. however Section 10.3, Problem 2). (16.7.7) (i) Let f : X -+ Y be an injective immersion, and let g : Z + X be a continuous mapping ( X , Y , Z being differential manifolds). For g to be of class c'it is necessary and suficient thatf g : Z + Y should be of class C'. (ii) Let f : X -+ Y be a surjective submersion and let g : Y 2 be a mapping ( X , Y , Z being dixerential manifolds). For g to be of class C' it is necessary and suficient that g f : X + Z should be of class C'. 0
-+
0
Only the sufficiency of these conditions requires proof, by (16.3.3). Also the questions are local on X, Y, and Z by virtue of the continuity of g (which follows from the continuity of g f i n (ii)). In case (i), we may therefore suppose (16.7.4) thatfis the mapping (tl, . . . , t")~ ( t '. .,. , t",0,. . . , 0) of R" into R" (with n 5 m). The assertion of (i) is then that a mapping z H ( g ' ( z ) , . . . , g"(z), 0, . . . ,0) of Z into R" is of class C' provided that the gi are of class C'. In case (ii) we may likewise suppose that f is the mapping (el, . . . , 5") . .., 5") of R" onto R" (with n 2 m). The assertion of (ii) is then that the mapping (tl,.. ., t " ) ~ g ( t ' , .. ., 5") is of class C' on R" if and only if the mapping (c', ..., y)t+g(tl, ..., t") is of class C' on R" (16.6.6). 0
Remark (16.7.8) The preceding results can be extended immediately to the situation where C'-mappings (r a positive integer) replace C"-mappings, subimmersions are replaced by mappings of locally constant rank, and submersions (resp. immersions) by mappings f such that T,(f) is surjective (resp. injective).
8. SUBMANIFOLDS
(16.8.1) Let X be a separable metrizable space, Y a differential manifold, f a mapping of X into Y . In order that there should exist on X a structure of diflerential manifold for which the underlying topology is the given topology
40
XVI
DIFFERENTIAL MANIFOLDS
on X and such that the mapping f is an immersion, it is necessary and sufficient that the following condition be satisfied: (16.8.1.1) For each a E X , there exists an open neighborhood U of a in X anda chart (V, $, m )of Y such thatf(U) c V andsuch that $ (fI U) is a homeomorphism of U onto the intersection of $(V) with a linear subvariety of R". When this condition is satisfied, the structure of diferential manifold on X satisfying the conditions above is unique. 0
The necessity of the condition follows immediately from (16.7.4). To prove sufficiency, consider for each a e X a neighborhood U, of a in X and a chart (V,, $, ma) of Y satisfying (16.8.1.1), and let E, denote the linear subvariety of Rmasuch that $,(V,) n E, = $,(f(U,)). (E, is unique because E, n $,(V,) is a nonempty open subset of E, .) Let n, = dim E, and let 1, be an affine-linear bijection of E, onto R".. Finally let cp, be the composition of 1, and $, 0 (fI U,). We shall show that the charts c, = (U,,cp, n,) form an atlas of X. Suppose therefore that U, n Ub # @; then (16.8.1 .I) implies that cp,(U, n ub) = W.6 and (pb(u, n ub) = Wb, are open sets in Rnaand Rnb,respectively. If $&(resp. $6,) is the restriction of $. (resp. $b) to V, n v b , and Vab (resp. %a) the restriction of cp, (resp. (pb) to U, n ub, it is immediate that (Pb, o c p ~ = l Ab 0 (i+hbao $,;I) o ((,I;l)l w,,). Hence to show that %a ' ;pc is indefinitely differentiable, it is enough to observe that the restriction of an indefinitely differentiable mapping 0
$ba
';$ ,
0
: V, n v
b+
R"
(where m = ma = mb)
to the intersection of the open set V, n v b in R" with a linear subvariety E, of R" is indefinitely differentiable on this intersection, which is obvious by (8.12.8). The uniqueness of the structure of differential manifold on X follows from (16.7.7(i)). For by replacing X by an open neighborhood of a point of X, we reduce to the case where f is injective, and apply (16.7.7(i)) to g = 1, (considering two structures of differential manifold on X satisfying the conditions of (16.8.1)). When the condition (16.8.1.1) is satisfied, the unique structure of differential manifold defined in (16.8.1) is called the inverse image under f of the structure of differential manifold on Y. In particular: (16.8.2) Let X be a separable metrizable space, Y a differential manifold, f a mapping of X into Y with the property that for each x E X there exists an
8 SUBMANIFOLDS
41
open neighborhood U of x such that f I U is a homeomorphism of U onto an open set in Y . Then there exists a unique structure of dflerential manifold on X for whichf is an immersion, andf is then infact ktale (16.5.6). (16.8.3) Let Y be a differential manifold, X a subspace of Y. I f the canonical injection f : X -+ Y satisfies the condition (16.8.1.1), then the space X, endowed with the structure of differential manifold which is the inverse image under f of that of Y, is said to be a submanifold of Y. We also say that X is a submanifold o f Y; this abuse of language is justified by the property of uniqueness in (16.8.2). The condition (16.8.1 .I), in the present situation, is that for each x E X there exists a chart ( V , $, m ) of Y such that x E V , $(x) = 0 and such that $(V n X ) is the intersection of the open set $(V) of R" and the vector subspace of R" given by the equations = 0, . . . , = 0; and then (V n X, $1 (V n X), n ) is a chart of the submanifold X. It follows that V n X is closed in V, and hence that X is locally closed in Y (12.2.3). Moreover, there is an open neighborhood W c V of x in Y which is diffeomorphic to (W n X) x Z, where Z is a submangoldof dimension n - m of Y ,containing x. To see this we reduce to the case where Y = $(V) c R" and X = $(V) n R", and then it is obvious.
r"
cn+'
(16.8.3.1) In view of (16.7.4), the condition for X to be a submanifold of Y can be expressed as follows :for each x E X there exists an open neighborhood U o j x in Y and a submersion g : U -+ R"-" such that X n U is the set ofpoints z E U such that g(z) = 0. (16.8.3.2) In particular, when Y = R", we may always assume (by translation if necessary) that OEX, and then, by permuting the coordinates, that if g = ( g l , .. . ,g"-"), the determinant of the matrix formed by the first n - m columns o f the Jacobian matrix of g is # O at the point 0. If we identify R" with R" x R"-", it follows from the implicit function theorem (10.2.2) that there exists an open neighborhood V of 0 in R" such that U n (V x Rn-") is the graph of a C"-mappingf= (f',. . . ,f"-") of V into R"-", or in other words this submanifold is the set of points x = ( tl,. . . , 5") such that r j E V for 1 m and lm+k -f k ( < ' ,. . ., 5") = 0 for 1 k 5 n - m.
sjs
'
s
(16.8.3.3) Every open subset of a manifold Y, endowed with the induced structure (16.2.4) is a submanifold of Y. Conversely, a submanifold X of Y such that dim,(X) = dim,(Y) for all x E X is open in Y. Every discrete subspace (necessarily at most denumerable) of Y is a submanifold of dimension 0. In a pure manifold Y of dimension n, a pure submanifold of dimension n - 1 is called a hypersurface.
42
XVI DIFFERENTIAL MANIFOLDS
(16.8.3.4) Let X be a submanifold of Y, X a submanifold of Y , and j : X + Y, j ' : X + Y' the canonical injections. Let g : Y + Y be a C'mapping such that go() c X'. Then we can write g o j = j' o f , where f is a C'-mapping of X into X'. This follows from (16.7.7(i)). (16.8.4) Let X, Y be two diferential manifolds, f : X -+ Y an immersion. I f f is a homeomorphism of X into the subspace f ( X ) of Y, then f ( X ) is a submanifold of Y, and f : X -,f ( X ) is a difleomorphism.
For each x E X we apply (16.7.4) (keeping the notation used there) with r = n = dim,(X) and m = dimf(x,(Y). Since f is a homeomorphism of X onto the subspacef(X), it follows that F(p(U)) is an open neighborhood of F(cp(x)) = $(f ( x ) )in $(V) n R",and hence there exists an open neighborhood T c $(V) of ~ ( f ( x )in) Rm such that T n R" = F(p(U)). Putting W = $-'(T), the chart (W, $1 W, m) on Y satisfies the condition of (16.8.3) relative to the subspace f(X). Furthermore, it follows from (16.7.4) that F is a diffeomorphism of q(U)onto the open set F(q(U)), and the second assertion of (1 6.8.4) follows. An immersion f which satisfies the hypotheses of (16.8.4) is called an embedding of X in Y.
Remark (16.8.5) It can happen that an immersion f : X -,Y is injective and that f ( X ) is closed in Y but that f is not an embedding. For example, take X to be the open interval ] - co, 1[ in R,Y = R2,andfto be the immersion t 2 - 1 t(t2 - 1) tH
(TiTi'7Tfl).
This immersion is not an embedding, because,f(- I)
= lim f ( t ) (cf. Problem 2). t-tl
If X is a submanifold of Y a n d j : X -+ Y is the canonical injection, it follows from the definition of an immersion (16.7.1) that for each x E X the linear mapping T,(j) : T,(X) ?-. Tx(Y) is an injection, by means of which we shall identifr canonically T,(X) with a vector subspace of T,(Y). In the particular case where Y is a vector space E of finite dimension n, we recall (1 6.5.2) that there is a canonical linear bijection 2, : T,(E) + E. The image of z,(Tx(X)) under the translation hl-, h + x is an afie-linear variety in E, passing through the point x , of dimension m = dim,(X). This is called the tangent asne-linear variety to X at the point x (or the tangent (16.8.6)
8 SUBMANIFOLDS
43
to X at x if m = 1, the tangent plane if m = 2, the tangent hyperplane if m = n - 1). It is the set of points x + z,(h,) in E, as h, runs through T,(X). It should be observed that the possibility of defining such a "tangent linear variety" as a submanifold of E depends essentially on the group-structure of E, and that there is no analogous definition when E is replaced by an arbitrary differential manifold Y. Remark
(16.8.6.1) If X is a differential manifold, a a point of X, and E a vector subspace of T,(X), then there exists a submanifold Z of X containing a and such that T,(Z) = E. T o see this, it is enough to consider the case where X is an open set in R" and a = 0, and then we may take Z to be the intersection of X with a vector subspace of R". (16.8.7) (i) Let Z be a differential manifold, Y a submanifold of Z, X a subspace of Y. Then X is a submanifold of Z if and only i f X is a submanifold OJY.
(ii) If X, (resp. X,) is a submanifold of Y, (resp. YJ, then XI x X, is a submanifold of Y, x Y, . Assertion (ii) follows immediately from the definitions of a submanifold (16.8.3) and a product manifold (16.6). As to (i), suppose first that X is a submanifold of Y. Using local charts, we reduce to the case where Z = R", Y is an open subset of R" (where n < m ) , and there exists a submersion f : Y R"-P such that X is the set of points y E Y satisfying f ( y ) = 0. We then extend f to a submersion g : Y x R"-" -+ Rm-Pby defining g ( y , t ) = ( f ( y ) ,t ) for y E Y and t E R"-". Then X is the set of points ( y , t ) satisfying g(y, t ) = 0, hence is a submanifold of Z. Conversely, suppose that X is a submanifold of Z. This time we may suppose that Z = R" and that X is an open set in RPcontaining the origin. The tangent space T,(Y) can be identified with a vector subspace E of R" containing RP. Let F be a supplement of RP in E, let G be a supplement of E in R", and let n be the projection of R" onto F parallel to G + RP.The restriction h of n to Y is a submersion, because by definition the rank of T,(h) is dim(E) - p = dim(F). The set X' of points y E Y such that h(y) = 0 is therefore a submanifold of Y , hence of Z , and of the same dimension as X. Since X' contains X and the canonical injection .j of X into X' is of class C" (16.7.7(i)),it follows that j is a local diffeomorphism (16.5.6); hence X is open in X' and therefore is a submanifold of Y. -+
(16.8.8) L e t f : X - + Y beasubimmersion (16.7.1),aapointofX,andb=f ( a ) .
44
XVI DIFFERENTIAL MANIFOLDS
(i) The subspace f -'(b) is a closed submanifold of X . The tangent space T,( f -'(b)) tof -'(b) is the kernel of T,(f), and hence we have an exact sequence
TQU)
0 T,(f -l(W--* TAX) TO). There exists an open neighborhood U of a in X such that.f(U) is a sub(ii) manifold of Y ,and we have +
(1 6.8.8.1)
dim,(X)
= dim,( f ( U ) )
+ dim,( f
-'(b)).
Furthermore, if E is any supplement of T,( f -'(b)) in T,(X), there exists u submanifold of V of U whose tangent space at the point a is E. For each submanifold V of U having this property, there exists an open neighborhood W of b in Y such that the restriction o f f to V' = V nf -'(W) is an isomorphism of V' onto W nf ( U ) , and such that T,(V) is a supplement of T,( f -'( f ( x ) ) ) in T,(X) for all x E V'. (iii) rS To(f ) is not surjective, V may also be chosen so thatf ( U ) is nowhere dense (12.16) in Y . (iv) I f f is an injective subimmersion, thenf is an immersion. I f f is a bijective submersion, then f is a diffeomorphism (cf. Problem 3). We can apply (16.7.4) to the point a E X, and then it is clear that
U nf -l(b) = (p-'((p(U) n F-'(O)). Hence we reduce to the case where X and Y are open sets in finite-dimensional real vector spaces and f is the restriction to X of a linear mapping. Parts (i)-(iii) of the proposition are now obvious ((16.5.2) and (12.16)). As to (iv), it follows from (i) that iff is injective, then To(f ) is injective, for each a E X, hence f is an immersion. If in addition f is a submersion, then f is a local diffeomorphism. Finally, iff is also bijective, then f is a diffeomorphism. (16.8.9) Let Y be a di3erential manifold and let be a sequence of real-valued Cw-functions on Y . Let X be the set of x E Y such thatfi(x) = 0 for 1 5 i 5 r. Suppose that, for each x E X , the differentials d, f i (1 5 i 5 r ) are linearly independent covectors in T,(Y)*. Then:
(i) X is a closed submanifold of Y , and for each x E X the tangent space T,(X) is the annihilator in T,(Y) of the subspace of T,(Y)* spanned by the differentials d,fi, and consequently is of dimension dim,(Y) - r. (ii) Let F be a Cw-function on Y which vanishes at all points of X . Then for each x,, E X there exists an open neighborhood U of x,, in Y and r functions Fj (1 $ j 5 r ) of class C" on U such that
8 SUBMANIFOLDS
45
for ally E U. If Y is a real- (resp. complex-) analytic mangold and thefunctions F and& are analytic, then the functions Fj may be chosen to be analytic. (i) Since the differentials dxfi are linearly independent, there exists an open neighborhood V(x) of x in Y such that the differentials d,. fiare linearly independent for all x' E V(x) (16.5.8.4). Replacing Y by the open set which is the union of the neighborhoods V ( x ) as x runs through X, we may therefore assume that the d, fiare linearly independent for all y E Y. However, then the mapping g :y ~ ( J ( y ) )s i,5 r of Y into R' is a submersion, by virtue of (16.5.7.2) and the definition (16.7.1). Since X = g-l(O), we can now apply (16.8.8).
(ii) By virtue of (16.7.4) we may limit ourselves to the case where Y is an open set in R" and f i ( x ) = xi, the j t h coordinate of x (1 S j 5 r), so that X = U n R"-' (where R"-' is identified with the subspace spanned by the last n - r vectors of the canonical basis of R"). Moreover we may take xo to be the origin. Then the assertion (for C"-functions) is a consequence of the following lemma: (16.8.9.1) Let F be a real-valuedfunction of class C" on an open cube I" c R" (where I is an open interval in R). Then in 1" we can write (16.8.9.2)
F(x', . . . ,x") = F(0,. . . ,0) + x'Fl(x', .. . , x") + x2F2(x2,. . ., x") + * - . + x"-'F,,-,(x"-', x") + x"F,(x"),
where F,, . . . , F, are P-functions on I". Assuming this lemma, since F(x', . . . ,x") = 0 whenever x1 = * * . = x'= 0 we obtain successively that F,, F,,-,, . . ., F,+, are identically zero: first we put all the x i except x" equal to zero, then all the x i except x"-' and x", and so on. To prove (16.8.9.1) we write
F(d, . . . , 2)= (F(x', x', . . . , x") - F(0, x', . . ., 2)) + F(0, x', . . . , x") so that by induction on n we are reduced to proving that the function (16.8.9.3) G(xl, . . . , x") = (xl)-'(F(xl,
. . .,x") - F(O, xz, ...,x")>
which is defined whenever x1 # 0, tends to a finite limit as x1 + 0, and that the function so extended is of class C" on I". Using Taylor's formula (8.14.2) we have, for x1 # 0 and any integer p 2 1,
46
XVI DIFFERENTIAL MANIFOLDS
(16.8.9.4) G(x',
...,X") = D,F(O, x', ...,X") + X1 D:F(O,
+-
(x1)P-l
P!
DSF(0, x2, .. . ,x")
x',
. . . ,2')+ -
* *
+ (xl)-'H(x', . ..,x"),
where (16.8.9.5)
'F(t, x2, ... , x") d t
H(xl, ..., x")=
is of class C" on I" by hypothesis. Differentiating under the integral sign (8.11.2) and replacing F by some derivative of the form DTDY...D;F, we are reduced to showing that, as x1+ 0, the derivatives (16.8.9.6)
D!((x1)-lH) =
k
j=O
(-l)j(k))J!(x')-'-'D:"H 1
(0 k S p - l), calculated by Leibniz's rule (8.13.2), tend to zero uniformly in (x', . . , ,2')on a neighborhood V of 0 in R"-'; but by (8.11.2) we have (16.8.9.7)
D;-jH(x',
.. . ,x") =
(p - k + j ) !
DP,+'F(t,x2, .. .,x") d t
and hence by the mean value theorem
where C is a constant, for all (x', . . . ,x") E V. Using this inequality in (16.8.9.6) now completes the proof of the lemma. In the case where F is (real- or complex-) analytic, the proof is much simpler, by considering the Taylor expansion of F at the origin, and it follows from (9.1.4) that the F, are analytic in a neighborhood of 0. We remark that (16.8.3.2) shows conversely that for each submanifold X of Y and each point a E X, there exists an open neighborhood V of a such that V n X is defined by equations satisfying the conditions of (16.8.9). Examples (16.8.10)
In (16.8.9) let us take Y = R"" - (0)and r = 1, and the sequence
(A) to consist of the single functionf: YH Ilyll, the Euclidean norm on Y.
8 SUBMANIFOLDS
47
Then X =f -'(l), as a topological space, is just the unit sphere S,; since Df is of rank 1, it follows that S, is endowed with a structure of a submanifold of Y. Let us show that this structure of differential manifold on S, is the same as that defined in (16.2.3). For this it is enough to observe, in view of (16.8.3), that the formulas (16.2.3.1) define a dzTeomorphism of the submanifold s,,- (eo} of R"" onto R". Now consider the mapping g : y-(y/llyll, Ilyll) of Y into S, x : R . This mapping is a bijection, whose inverse is (z, [)I+ [z; also g is a submersion, because Df # 0 and the restriction of the mapping y~-+y/llyll t o a sphere I S , (where 1 > 0) is a homothety of this sphere onto S,, hence a diffeomorphism. It follows that g is a diffeomorphism, by virtue of (16.6.4) and (16.8.8(iv)). (16.8.11) Suppose that the conditions of (16.6.8) are satisfied, so that f : X , x X , --f Z is a submersion at the point ( a l , a,). Then there exists an open neighborhood W of (a,, a,) in X, x X, such that, if Y is the set of points ( x , , x,) satisfying f ( x l , x , ) = c, then Y n W is a submanifold of X, x X,, and the restriction of pr, to Y n W is an isomorphism of this submanifold onto an open subset of X,. If X, = X2 = Z = C and i f f is holomorphic, the set Yo of points of Y where D, f ( x , , x,) # 0 is an open subset of Y; this complex-analytic submanifold of C2 is called the Riemann surface (relative to the second coordinate) defined by the holomorphicfunction f.For example, if f(x17
x2> =
- ex',
we have Y = Y o , and Y is an analytic subgroup of the complex-analytic group C* x C (16.9.10). This surface Y is called the Riemann surface of the logarithmic function (cf. Problem 12). The restriction of pr, to Y is an &ale morphism of Y onto C* = C - (0). For each f E Y we write log(t) = pr, t , so that we have log(tt') = log(t) log(t') for t , t' in Y (the law of composition in Y being ( x ,y)(x', y ' ) = (xx',y + y')), and the mapping t~ log t is a holomorphic mapping (16.3.5) of Y into C.
+
(16.8.12) Let f : X Y be a submersion,Z any submanifold of Y .Thenf - ' ( Z ) is a submanifold of X , and the restriction f - '(2) + Z o f f is a submersion (cf. Problem 17). --f
Using a chart satisfying (16.7.4), we may assume that Y is an open set in R" and X an open set in R" ( n 5 m),and that f is the restriction to X of the canonical projection ((l, . . . , trn)++(t', . . . , Y). Let x Ef -'(Z) and let y E 2 be its projection. Then by hypothesis there exists a chart (U, $, n) on Y at the point y such that $(U n 2) = $(U) n RP,where p =< n. If we
48
XVI DIFFERENTIAL MANIFOLDS
denote by cp the restriction to f -'(U) of J/ x lRm-": U x Rm-"+ R"', then
cf-'(TJ), cp, m) is a chart on X at the point x , such that cp(s-'(U) nf-'(Z)) = cp(f'(U))
n RP+"'-".
Hence the result. (16.8.1 3) Let X, Y be two diyerential manifolds and f:X + Y a mapping of class C". Then the graph rf off in X x Y is a closedsubmanifold of X x Y, the mapping g : x ~ ( xf(x)) , is an embedding (16.8.4) and the vector subspace TCx, ,(,),(Tf) of T(,, ,(,.,(X x Y) is the graph of the linear mapping T,(f).
We know that g is a homeomorphism of X onto r, (the inverse of g being the restriction to rf of the projection prl), and that rf,being the set of points z E X x Y such that pr,(z) =f(pr,(z)), is closed in X x Y. Hence (16.8.4) it is enough to prove that g is an immersion, but since T,(g) = (Tx(lx),T,(f)) by virtue of (16.6.4), it is clear that T,(g) is a linear mapping of rank equal to dim,(X). Hence the result. In particular, when Y = X , the diagonal A of X x X is a closed submanifold of X x X, and the diagonal mapping XH(X, x) is a diffeomorphism of X onto A. In R x R,the set of pairs (c, q) such that 5 # 0 and q = sin(l/c) is an (analytic) submanifold whose closure is not locally connected. Remark (16.8.14) Let Z be a submanifold of X x Y such that at a point (a, b) E Z the restriction to T(,, b)(Z) of the projection T,(X) x Tb(Y)+ T,(X) is a bijection onto T,(X). Since this restriction is equal to T(,, b)@), where p : Z -+ X is the restriction of prl : X x Y + X, it follows (16.5.6) that there exists an open neighborhood U of (a, b) such that p [U is a diffeomorphism onto an open neighborhood V of a in X. Sincep(x, y ) = x , the inverse diffeomorphism g : V + U is of the form XH (x,f(x)), where f:V 3 Y is of class C", and U is therefore the graph off in V x Y.
PROBLEMS
1. (a) Let f : X + Y be a submersion and g : Y + Z an immersion. Show that g f : X Z is a subimmersion. (b) The mapping f:t H(t, t z, t 3, of R into R3 is an immersion, and the projection g : (x, y, z)H (y, z) of R3into R2 is a submersion, but g 0 f is not a subimmersion oi R into R2,although it is injective. 0
--f
8 SUBMANIFOLDS
49
2. Let f :X +-Y be. an injective immersion which is proper (Section 12.7, Problem 2).
Show that fis an embedding (observe that the image of a closed subset of X is closed in Y). Give an example of an embedding of R into RZwhich is not proper (consider a ''spiral ").
3. (a) In R3,the union of the line z = 1,y = 0 and the complement in the plane Y :z = 0 of the line z = 0, y = 0 is a nonconnected manifold X. Show that the restriction to X of the projection (x, y, z) H(x, y) of R3onto Y is a bijective immersion of X onto Y which is not a diffeomorphism. (b) Let X be a connecteddifferential manifold, Y a differential manifold, andf :X --+ Y a bijective subimmersion of X onto Y.Show thatfis a dfleomorphism. (Observe that the set of points x E X at which f is a submersion is both open and closed in X;to show that this set is nonempty, use (16.8.8(iii)), (12.6.1), and Baire's theorem (12.16.1).) 4. Give the analogs of the results of Sections 16.3-16.8 for manifolds of class C (r 2 I), which were defined in Section 16.1, Problem 2. Show that the analog of (16.8.9(ii)) is
false.
5. Let E be a finite-dimensional real vector space, F a closed subset of E,and IIxII a norm defining the topology of E (12.13.2). If a is a nonisolated point of F,the contingenf of F at a is the union of the rays through 0 whose direction vectors of norm 1 are l i t s of sequences of the form ((x. u)/llx, - all), where (xJ is a sequence of points of F, distinct from a and with a as limit. The paratingent of F at (1 is the union of the lines
-
through 0 whose direction vectors are limits of sequences of the form
((xn-
ym)/lkn-~nll),
where (x,,) and (y,,) are two sequences of points of F, distinct from a and with a as limit, and such that xn # y,,for all n. Show that F is a submanifold of class C1 of E if and only if, for each nonisolated point a E F, (i) the paratingent of F at a is a vector subspace P of E, and (ii) if N is a supplement of P in E and p : E +- P is the projection parallel to N, then the image under p of any neighborhood of a in F is a neighborhood of p(a) in P. (Show first that there exists a compact neighborhood U of a in F such that p ] U is a bijection onto a compact neighborhood V ofp(a), by contradiction. If we identify E with P x N, then U is the graph of a continuous mapping f :V +N. Show that f is of class C' by using condition (i) above, Problem 3 of Section 8.6, and arguing by contradiction.) Give an example in which the contingent and the paratingent of F at a are each equal to the whole of E but the condition (ii) above is not satisfied. (Take E = R2.) 6. In a differential manifold X,let Y1, . . . , Y,be submanifoldswith a common point a.
Show that the union of the Y t cannot be a submanifold of X unless it has the same dimension at a as one of the Y1. (UseProblem 5.)
7. Show that a complex-analytic submanifold of C? which is compact and connected consists of a single point (seeSection 16.3, Problem 3).
50
XVI DIFFERENTIAL MANIFOLDS
8. Let X, Y be. two differential manifolds, f : X+Y a mapping of class C", and U a connected open subset of X. If r is the least upper bound of rk,(f) as x runs over U, show that r is finite and that the set of points x E U at which rk,Cf) = r is open. Deduce
that the set of points at which f is a subimmersion is a dense open subset of X (argue by contradiction). Iffis an open mapping, the set of points at whichfis a submersion is dense in X.
9. Let X, Y be two differential manifolds, f : X +Y a mapping of class C'. If Z is a sub-
manifold of Y, the mappingfis said to be trunsuersal ouer Z at x E f-'(Z) if the tangent space T,(,,(Y) is the sum of T,(,)(Z) and T,(f)(T,(X)), and f is said to be transyersul ouer Z if this condition is satisfied for all x ~ f - l ( Z ) .If so, thenf-I(Z) is a submanifold of X, and for each x ~ f - l ( Z )the tangent space T,Cf-'(Z)) is the inverse image under T,Cf) of T,(,,(Z). (Since the question is a local one, we may take Z to be a submanifold given by an equation g ( y ) = 0, where g : Y +RP is a submersion; consider the composite mapping g f.) In particular, if X and Z are submanifolds of Y, we say that X and Z are transversal at u point x E X n Z if the canonical injection of X into Y is transversal over Z at x , or equivalently if T,(Y) = T,(X) T,(Z), which is symmetrical in X and Z. The submanifolds X and Z are said to be trunsuersal if they are transversal at all points x f X n Z; in that case, X n Z is a submanifold of Y . 0
+
10. Let f :X +Z and g : Y +Z be two mappings of class C", and consider their product f X g : X X Y +Z x Z, which is also of class C". Show that f x g is transversal over the diagonal A of Z x Z if and only if, for each pair (x, y ) E X x Y such thatf(x) = g(y), we have (*)
Tz(Z) = TxW(Tx(X))
+ T&)(TAY)),
where z =f(x) = g(y). This condition is always satisfied if either f o r g is a submersion. When condition (*) is satisfied, the set of points (x, y ) E X x Y such thatf(x) = g ( y ) is a submanifold of X x Y, which is called the fiber product of X and Y over Z and is written X x = Y .The tangent space at the point (x, y ) E X x Y to the fiber product is the subspace of T,(X) x T,(Y) consisting of the pairs (h, k) such that TX(n .h = T y ( g ).k.
In this situation, f and g are said to be transversal mappings into Z. Show that if f i s a submersion (resp. an immersion, resp. a subimmersion), then so is the restriction X x z Y + Y of pr2. 11. Let Y be a differential (resp. real-analytic, resp. complex-analytic) manifold, X a Hausdorf€ topological space, and p : X +Y a mapping with the following property: For each x E X there exists an open neighborhood V of x such that p I V is a homeo-
morphism of the subspace V onto a submanifold of Y .
(a) Show that X is locally connected, that each point of x has a closed neighborhood which is homeomorphic to a closed ball in R",and that for each y E Y the fiber p - ' ( y ) is a discrete subspace of X. (b) Let b be a denumerable basis for the topology of Y . A pair (W, U ) is said to be distinguished if U E b and if W is a connected component of p - ' ( U ) such that QI' is compact and metrizable and p I is a homeomorphism of %' onto a subspace of Y .
8 SUBMANIFOLDS
51
Show that for each x E X there exists a distinguished pair (W, U) such that x E W. Show also that if (W, U) is a distinguished pair, the set of distinguished pairs ( W , U') such that W n W' # @ is denumerable (use the fact that W is separable). (c) Deduce from (b) that each connected component Xo of X is metrizable and separable. (Consider the following relation between two points x, x' in X: There exists a finite sequence of distinguished pairs (W,, U,) (1 5 i 5 r ) such that x E W, ,x' E W,, and W t n Wt+l # @ for 1 5 i 5 r - 1. Then apply (12.4.7).) (d) Show that there exists on Xo a unique structure of differential (resp. real-analytic, resp. complex-analytic) manifold such that plXo is an immersion of Xo into Y (Poincari- Volterra theorem).
h = (PA, f~),where PAis a nonempty open polydisk in C" (resp. R")andf, is a complex (resp. real) analytic function on PA.For each pair of elements A, p in L, define a set A,, as follows: A,, = 0 if PAn P, = @ or if the restrictions off, andf, to PAn P,, are distinct; A,, = PAn P, if the restrictions of f A and f , to PAn P,, are equal. Let hrrnbe the identity mapping of AA, onto itself. Show that the mappings ,I/ satisfy the patching condition (12.2.4.1), and hence that we obtain a topological space X by patching together the PA along the AA, by means of the ha,,. Let nA: P A+X be the canonical mapping and let XA= TA(PJ be its image, which is an open subset of X. IfjA : PA+ C" (resp. j , : PA+R")is the canonical injection, show that there exists a unique mapping p : X .+C" (resp. p : X -+R")such that p T , =in for all h. The restriction p I X, is a homeomorphism of X, ontop,, and n, is the inverse homeomorphism. Show that X is Hausdoff (use (9.4.1)). Deduce that the results of Problem 11 apply to X and p: If (Y,)is the family of connected components of X, then there exists on each Y. a unique structure of a complex (resp. real) analytic manifold such that p I Y . is a local isomorphism of analytic manifolds. For each index a, there exists a complex (resp. real) analytic function F, on Y . such that, for each index h for which X I c Y , , the restriction of F. to X, is equal t o h (PI X,); for each such index A, Y , is said to be the analytic manifold defined by f, (the Riemann surface of fi in the complex case when n = l), and F, is the natural continuation of f, .
12. Let L be the set of all pairs
0
0
13. (a) Let X, Y be two pure differential manifolds of the same dimension, and let f : X 4 Y be a mapping of class C". Let S C X be the closed set of points at which f is not a local diffeomorphism, and suppose that the set of nonisolated points of S is discrete. Show that, for each point xo E X, the image underfof any neighborhood of xo is a neighborhood off(xo), and hence thatfis an open mapping. (Reduce to the case X = Y = R",and show that it is impossible that on each sphere Ilx - xo j l = p in R" there should exist a point x such that f ( x ) =&), by a compactness argument. Let IJx- xoll= p be a sphere on whichf(x) # f ( x o ) , so that l/f(x)-f(xo)II 2 cc > 0; let D be the open ball Ilx - xoll < p and D' the open ball lly -f(xo)ll < cc; and let G = D n CS and H = D' n Show that f(Fr(G)) does not intersect H; deduce thatf(G) 2 H and hence thatf(D) is a neighborhood o f f ( x o ) . ) (b) Deduce from (a) that Fr(f(U)) cf(Fr(U)) for every relatively compact open set U in X. Deduce that if in addition f is proper (Section 12.7, Problem 2) and X, Y are connected, thenf(X) = Y . (c) If Y = R",show that for each a E R", inf llf(x) - a I1 is not attained a t a point IE x xo E X unless f ( x o ) = a. (d) Suppose that Y = R2and that X is an open neighborhood of the disk D : I z I 5 I in RZ.Show that iffsatisfies the conditions of (a), thenf(S,) cannot be a Bernoulli
Cfo.
52
XVI DIFFERENTIAL MANIFOLDS
+
lemniscate B (with equation (5: = 5: - fg). (Observe that the image of the intezior of D cannot contain any point of the unbounded connected component of the complement of B.) 14. Let X be a connected complex-analytic manifold of dimension n, and let Y, (1 5j 5 r) be a finite number of closed submanifolds of X,of dimensions - 1. Show that the complement in X of the union of the Y, is a connected dense open set (use (12.6.1) and Section 16.3, Problem 3(c)).
sn
15. Let X be a real-analytic manifold and Yo a differential manifold whose underlying set is contained in X. Suppose that, for each point x E YO,there exists a chart c = (U,9,n) of X at x and a neighborhood V c U of x in YO,such that Q I V maps V onto cp(U) n Rm and (V, Q 1 V,m) is a chart of Y o .In these conditions show that there exists a unique real-analytic submanifold Y of X whose underlying differential manifold is Yo. 16. Let Xo be a differential manifold, Y a real-analytic manifold, and suppose that the differential manifold Yo underlying Y is a submanifold of Xo Show that for each y E Y there exists a chart (U, 9,n) of Xo and an open neighborhood V of y in Y contained in U, such that (V,Q I V, m) is a chart of Y for which fl) = #) n R".
.
17. If j :R +Rf is the canonical injection, give examples of submanifolds Y c R2 such that j-'c y ) is not a submanifold of R.
9. LIE GROUPS
(16.9.1) Let G be a set endowed with a group structure and a structure of a differential manifold. These two structures are said to be compatible if the mappings (x, y)t+xy of G x G into G and x w x - ' of G into G are of class C". It comes to the same thing to require that the mapping (x, y)~+xy-' (or (x, y ) ~ x - ' y )should be of class C", by virtue of the relations x - l = ex-' and xy = x(y-')-'. A group endowed with a structure of differential manifold which is compatible with its group structure is called a Lie group (or a real Lie group). It is clear that the topology of a Lie group G is compatible with the group structure, and the topological group so defined is metrizable, separable, locally compact, and locally connected (16.1.3), and the set of its connected components is therefore at most denumerable. Moreover, this metrizable group is complete (12.9.5). An isomorphism of a Lie group G onto a Lie group G is by definition an isomorphism of the group G onto the group G' which is also a diffeomorphism (16.2.6). If G = G , we say automorphism in place of isomorphism. ForeachaeG, theleft andrighttranslationsy(a):xwaxand6(a-') :x w x a
9
LIE GROUPS
53
are diffeomorphisms of G onto itself (16.6) In particular, it follows that a Lie group is a pure differential manifold (16.1.3). For each a E G , the inner automorphism Int(a) : X H U X U - ' is a Lie group automorphism of G. Examples (1 6.9.2) If E is a finite-dimensional real vector space, the canonical structure of differential manifold on E (16.2.2) is compatible with the additive group structure of E. Hence E is endowed with a canonical structure of (commutative) Lie group. (16.9.3) Let A be an R-algebra of finite dimension with unit element. Then A is normable (15. I .8) by virtue of the continuity of polynomial functions on R" and hence (because R" is complete) the multiplicative group A* of invertible elements of A is a nonempty open set in A (15.2.4). The structure of differential manifold induced on A* by the canonical structure on the vector space A (16.2.2) is compatible with the group structure of A*, by the argument of (8.12.11), which applies to any Banach algebra, not merely to Y(E; E). In particular, if E is a real vector space of dimension n, the algebra A = Y(E; E) = End(E) may be identified, together with its canonical structure of differential manifold, with the vector space R"', and therefore the linear group GL(E) = A* is a Lie group of dimension n2. Likewise, if E is a vector space of dimension n over the field of complex numbers C (resp. the division ring of quaternions Ht), then GL(E) is a (real) Lie group of dimension 2nZ (resp. 4nz). When E = R" (resp. E = C", resp. E = €P' (the Zeft vector space)), we write G u n , R) (resp. GL(n, C), resp. GL(n, H)) instead of GL(E).
If G I , G,, . . ., G , are Lie groups, G = GI x G , x * - - x G, is a Lie group when endowed with the product group structure and the product manifold structure; this follows immediately from (16.6.5). The Lie group G so defined is called the product of the Lie groups G i. (16.9.4)
(16.9.5) If G is a Lie group, then the set G endowed with the same structure of differential manifold and the opposite group structure is again a Lie group, by virtue of (16.6.5); it is called the opposite of the Lie group G, and is denoted by Go.
"
t For the elementary algebraic properties of quaternions, see the author's book Linear Algebra and Geometry," Houghton, Boston, Massachusetts, 1969.
54
XVI
DIFFERENTIAL MANIFOLDS
(16.9.6) Let G be a Lie group and H a subgroup of G which is a submanifold of G (16.8.3). Then the group structure and manifold structure of H are compatible. For H x H is a submanifold of G x G , and if
and
j:HxH+GxG
j':H+G
are the canonical injections, the mapping g : ( x , y ) ~ x y - 'of H x H into H is such that j' 0 g = f o j , where f is the mapping ( x , y ) ~ x y - ' of G x G into G; our assertion now follows from (16.8.3.4). The set H, endowed with its structures of group and differential manifold, is called a Lie subgroup of G . It is closed in G by (12.9.6). Every open subgroup of a Lie group G is a Lie subgroup. In particular, the neutral component Go of G is a Lie subgroup, because G is locally connected (12.8.7). Every discrete subgroup of G is a Lie subgroup. In a product G , x . x G, of Lie groups, if Hi is a Lie subgroup of Gi for 1 i S m, then HI x . . * x H, is a Lie subgroup of GI x . * * x G, . For each n 2 1 , the Lie group GL(n, R) is a Lie subgroup of GL(n, C ) , which in turn is a Lie subgroup of GL(n, H) (if we identify C with the subfield of H generated by 1 and i ) . For each pair of integers p 2 1, q >= 1, the product group GL(P?R) x GL(9, R)
may be canonically identified with the Lie subgroup of GL(p + q, R) consisting of matrices of the form
:( :),
where S E GL(p, R) and T E GL(q, R);
likewise when R is replaced by C or H. Remark (16.9.6.1) To verify that a subgroup H of a Lie group G is a Lie subgroup
of G, it is enough to verify that at one point xo E H there exists a chart ( V , $, m) on G such that xo E H, $ ( x 0 ) = 0 and $(V n H) is the intersection of the open set $ ( V ) of R" with a vector subspace of R". For if x is any other point of H, we shall obtain a chart having analogous properties by taking ( ( x x ; ' ) ~ ,$ y(xo x - I ) , m), because (xox - l ) . ((xx;')V n H) = V n H. 0
(16.9.7) Let G , G' be two Lie groups. A mapping u : G G' is said to be a Lie group homomorphism (or simply a homomorphism) if u is a homomorphism of groups and a morphism of differential manifolds. in order that a group homomorphism u : G + G' should be a Lie group homomorphism, it is necessary and sufficient that there should exist an open neighborhood U of e in G such that ulU is of class C m ; for then u is of class C" on aU for all a E G, because u(x) = u(a)u(a-'x) for all x E aU. --f
9
LIE GROUPS
55
A Lie group homomorphisni of a Lie group G into a linear group GL(E) (where E is a finite-dimensional real vector space) is called a linear representation of G on E. (16.9.8) Let G be a Lie group and s an element of G . We have already seen that the translations 6(s) = xl+xs-I
y(s) : X H S X
are diffeomorphisms of G onto itself. Hence for each x E G we have tangent linear mappings Tx(Y(s)) : TAG)
+
Tx(Ws)) : TAG) + TXs-1(G)
Tsx(G),
which'are bijections (16.5.6). The image of a vector h, E T,(G) under T,(y(s)) (resp. under T,(6(s))) is denoted by s . h, (resp. h, . s-') when there is no risk of confusion. I f s, t E G and h, ET,(G), it is clear that s * (h, . t-') = (s * h,) . t ; this is an element of T,,,- {(G),which we denote by s . h, * t-'. I f s, t E G and h, E T,(G), we have
-'
(st) * h, = s . ( t . h,), h,) = e * h, = h, . This follows from (16.5.4).
and in particular s * (s-'
(16.9.9) (i) Let i : G + G be the mapping for all x E G and h, E T,(G) we have
Tx(i)* h, = -x-'
*
on a Lie group G. Then
XWX-'
h, ' x-'.
(ii) The C" mapping m : (x,y ) ~ x oyf G x G into G is a submersion, and with the identijication (1 6.6.2) we have
T(,, y)(m). (h, h,) = x . h, + h, * Y . 2
(iii) If u : G -+ G' is a Lie group homomorphism, then u is a subimmersion of constant rank; Ker(u) is a normal Lie subgroup of G, and for all x E G, we have
-
T,(u)
. h,
= U ( X ) * (T,(u) * (x-'
. h,)),
where hxE T,(G) and e is the identity element of G. (iv) If u : G G' is a surjective Lie group homomorphism, then u is a submersion and we have (16.9.9.1)
dim(G) - dim(G')
= dim(Ker(u)).
In particular, a bijective homomorphism is an isomorphism.
56
XVI
DIFFERENTIAL MANIFOLDS
At a point (xo,yo) E G x G, the partial mappings m( . ,yo) and m ( x o , * ) are just the translations S(y0') and y(x,), so that (ii) follows from (16.6.6). Next, we have m(x, i(x)) = e for all x E G, so that from (ii) and (16.5.4) we deduce that h,
*
X-'
+x
*
(T,(i)
*
h,.)
=0
for all h, E T,(G); this establishes (i). Assertion (iii) follows from the relation u 0 y(x-') = y(u(x-')) u, which by (16.5.4) leads to 0
T,(u). (x-' . h,) = u ( x ) - ~. (T,(u) * hx)
for all h,ET,(G). It follows that T,(u) and T,(u) have the same rank, hence u is a subimmersion. The assertion about Ker(u) then follows from (16.8.8).
To prove (iv) we argue by contradiction. If u is not a submersion, then there exists a point xo E G and a compact neighborhood V ( x o ) of xo in G such that u(V(xo))is nowhere dense in G' (16.8.8). If follows that for all x E G the set u ( ( ~ x , ~ ) V ( x , = ) ) u ( x x ~ ' ) u ( V ( x o )is) nowhere dense in G'. Since there exists a denumerable open covering (A,,) of G which is finer than the covering formed by the sets (xx;')V(x,) (12.6.1), we conclude that G' is a denumerable union of nowhere dense subsets, which is absurd (12.16.1). Examples (16.9.9.2) The mapping X++det(X)is a homomorphism of the Lie group GL(n, R) (resp. GL(n, C)) onto the Lie group R* (resp. C*). The kernel,
which is denoted by SL(n, R) (resp. SL(n, C)), is a real Lie group of dimension n2 - 1 (resp. 2(n2 - I?), called the unimodular group (or special linear group) in n variables. Remarks (16.9.9.3) The image u(G) is not necessarily a Lie subgroup of G'. This is shown by the example where G = Z x Z and G' = R, the homomorphism u being defined by u(m, n) = m + n8, where 8 is a fixed irrational number (12.8.2.1). (16.9.9.4) Let G, G' be two topological groups. A local homomorphism from G to G' is by definition a continuous mapping h of an open neighborhood U of the identity element e of G with values in G , such that h(xy) = h(x)h(y) whenever x , y and xy all lie in U. Since there exists a symmetric open neighborhood V of e such that V z c U (12.8.3), these conditions are satisfied whenever x E V and y E V . A local isomorphism from G to G' is defined to
9 LIE GROUPS
57
be a local homomorphism h which is a homeomorphism of U onto a neighborhood U' of the identity element e' of G'. If we put V' = h(V), then V' is an open neighborhood of e' in G'. Putting x' = h(x) and y' = h(y) with x , y E V , we have x'y' = h(x)h(y)= h(xy), so that x'y' E U' and
'
'
h - (x')h- ' ( y') = h - (X'Y'). This shows that h-' IV' is a local isomorphism from G to G. Two topological groups G, G are said to be locally isomorphic if there exists a local isomorphism from G to G'. This relation is an equivalence relation; for we have just shown that it is symmetric, and it is clearly reflexive and transitive. For example, if H is a discrete normal subgroup of G, then G and G/H are locally isomorphic (12.11.2(iii)). If G and G' are Lie groups and h is a local homomorphism of class C" from G to G , the argument of (16.9.9(iii)) shows that h is a subimmersion at all points of V , and the same calculation gives its tangent linear mapping T,(h) at all x E V . (16.9.10) Let G, G' be twoliegroups, u : G + G a homomorphismof (abstract) groups. In order that u should be a Lie group homomorphism, it is necessary and suficient that the graph rushould be a Lie subgroup of G x G'. The necessity of the condition follows from (16.8.13). Conversely, if the condition is satisfied, let h be the restriction of the projection pr, to the submanifold r,,. Then clearly h is a bijective Lie group homomorphism, hence an isomorphism (16.9.9(iv)). Remarks (16.9.1 1) Everything in this section remains valid, mutatis mutandis, when we replace differential manifolds by real- or complex-analytic manifolds (16.1.4) and C"-mappings by analytic mappings (16.3.5). This leads us to the notions of a real- (resp. complex-) analyticgroup. Such a group has an underlying structure of (real) Lie group (resp. real-analytic group). We shall prove later (in Chapter XXI) that every real Lie group can be endowed with a structure of a real-analytic group, such that the given Lie group structure is the underlying structure. On the other hand, a real Lie group cannot necessarily be endowed with a structure of a complex-analytic group for which the given structure is the underlying one, even when the group is of even dimension. (16.9.12) We shall prove in Chapter XIX (19.10.1) that every closed subgroup of a (real) Lie group is a Lie subgroup. The example of C and its subgroup R shows that the corresponding statement for complex-analytic groups is false.
58
XVI
DIFFERENTIAL MANIFOLDS
PROBLEMS
1. Let X, Y, Z be three differential manifolds and f : X + Y, g : Y Z two mappings of class C' (k 2 I); let x be a point of X, and put y = f ( x ) . Then the jet J:(g of) (Section 16.5, Problem 9) depends only on J:(f) and J:(g); it is called the composition of these two jets and is written J:(g) J:(f). A jet u of order k from X to Y is said to be invertible if there exists a jet v of order k of Y into X such that u u and u 0 v are defined and equal to the jets of the identity mappings 1 and 1y , respectively. A jet is invertible if and only if it is the jet of a local diffeomorphism of X into Y. The set Gk(n)C L!& of invertible jets from R" to R" with source and target at the origin is a group with respect to composition of jets, and is an open subset of the vector space L:," endowed with its canonical topology. Show that the structure of analytic manifold induced on G'(n) by that of L:," is compatible with the group structure, and that Gk(n)acts analytically on the left on L!, ", , and that Gk(m)acts analytically on the right on Li,,,,. If r =< s, the jet J:(f) of a C'-mapping f:X -+ Y depends only on the jet J:(f) of order s, so that we have canonical surjections J:(X, Y), --f J:(X, Y), and P:(X) + P;(X), the latter being an R-algebra homomorphism. In particular, we have a surjective mapping L;, +L;.,,, ,which is linear and whose kernel is the set of jets of order s from R" to R'" with source and target at the respective origins of these spaces and which have contact of order Z r with the zero mapping. In particular, by restricting the canonical mapping L;.,, C,nto G'(n), we obtain a surjective Lie group homomorphism G"(n)+ G'(n). If s = r I and r 2 1, show that the kernel of this homomorphism is an additive group RN,and calculate N ; show also that the group G1(n)is isomorphic to GL(n, R). --f
0
0
-+
2.
+
Let G be a Lie group, e its identity element, and let A be a commutative Lie group, written additively. Suppose that there exists a C"-mapping B : G x G --f A satisfying the relations
B(x, e ) = B(e, x) = 0 B(x, y)
for all X E G ;
+ B(xy, z) = B(x, yz) + B( y , z)
for all x , y, z E G .
Show that on the product manifold G x A the law of composition (x, U)(X u) = (XY, t- u
+ B(x, A)
defines a Lie group structure. The identity element is (e, 0), and the set {(e,u ) : u E A} is a Lie subgroup isomorphic to A. The center of the group is N x A, where N is the set of all elements n belonging to the center of G such that B(n, x ) = B(x, n) for all x E G . The quotient group (G x A)/A is isomorphic to G. When G and A are real vector spaces (G now being written in additive notation) we may take B to be a bilinear function on G x G with values in A. This generalizes to the situation where G is an arbitrary Lie group and B is a C"-mapping such that the mappings x +P B (x, y) and x ++ B( y , x ) are homomorphisms of G into A for each y E G.
3. (a) Let M be a differential manifold, e a point of M, U an open neighborhood of e, and rn : U x U + M a mapping of class C", satisfying the following conditions:
10 ORBIT SPACES AND HOMOGENEOUS SPACES
59
(1) m(e, x ) = m(x, e) = x for all x E U; (2) there exists an open neighborhood V of e contained in U such that m(V x V) c U and such that m(m(x, y ) , z ) = m(x, m ( y , z))
for all x , y , z in V. Show that there exists an open neighborhood W of e contained in V and a diffeomorphism 0 : W + W such that &) = e, e(O(x)) = x and m(x, @)) = m(O(x), X ) = e
for all x E W. (Apply the implicit function theorem (16.6.8) to the equations m(x, y) = e and m( y, x) = e.) (b) Let M be a differential manifold and m : M x M -+ M a law of composition of class C" on M, which is associative and admits an identity element. Let G be the set of elements of M which are invertible with respect to m. Show that G is open in M and is a Lie group for the structure of differential manifold and law of composition induced from M. (Remark that if s E G, then x ~ m ( sx,) is a diffeomorphism of M onto M, and use (a) above.)
10. ORBIT SPACES A N D H O M O G E N E O U S SPACES
(16.10.1) Let G be a Lie group, X a differential manifold. We say that G acts diferentiubly on the left on X if we are given a left action (s, x ) ~ .sx of G on X (12.10) which is a C"-mapping of G x X into X. Usually we shall omit the word "differentiably" when there is no risk of ambiguity. Similarly we define a (differentiable) right action of G on X. For example, if p : G + GL(E) is a linear representation of G on a finitedimensional real vector space (16.9.7), then G acts differentiably on E by (s, X)H p(s) . x. Conversely, if G acts differentiably on E in such a way that, for each s E G , the mapping p(s) : XHS . x is linear, then if (aj), 6 j 6 m is a basis of E, the mapping SH p(s) . aj is a C"-mapping of G into E, and hence the entries in the matrix of p(s) relative to the basis ( a j )are C"-functions on G . This shows that sr-,p(s) is a Lie group homomorphism of G into GL(E). Suppose that G acts on the left on X, and put m(s, x) = s . x.We denote the tangent linear mappings
T,(m(s,
*
1) :T,(X) -,T,.,(X)
and
T,(m( . , x ) ) : T,(G) + TS.,(X),
respectively, by k,Hs
*
k,,
h,++ h, . x.
Since m(s, . ) : X H S . x is a difeomorphisnz of X onto itself for each s E G (12.10.2) it follows that k,++s. k, is a linear bijection, and by (16.5.4) we have the formulas (16.10.1.1)
(st) * k,
= s * ( t * k,),
t * (h, . X) = ( t . h,) . x,
h, . ( t . X) = (h, . t ) . x
60
XVI
DIFFERENTIAL MANIFOLDS
-
for all s, t in G and x and X: the notation t h, and h, * t was introduced in (16.9.8). Further, it follows from (16.6.6) that the tangent linear mapping to m is given by (16.10.1.2)
T(,,,)(m)
@,, k,)
=s
k,
+ h,
*
x
which proves that m is a submersion. (16.10.2) For each x E X, the mapping SHS . x of G into X is a subimmersion of constant rank. The stabilizer S , of x is a Lie subgroup of G .
The first assertion follows from the facts that u : h , H ( t s - l ) * h, is a * k ,., a bijection of bijection of T,(G) onto T,(G) (16.9.8), v : k,.,-(ts-') T,.,(X) onto Tt.,(X), and the diagram
(in which f:h,H h, * x and g : h , H h, . x are the tangent linear mappings defined above) is commutative. The second assertion follows from the first and from (16.8.8). (16.10.3) In order that there should exist on the orbit space X/G (12.10) a structure of diyerential manifold for which the underlying topological space is the topological space X/G and for which the canonical mapping n: : X -+ X/G is a submersion, it is necessary and suficient that the set R of pairs (x, y ) belonging to the same orbit (that is to say, the graph (1.3) of the equivalence relation "there exists s E G such that y = s * x") should be a closed submanifold of the product manifold X x X . The structure of diferential manifold on X/G satisfying these requirements is then unique.
If n: is a submersion, then so also is n x n: : X x X -+ (X/G) x (X/G), and we have R = (z x n:)-'(A), where A is the diagonal of (X/G) x (X/G), which is a closed submanifold of this product (16.8.13). Hence R i s a closed submanifold of X x X by (16.8.12). Conversely, suppose that R is a closed submanifold of X x X. Then (12.10.7) the space X/G i s Hausdorff. We shall prove: (16.10.3.1) For each x E X , there exists a chart ( U , cp, n) on X at the point x, such that q(x)= 0 and q(U) = V x W, where V , W are open subsets of R", R"-", respectively, and such that the relation (z, z') E R n (U x U) is equivalent to pr,(cp(z)) = pr2(q(z')).
10 ORBIT SPACES A N D HOMOGENEOUS SPACES
61
This result is a consequence of the following: (16.10.3.2) For each x E X, there exists an open neighborhood U of x in X, a submanifold S of U containing x anda submersion s : U + S such that for each z E U, s(z) is the onlypoint of S for which (z, s(z)) E R.
Let us assume (16.10.3.2) for the moment. Replacing U by a smaller open neighborhood of x if necessary, we can assume that there exists a chart (U, cp, n) on X at x for which the submersion s is of the form indicated in (16.7.4), and it is then clear that this chart (again restricted, if necessary, to the inverse image of a product of open sets in R" and R"-") satisfies the conditions of (16.10.3.1). For the proof of (16.10.3.2), let n = dim,(X); observe that since the submanifold R contains the diagonal of X x X, the dimension of R at the point ( x , x) is of the form m + n where 0 5 m 5 n, and T(,,,)(R) contains the diagonal A of T,,,,,(X x X) = T,(X) x T,(X). Hence T(,,,)(R) is the direct sum of A and a subspace (0) x E = T(,,,)(R) n ((0) x T,(X)), of dimension m. By (16.8.3) there exists an open neighborhood U, of x in X and a submersion f : U, x U, -+ R"-", such that R n (U, x U,) is the set of pairs (z, 2') E U, x U, satisfying f ( z , z') = 0. Furthermore ((16.8.7) and (16.8.8)), the intersection R n ((x} x U,) is a submanifold of { x } x X, of dimension m at the point (x, x ) , and consisting of the points ( x , z ) E { x } x U, such that f ( x , z ) = O ; the tangent space to this submanifold at the point ( x , x ) is (0) x E, the mapping Z H ~ ( X , Z ) of U, into R"-" being a submersion at the point x , because A is contained in the kernel of T,,,,,(f). Observe that the mapping Z H ~ ( X . , z) is also a submersion of U, into R"-" at the point x ; for since the relation R is symmetric, the kernel of T ( , , , ) ( f ) , which is the tangent space to R at ( x , x ) , is invariant under the mapping (u, V)H(V, u) of T,(X) x T,(X) onto itself, and hence its intersection with T,(X) x (0) is of dimension m , which proves the assertion. Replacing Uo if necessary by a smaller neighborhood of x , we can assume (16.8.8) that there exists a submersion g : U, -+ R" such that, if N is the submanifold of U, given by the equation g(z) = 0, then x E N and the tangent space at ( x , x) to {x} x N is a supplement (0) x F of (0) x E in (0) x T,(X). This being so, consider the mapping u : (z, z ' ) ~ ( f ( z , z'), g(z')) of U, x U, into R" = R"-" x R". The choices off and g show that the partial mapping z' t,( f ( x ,z'), g(z')) of U, into R" has a bijectioe tangent linear mapping at the point x . For if a vector h E (0) x T,(X) runs through (0) x E, then its image under T,(f(x, .)) is zero and its image under T,(g) runs through R"; and if, on the other hand, h runs through (0) x F, its image under T,(g) is zero and its image under T,(f(x, runs through R"-". By the implicit function theorem (16.6.8) there exist two open neighborhoods U 1 , U2 of x in X and a C"-mapping *>>
62
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DIFFERENTIAL MANIFOLDS
u : U, -+ U, such that for each z E U 1 ,the only solution z' of equations
f(z,z') = 0,
E U2 of
the system
g(z') = 0
is z' = u(z). Moreover, since z ~ f ( zx), is a submersion of U, into R"-" at the point x, it follows that T,(f( . , x)) is of rank n - m,hence (16.6.8.1) T,(u) is of rank n - m, or in other words u is a submersion, at the point x, of U, into the submanifold N of U, . Replacing U, by a smaller neighborhood we can therefore assume that u is a submersion of U, into N , so that zi(U,) c U, n N
is open in N. If S = u(Ul) n U1 and U = v - ' ( S ) n U,, then U, S and s = u I U satisfy the conditions of (16.10.3.2). Now that (16.10.3.1) is established, we shall return to the proof of the sufficiency of the condition in (16.10.3). For each x E X, let (Ux,cp, n,) be a chart of X at x satisfying the condition of (16.10.3.1), where V, W, m are replaced by V, , W, , m, . Observe first that if K, is the image under cp'; of a compact neighborhood of 0 in (0) x W,, then n(KJ is a neighborhood of n(x) in X/G and the restriction of n to K, is injective. Hence (12.10.9) X/G is metrizable, locally compact, and separable. Moreover, the open sets in X/G contained in n(U,) are the sets of the form n(q;'(V, x T)), where T runs through the open sets in W, (12.10.5); in other words, there is a homeomorphism w, : n(U,) + W, such that w;'(w) = n(q;'(O, w)). We have to show next that the charts (n(lJ,), ox, n, - m,) are mutually compatible. So let (U,, q,, n,) and (U,,, q,.. n,,) be two charts of X of the family considered above; put S = x-'(n(U,) n n(U,.)) n U,,
S' = n-'(n(u,) n ~(IJ,,)) n U,.
which are open sets, and put
Q = cp,(S)
c V, x W,,
Q'
= cp,.(S') c
V,, x W,,
.
The projections
P = pr,(Q)
c
W,,
P'
= pr,(Q') c
W,,
are then open sets such that Q = V, x P and Q' = V,, x P'. Since n(S) = n(S') = n(V,) n n(V,.) by definition, for each p E P, there exists a unique point p' =f(p) E P such that
n((P;'(v, x {PI)) = .(cp,.'(VX, x {P", and what has to be shown is that the bijectionf: P -+ P' so defined is of class C" in a neighborhood of each point p E P. Now, let q E Q (resp. q' E Q')
10
ORBIT SPACES AND HOMOGENEOUS SPACES
63
be such that pr,(q) = p (resp. pr,(q') = p ' ) ; if z = q;'(q) and z' = q;'(q'), there exists an element s E G such that z' = s z. Consider the diffeomorphism g : U H S * u of X onto itself; since it maps z to z', there exists an open neighborhood T c U, of z such that g(T) c U,, , and the composite mapping U H rp,,(g(u)) defined on T furnishes a chart on X at the point z, with domain T, which is therefore compatible with that defined by rp,. The mapping r++q,,(g(q;l(r))), defined on the neighborhood q,(T) of q, is therefore of class C". However, by its definition it is of the form r H ( e ( r ) ,f(prz(r))), which shows thatfis of class C" in a neighborhood of pr,(q) = p . Finally, the uniqueness of the structure of differential manifold on X/G in (16.10.3) is a consequence of (16.7.7(ii)) (take f = n and g = lX,Gthere). Q.E.D. When the condition of (16.10.3) is satisfied, the space X/G endowed with the structure of differential manifold defined in (16.10.3) is called the orbit manifold of the action of G on X. If n : X X/G is the canonical submersion, then we have n(s * x ) = n(x) for all s E G and all x E X; taking tangent mappings and using the notation introduced earlier. we obtain --f
(16.10.3.3)
TS.,(n) *
(S
*
h,) = T,(n)
. h,
for all h, E T,(X). (16.10.3.4) It should be noted that the condition in (16.10.3) is not always satisfied, even when G is afinite group. Consider for example the case where X = R and G is the multiplicative subgroup (1, - l} of R*,the action of G on X being multiplication (cf. Problem 1). (16.10.4) Suppose that the orbit manifold X/G exists. Then a mapping @ : X/G --* Y, where Y is a differential manifold, is of class C' (resp. a subimmersion, resp. a submersion) i f and only i f the composite mapping Q, n : X -+ Y has the same property. 0
The assertion relative to class C" is a particular case of (16.7.7(ii)). The other assertions follow from the relation rk,(@ n) = rk+)(@), which comes from the fact that n is a submersion. 0
(16.10.5) Let G (resp. G') be a Lie group acting differentiably on a differential manifold X (resp. XI). Then G x G' acts differentiably on X x X'.
64
XVI
DIFFERENTIAL MANIFOLDS
If the orbit manifolds X/G
and X'/G exist, then so does the orbit manifold (X x X')/(G x G), and the canonical mapping
(X x X')/(G x G')
--f
(X/G) x (X'/G')
is a difeomorphism. The first two assertions follow from (16.6.5), (16.10.3), and (16.8.7(ii)). The third is a consequence of (16.10.4) and (16.5.6). (16.10.6) Let H be a Lie subgroup of a Lie group G, and consider H as acting on G on the right by translation. Then the orbit manifold G/H exists, G acts differentiably on the left on G/H, and we have
dim(G/H) = dim(G) - dim(H).
If H is normal in G, the manifold structure of G/H is compatible with its group structure. To verify that the condition of (16.10.3) is satisfied in the present situation, we observe that the set R c G x G is here the set of pairs (x, y) such that x-'y E H. Now the mapping ( x ,y ) ~ x - ' yof G x G into G is a submersion (16.9.9), and H is a submanifold of G, hence (16.8.12) R is a submanifold of G x G. To show that G acts differentiably on G/H, let p denote the mapping ( X , Y ) H X ~ of G x G into G, fi the mapping ( x , j ) ~ x -ofj G x (G/H) into G/H, and n : G + G/H the canonical mapping. Then we have a comniutative diagram GxG P ' G lc,xn
I
G x (G/W
7 G/H
and 0 x (G/H) may be identified with the orbit manifold (G x G)/({e}x H) (16.10.5). The fact that fi is of class C" now follows from (16.10.4), since p and n are C"-mappings. When H is normal in G, let m denote the mapping (x, y)t+xy-' of G x G into G, and m the mapping (a, j ) w i j - ' of (G/H) x (G/H) into G/H. Then we have a commutative diagram GxG
rn
- G
10 ORBIT SPACES A N D HOMOGENEOUS SPACES
65
Identifying (G/H) x (G/H) with (G x G)/(H x H), it follows as above that m is of class C". Finally, the dimension formula follows immediately from (16.10.3.1). Examples (16.10.6.1) If H is a discrete normal subgroup of G, the canonical mapping n : G -+ G/H is a local direomorphism (16.5.6). As an example, consider the
n-dimensional torus T" = R"/Z", which is a compact connected commutative Lie group, being the canonical image of the cube [0, l]" in R".
(16.10.7) Let G be a Lie group acting diflerentiably on a diflerential manifold X . Ifapoint x E X is such that the orbit G * x is a locally closedsubspace of X , then G x is a submanifold of X , and the canonical mapping f , : G/S, + G x (12.1 1.4) is an isomorphism of differential manifolds. In particular, the above condition is satisfied for all x E X wherever the orbit manifold X/G exists.
-
-
Every point of the subspace G . x of X has by hypothesis a neighborhood homeomorphic to a complete metric space (3.14.5), hence (12.16.12) f, is fx a homeomorphism. Next, since the composite mapping h, : G 3 G/S,+ G * x is a subimmersion (16.10.2), it follows from (16.10.6), (16.10.4), and (16.8.8) that f, is an immersion; hence the first two assertions follow from (16.8.4). Finally, if the manifold X/G exists, then since G . x is the section R(x) of the set R defined in (16.10.3) it follows that G . x is closed in X (3.20.12).
-
The example of the group Z acting differentiably on T by the rule n x = x + rp(ne), where rp : R + T is the canonical homomorphism and f3 is irrational, shows that the hypothesis on G . x in (16.10.7) is not always satisfied. In particular: (i) IfG is a Lie group which acts diflerentiably and transitively on a differential manifold X , thenfor each x E X the canonical mappingf , : G/S, -+ X is a diffeomorphism. (ii) If u ; G --+ G' is a surjective homomorphism of Lie groups with kernel H , then the canonical mapping G/H + G' is an isomorphism of Lie groups. (16.10.8)
The assertion (ii) follows from (i) by considering G as acting transitively on G' by means of the mapping ( x , x')Hu(x)x'.
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XVI DIFFERENTIAL MANIFOLDS
(16.10.9) Let u : G 4G' be a homomorphism of Lie groups, H a Lie subgroup of G, and H' a Lie subgroup of G such that u(H) c H'. Then the unique mapping ii : G/H --+ G / H ' for which the diagram
G
G'
It
(a,n' being the canonical mappings) is commutative, is of class C".
We have n' 0 u = ii 0 a, and it is clear that n' 0 u is of class C"; now apply (1 6.1 0.4) and (1 6.10.6). (16.10.10) Let G (resp. G') be a Lie group acting differentiably on a differential manifold X (resp. X'). If p : G + G is a Lie group homeomorphism and f : X -+ X a C"-mapping, then G and G' are said to act equivariantly (relative to p andf) on X and X if the diagram
GxX-X
m
is commutative (where m,m' define the actions of G on X and G' on X' respectively). This leads, for each pair (s, x ) E G x X, to a commutative diagram of tangent linear mappings (16.5.4) :
Remarks (16.10.11~ We leave it to the reader to transpose the results of this section
to the context of real- (resp. complex-) analytic groups acting analytically on real- (resp. complex-) manifolds.
10 ORBIT SPACES A N D HOMOGENEOUS SPACES
67
(16.10.12) Let E be a set (not a priori equipped with a topology), G a Lie group acting transitively on E, and suppose that the stabilizer of each element of E is a Lie subgroup of G (in fact it is sufficient that this should be the case for one point of E). Then there exists on E a unique structure of differential manifold such that G acts differentiably on E: this follows from (16.10.6) and (16.10.8).
PROBLEMS
1. If a Lie group G acts differentiably and properly (Section 12.10, Problem 1) on a differential manifold X,the orbit-manifold X/G does not necessarily exist (cf. (1 6.10.3.4)). Show however that the orbit-manifold X/G does exist if, in addition to the hypotheses above, G acts freely on X. 2. Let G be a Lie group, H a Lie subgroup of G, and X a differential manifold on which H acts differentiably (on the left).
(a) The Lie group H acts differentiably on the right on G x X by the rule (s, x ) . t = (sr, t
-l
. x).
Show that with respect to this action the orbit manifold Y = H\(G x X) always exists. If H = G, the orbit manifold is diffeomorphic to X.If n : G x X -+Y is the canonical mapping, show that G acts differentiably on the left on Y by the rule s‘ . n(s, x ) = Tr(s‘s, x).
(b) Show that the mapping h : x Hn(e, x ) is a diffeomorphism of X onto a submanifold X‘of Y, that X’is stable under the action of H c G, and that H acts equivariantly (relative to h) on X and X . If s E G is such that s . X’n X # 0, then s E H. The stabilizer of n(e, x ) under the action of G is equal to the stabilizer of x under the action of H. (c) Show that the mapping (s, y ) Hs . y is a surjective submersion of G x X’onto Y (cf. 16.14.8). 3. Let G be a Lie group, H a Lie subgroup of G, and n : G +G/H the canonical submersion.
(a) For each Lie subgroup G’ of G, show that G’ n H is a Lie subgroup and that WIG’is a subimmersion of G’ into G/H. If n(G’) is closed in G/H (which will be the case if either H or G’ is compact), then n(G’) is a submanifold of G/H diffeomorphic to G‘/(G’n H).(Use (16.10.7).) (b) Consider the Lie group G = R x T2and the Lie subgroup H = R x {0} of G. If : R +T is the canonical homomorphism, let G’ be the subgroup of G which is the , where 0 is a fixed irrational image of R under the homomorphism x t-+ ( x , ~ ( x )V(Ox)), number. Show that G’ is a Lie subgroup of G but that n(G’) is dense and not closed in G/H = T2. (c) If dim(G’) - dim(G’ n H) = dim(G) - dim(H), the restriction of n to G’ is a submersion into G/H, and hence factorizes into G’ + G’/(G’ n H)+ G/H, where u is
68
XVI
DIFFERENTIAL MANIFOLDS
a diffeomorphism of G'/(G' n H) onto an open submanifold of G/H. If either G' or H is compact and G is connected, then u is a diffeomorphism of G'/(G' n H) onto G/H. Give an example in which the image of u is a nondense open set in G/H. (Take G = SL(2, R)andG'to be the subgroupof upper triangular matrices
(0"
x?l), where x > 0,
and H the subgroup of lower unitriangular matrices
11. EXAMPLES: U N I T A R Y GROUPS, STIEFEL MANIFOLDS, GRASSMANNIANS, PROJECTIVE SPACES
(16.11 .I) Let E be a real vector space of dimension n, and let X(E) be the set of all symmetric bilinear forms on E x E, which is a real vector space of dimension in(n + 1). For each pair (p, q) of integers 50 such that p + q = n, the subset 3 f P J E ) of symmetric bilinear formsof signature (p, q) on E is open in the vector space X(E). To see this, let m0be a form belonging to X,JE); then there exists a direct sum decomposition E = P 0 N, where P and N are vector subspaces of dimensions p and q, respectively, such that Q0(x, x) > 0 for x # 0 in P, and O0(x, x) < 0 for x # 0 in N. If llxll is a norm which defines the topology on E, then there exist two real numbers a > 0 and b > 0 such that Qo(x, x) 2 ~11x11~ for all X E P and @,(x, x) 6 -bllxl12 for all x E N, because spheres are compact (3.17.10). If Y is a sufficiently small symmetrical bilinear form such that 1 Y(x, x) I I inf(a, b)IlxllZfor all x E E, and if Q, = Q 0 Y , then we shall have @(x,x) L +a11x112 for X E P and @(x,x) 5 - 3 b l l ~ 1 )for ~ x E N ; this shows that Q, has signature (p, q), by virtue of the law of inertia.
+
+
The group GL(E) acts differentiably (indeed analytically) on X ( E ) and on each of the X P J E ) . Namely, if Q, is any symmetric bilinear form and s E GL(E), then s . Q is the form (16.11.2)
(x, y)H@(s-l . x, s - '
. y).
Moreover, each of the open sets X p J E ) is an orbit of this action. Hence it follows from (16.10.2) that, for each Q E Hp, JE), the subgroup of elements s E GL(E) such that s * Q, = 0 is a Lie subgroup of GL(E) of dimension nz - 3n(n + 1) = $n(n - 1). This Lie group is called the orthogonal group of the form Q, and is denoted by O(Q,).When p = 3 and q = 1 it is called the Lorentz group. When p = n and q = 0, it is called simply the orthogonalgroup in n variables. All orthogonal groups O(Q,) with Q, of signature (n,0) are isomorphic to the group corresponding to E = R" and @ the Euclidean scalar product, namely n
Q,(x,Y)=(xIY)=
C
j= 1
tjqj,
11 EXAMPLES
69
where x = (ti),y = (qj). This group is also denoted by O(n, R), or simply O(n) if there is no risk of ambiguity. It is compact, because the matrices S = (aij) belonging to O(n) are characterized by the relation 'S . S = I and therefore in particular satisfy the relations j= 1
at.= 1 for 1 5 i
5 n ; hence
they form a bounded closed subset of R"'. The kernel in O(n) of the homomorphism s H det(s) is a Lie subgroup of O ( n ) of index 2 (because det(s) = - 1 if s is a reflection in a hyperplane) called the rotation group or special orthogonal group in n variables, and denoted by SO@,R) or S O ( n ) ; it is an open subgroup of O(n). (16.11.3) There are analogous definitions and results when E is taken to a vector space of dimension n over the field of complex numbers C, or a left vector space of dimension n over the division ring of quaternions H. In either case 2 ( E ) now denotes the set of hermitian sesquilinear forms Q, on E x E, that is to say forms Q, satisfying
+
Q,(x x', Y) = @(x, y) + W',y), @(Ax,y) = AQ,(x, y) (A E C (resp. H)), @(Y, x) = Q,k Y).
It follows that @(x,x) is always real, and therefore X ( E ) is a real vector space of dimension n + n ( n - 1) = n2 in the complex case, and of dimension n + 2n(n - 1) = 2n2 - 17 in the quaternionic case. Just as in (16.11.1), we can show that for each signature ( p , q ) such that p q = n, the subspace Z,,,,(E) of forms of signature (p, q ) is an open subspace of X ( E ) . If Q, E X,,,&E), we see as in (16.11.2) that the subgroup of elements s E GL(E) such that s . Q, = d) is a Lie subgroup of dimension 2n2 - n2 = n2 in the complex case, and of dimension4n2 - (2nZ- n) = n(2n 1) in the quaternionic case. This subgroup is called the unitary group of the form @, and is denoted by U(Q,). When (p, q ) = (n, 0), it is called simply the unitary group in n variables. All unitary groups U(@) with @ of signature (n, 0) are isomorphic to the group corresponding to E = C" (resp. E = H") and
+
+
n
WX,Y)=(XIY)=
C
j= 1
(cj),
t j r j ,
where x = y = (qj). This group is also denoted by U(n, C) or U(n)(resp. by U(n, H)). It is conipuct, because the matrices S = ( a i j ) belonging to U(n,C) (resp. U(n, H)) are characterized by the relation S * = I and therefore in particular satisfy the relations
n
's
1 ( a i j [ ' = I for I 5 i 6 n ; consequently they
j= I
form a bounded closed subset of Cn2(resp. H"').
70
XVI
DIFFERENTIAL MANIFOLDS
The homomorphism sHdet(s) of U(n,c) onto U(1,C) = U (the unit circle in C) is surjective, because if 5 E U and if (ej)lsjs,,is an orthogonal basis of C",the automorphism s of C" defined by s(e,) = Gel, s(ej) = ej for 2 5 n is unitary and has determinant 5. Hence (16.9.9) the kernel of this homomorphism is a normal Lie subgroup of U(n) of dimension n2 - 1, called the special unitary group and denoted by SU(n).
sj
(16.11.4) Let n , p be two integers 2 1. The space RnPofsequences (Xk),Sk,, of @ vectors in R" can be identified with the set of real matrices X with n rows
and p columns, the kth column being the vector x k . The group GL(n, R) acts differentiably (indeed analytically) on the left on RnPas follows: the automorphism s E GL(n, R) transforms the sequence ( x k ) into the sequence (s * x k ) . Equivalently, if we identify s with its matrixS relative to the canonical basis of R", the action of GL(n, R) on RnPis left multiplication (S, X) -,S * X of matrices. Now let p 6 n and let S,,, (or Sn,,(R)) be the subset of RnPconsisting of sequences ( x k ) l s k $ , which are orthonormal relative to the Euclidean scalar product (6.5). This set may also be described as follows: the orthogonal group O(n, R) acts differentiably on RnPby restriction of the action of GL (n, R) defined above, and S,,, is the orbit, under this action of O(n, R), of the orthonormal sequence (ek)lsksp consisting of the first p vectors of the canonical basis ( e k ) l s k < , , of R". Since, by virtue of (12.10.5), this orbit is compact and hence closed in RnP,it follows from (16.10.7) that S,,, is a compact submanifold of RnP,called the (real) Stiefel manifold of orthonormal systems of p vectors (sometimes called p-frames) in R". It is clear that the subgroup of O(n, R) which stabilizes the p-frame (ek)l5 k s p may be canonically identified with the orthogonal group O(n - p, R), by identifying R"-, with the subspace of R" spanned by e p + l ., .., e,, (when n = p , this group consists only of the identity element). Hence S,,, is isomorphic to the homogeneous space
,
,
O(n,R>/O(n- p, R). When p = n the Stiefel manifold S,,,,(R) may be identified with O(n, R), and when p = 1 the manifold Sn,,(R) may be identified with the sphere S,,-, (16.2.3). When 1 = < p5 n - 1, S,,,(R) is also the orbit of the p-frame (ek),, k s p under the action of the rotation group SO@,R). Since the p be identified with SO(n - p , R), it follows that stabilizer of ( e k ) l g k smay S,,,,(R) may be identified with the homogeneous space SO@,R)/SO(n- p , R) forlgppn-1. (1 6.11.5)
In the considerations of (16.11.4) the field R can be replaced everywhere by C or H, the Euclidean scalar product being replaced by the Hermitian
(16.11.6)
11 EXAMPLES
71
scalar product. In this way we define complex and quaternionic Stiefel manifolds S,,,,(C) and S,,,J H ) . They are isomorphic, respectively, to the homogeneous spaces U(n, C)/U(n- p , C ) and U(n, H)/U(n- p , H); and if lspsn-I, S,,,,(C) is also isomorphic to SU(n, C)/SU(n- p , C ) . They are therefore compact manifolds. When p = 1, S,,,, ( C ) may be identified with the sphere S2,,-,,and S,,,,(H) with S4,,-,. (16.1 1.7) The groups SO@, R), SU(n, C ) , U(n, C ) , and U(n, H) are connected i f n 2 1 ;so also are the Stiefel manifolds S,,,,(C) and S,,,JH) if 1 p 5 n - 1, andSn,,(R)ij"1~pjn-1andn22.
When n = I , the groups SO(1, R) and SU(1, C ) consist of the identity element alone. The group U(1, C ) may be identified with the unit circle U = S, in C , and U(1, H) with the multiplicative group of quaternions of norm 1, which as a topological space is the sphere S,; hence these two groups are connected (16.2.3). It follows also from (16.2.3) that the Stiefel manifolds S,,,,(R) = S n - , are connected if n 2 2, and that S,,,,(C)= S2,,-]and S,,, ,(H) = S4,,-, are connected if n 2 1 . Consequently the homogeneous spaces SO(n)/SO(n- l),
W ,C)/U(n- 1 , C ) ,
SU(n)/SU(n- I), U(n, H)/U(n- H )
are connected if n 2 2. The first assertion of (16.11.7) now follows by induction on n , by virtue of (12.10.12). The second is an immediate consequence, by (3.19.7). (16.11.8) If E is a vector space over a field K, we denote by G,(E) the set of vector subspaces of dimension p in E. The set G,(R") is denoted by G,,,,(R) or simply G,,, ,. It is clear that the orthogonal group O(n, R) acts transitively on GnJR). Furthermore, if F is the subspace of R" generated by the first p vectors in the canonical basis, then the stabilizer of F under this action leaves fixed (as a whole) the orthogonal supplement of F, namely the subspace spanned by the last n - p vectors of the canonical basis. Hence the stabilizer of F is a Lie subgroup of O(n,R) which may be identified with the product O(p,R) x O(n - p , R) ((16.9.8) and (16.8.8(i))). It follows (16.10.12) that there exists on the set G,,, a unique structure of differential manifold for which O(n, R) acts differentiably on G,,, ,. The set G,,,,endowed with this structure is called the (real) Grassmannian with indices n, p . When p = 1, the Grassmannian G,,,, is also denoted by P,-,(R) or simply P,,-] , and is called (real) projective space of dimension n - 1. The differential manifold G,,,,(R) is
,
72
XVI
DIFFERENTIAL MANIFOLDS
diffeomorphic to the homogeneous space O(n,R)/(O(p, R) x O(n - p, R)). If n - 1, it is also diffeomorphic to SO@,R)/H,, where H, is the subgroup of O(p,R) x O(n - p , R) consisting of pairs (t, t ' ) such that det(t) = det(t'). We remark that G,,, may also be considered as the space of spheres with center 0 and dimension p - 1 contained in the sphere S,,- : these spheres correspond one-to-one with the vector subspaces of dimension p in R".
p
,
(16.11.9) The orthogonal group O(p,R) acts differentiably on the right on the Stiefel manifold S,,, ,(R) by matrix multiplication ( X , T)H X T. For each matrix X E S,,, can be written as S E, where S E O(n, R) and E is the matrix whose columns are the vectors el,.. ., e,; thus X consists of the first p columns of S. The columns of E * Tare the images of el,. . . , ep under the element of the orthogonal group O(p,R) whose matrix relative to (ek)l sksp is T; hence these columns form ap-frame, that is to say E * T ES,,, and therefore also X . T = S . E . T ES,,,,. This shows also that the set of orbits for the above action may be identified with G , , , p .If we endow G,,, with the structure of differential manifold defined in (1 6.11.8) and identify Sn, with O(n, R)/O(n - p , R) and G,,,,with O(n, R)/(O(p, R) x O(n - p , R)), then the canonical mapping x : S,,, --t G,,, is a submersion (16.10.4); consequently, for the action of O(p,R) on S,,, defined above, the orbit-manifold exists and can be identified with the Grassmannian G , , , (16.10.3). Moreover, the orbits are each diffeomorphic to O(p,R). It follows from (16.11.6) and (16.11.7) that G,,,,(R) is compact and connected if n 2 1 and 1 5 p 5 n. There are analogous definitions and results for the complex and quaternionic Grassmannians G,,,,(C) and G,,,,(H), and in particular for the complex and quaternionic projective spaces P,,-l(C) and P,,-l(H). The dimensions of the differential manifolds G,,,,(R), G,,,,(C),and G,,,,(H) are therefore, respectively,
-
,
,
,
,
,
jn(n
- 1)
-jp(p
,
- 1) - t ( n - p)(n - p - 1) = p(n - p ) , nz - p2 - ( n - p)Z
n(2n
= 2p(n
-p),
+ 1) - p ( 2 p f 1) - ( n - p)(2n - 2p + 1) = 4p(n - p ) .
(16.11.10) There is another, equivalent, definition of the structure of differential manifold on the Grassmannian G,,,,(R).For the group GL(n, R) also acts transitively on G,,,,(R), and the stabilizer of the subspace F considered in (16.11.8) is the subgroup H of GL(n, R) consisting of matrices of the form
(0"
:),
where A is a square matrix of p rows and p columns. Since H is
clearly a Lie subgroup of GL(n, R) (it is diffeomorphic to the product GL(p, R) x GL(n - p , R) x RP("-,)) we obtain (16.10.12) a structure of dif-
11 EXAMPLES
73
ferential manifold on G,,,p . Since the action of O(n, R) on G,,, is obtained by restriction of the action of GL(n, R), the structure so obtained is the same as that defined in (16.1 1.8). Now let L,,, denote the subset of RnPconsisting of the matrices X of rank p , that is to say matrices X whose p columns xk (1 g k g p ) are linearly independent. This set Ln,p is open in RnP. More precisely, for each subset J consisting of p elements il < iz < . < ip of the set I = { 1,2, . . . , n}, let Tj be the set of matrices X such that the matrix X, formed by the i,th, . .., ipth rows of Xis invertible; then it is clear that Tj is open in RnPand is canonically
-
diffeomorphic to GL(p, R) x Rp(n-p),and L n , pis the union of the
(3
sets
Tj . Note that GL(p, R) acts differentiably on the right on L,,, p , by matrix multiplication ( X , T )H X T. We assert that G,,, can be identified with the orbit-manifold of this action. Firstly, the orbit-manifold GL(p, R)\L,,, exists : for if R is the set of pairs ( X , Y ) of elements of L,,,p belonging to the same orbit, then the intersection R n (Tj x Ln,pJis the graph of the C"-mapping ( X , T ) H ( X X ~ - ~ T of ) ~Tj - ~x GL(p, R) into Rp(n-p),and the existence of the orbit-manifold now follows from (1 6.8.1 3) and (16.10.3). It is clear that there is a canonical bijection w of GL(p, R)\L,,, onto G,,, such that o(n(S . X)) = S n ' ( X ) for S E GL(n, R) and X E L n , P ,where n and n' are the canonical mappings of Ln,p onto GL(p, R)\L,,,, and G , , p , respectively. It follows (16.10.12) that o is a diffeomorphism, and our assertion is proved. From these considerations we can construct a convenient atlas for G,,,p . Let V, be the subset of Tj consisting of the matrices X E Tj such that XJ = Zp (the unit matrix). Then it is immediately seen that the restriction of n to V, is a bijection of V, onto U, = n(Tj). If 'pJis the inverse of this bijection, we have q,(n(X)) = XXJ-', from which we conclude (16.10.4) that 'pJ is of class C"; since nlVJ is also of class C", 'pJ is a dzfleomorphism of UJ onto VJ . AS VJ may be identified with Rp(n-p),we have an atlas of GL(p, R)\L,,, consisting of the charts (U, , 'p,, p(n - p ) ) . There are of course analogous results for complex and quaternionic Grassmannians. The real and quaternionic Grassmannians are real-analytic manifolds, and the complex Grassmannians are complex-analytic manifolds. (16.11.11) By virtue of (16.11.9), the projective space P,,(R) (resp. P,,(C), resp. P,,(H)) may be identified with the orbit manifold of the group consisting of the identity and the symmetry XH - x acting on S,, (resp. the group of rotations X H X ~ with 151 = 1 acting on SZn+lc C"", resp. the group of rotations x H x q , where q is a quaternion of norm 1, acting on S4n+3c H""). (16.11.12) The diflerential manifolds P,(R), P1(C), and Pl(H) are dzgeomorphic to S1, S2, and S4, respectively.
74
XVI
DIFFERENTIAL MANIFOLDS
We shall prove the assertion for Pl(H). For each pair of quaternions x = xo
+
+ ix, + j x , + kx, ,
y =yo
+ iyl + j y 2 + ky3
1 y I = 1 (where I x I is the Euclidean norm of x in R4),let such that 1 x I z =f(x, y ) = (zo,z l , z 2 , z 3 , z,) be the point of R5 (identified with H x R) defined by 2xJ = zo
+ iz, + j z , + k z , ,
24
= 1x1’
- IyI’.
Since Z%
+ Z : + Z $ + Z:
= 41xJI2= 4(xIZly1’,
it is clear that z E S,. Moreover, z, = 2 1x1 - 1 can take all values in the interval [ - 1, 11, and for a given value of z4 # & 1, the quaternion 2xJ can take all values on the sphere of radius 1 - 1 z4 I ’. Hence f is a surjective C”mapping of S7 onto S , . Next, the relation f ( x , y ) =f(x’, y’) implies firstly that lx’l = 1x1, so that we may write x’ = xq, where q is a quaternion of norm 1; also it implies that x’y’ = x J , which gives y’ = y4-I = yq. Hence the mapping f factorizes as’follows :
+
s7
+
Pl(W
9 -+
s4
9
where g is bijective. It remains to show thatfis a submersion. Since the rotations x w q ’ x , y - q ” y (where q’,q” are quaternions of norm 1) are diffeomorphisms of S, which fix z , and transform X J into q’xJij”, it is enough to check thatfis a submersion at the points (x,y ) E S7 such that the quaternions x, y are scalars xo ,y o . It is then immediately verified (since xo , y o are not both simultaneously zero) that the Jacobian matrix off(extended to H2 = RE by the same definition) is of rank 5. This fact, together with the relation f ( t x , ty) = t2f(x, y) for all scalars t, proves that g is a diffeomorphism of P,(H) onto S4 (16.8.8). The proofs for P,(R) and Pl(C) are analogous but simpler: In the case of P1(C), we map the point (x, y ) E S, c R4 = Cz to the point f ( X ? Y > = (2XK 1 X l 2 -
blZ)
on S, c C x R = R3,and the mappingffactorizes as
where g is a diffeomorphism. We may therefore tfansport to S, by means of g the structure of complex-analytic manifold of P1(C).The sphere S, , endowed with this structure, is called the Riemann sphere.
11
EXAMPLES
75
PROBLEMS
(a) For any two points x , y
E S2,-,
c C", put
m(x, y) = arc cos(W(x Iy))
which is a real number between 0 and n-. If s, t are any two elements of the unitary group U(4, Put d(s, t ) =
sup
a ( s . x , t . x).
XES2n-1
Show that d is a bi-invariant distance on U(n). (b) For s E U(n), let eiei (I 5 j 5 m) be the distinct eigenvalues of s, so that C" is the Hilbert sum of the eigenspaces V, of s (1 5 j 5 m), the restriction of s to Vj being the homothety with ratio eieJ;we may assume that - n < 0 , s n for each j. Show that if 0(s) = sup I0,I, then d(e, s) = @s). (Minorize W ( xI s . x ) by using the decomposition 1SjSm
of x as a sum of vectors x j E V, .) (c) Let s, t be two elements of U(n) such that s and the commutator (s, t) = s t s r ' t - ' commute, or equivalently such that s and u = tst-' commute. Show that if B ( t ) < &r, then s and t commute. (With the notation of (b), observe that if Vj is the orthogonal supplement of Vj ,and if W, = t(V,), then Wj is the direct sum of W, n V, and W, n Vl , and deduce from the hypothesis on t that W, n Vj = {O}.) When n = 2, the two charts q ~ , ,p), on S, defined in (16.2.3) are such that and p2 define on S , the structure of complex analytic manifold defined in (1 6.11.12). Let U be an open neighborhood of 0 in R". In the real-analytic manifold U x Pm-,(R), let U' be the subset consisting of points (x, z) such that for some system (z', . . ., z") of homogeneous coordinates for z we have xlzk - x*z' = 0 for all pairs of indices j , k (in which case these relations are satisfied for all systems of homogeneous coordinates for z). Show that U' is a closed analytic submanifold of dimension n in U x P,,-,(R) (consider the atlas of P.-,(R) defined in (16.11.10)). The restriction nu of the projection pr, to U' is a surjection of U' onto U and is proper; n, '(0) is a submanifold of U' isomorphic to Pn-,(R), and the restriction of n-u to U' - nU'(0) is an isomorphism of this open set onto U - {O}. Let r be the inverse isomorphism of U - {0}onto U' - ncl(0), and let f be any C'-function defined on an open neighborhood I of 0 in R and with values in U, such thatf(0) = 0 and T,(f) # 0. Then the function t Hr ( f ( t ) ) ,defined on 1 - {O} and with values in U', extends by continuity to a mappingf' : 1 + U', such that f'(0) is the canonical image in P,.. ,(R) of the vector T,(f) . I . Furthermore, iff is of class C', thenf' is of class C-'; and iff, g are two C'-functions defined on I, such that f(0) = g(0) = 0, which have contact of order >k 2 1 at the point 0, thenf' and g' have contact of order z k - 1 at the point 0 (Section 16.5, Problem 9). If V is another open neighborhood of 0 in R" and if u : U + V is an isomorphism of analytic manifolds (resp. a diffeomorphism), then if V' and nv are defined as above, there exists a unique isomorphism u' : U' + V' of analytic manifolds (resp. a unique diffeomorphism) such that nv u' = u 0 nu. Deduce that if X is a pure differential (resp. analytic) manifold of dimension n, and x a point of X, there exists a differential (resp. analytic) manifold X' of dimension n 0
76
XVI DIFFERENTIAL MANIFOLDS
and C m- (resp. analytic) surjection 7rx : X' + X with the following properties: (a) the restriction of 7rx to X - T?,'(x) is an isomorphism of X' - T?'(x) onto X - {x}; (b) there exists a chart (W, v, n) of X at the point x such that d x ) = 0 and @Y) =U is an open neighborhood of 0 in R", and a diffeomorphism (resp. an isomorphism of analytic manifolds) u of 7rx'(W) onto U' (with the notation introduced above) such that v(vX(x')) = T&(x')) for all x' E n<'(W). Moreover, these properties determine X' up to isomorphism, and X' is said to be obtained from X by blowing up the point x. Extend this construction to complex-analytic manifolds. 4.
Let p. q be two integers >0 such that p of signature (p, q) on R".
+ q = n, and let CP be a symmetric bilinear form
(a) Show that the set of vectors x E R" such that @(x, x) = 1 is a connected submanifold of R" except in the case p = 1, when there are two connected components. (b) Let SO(@)denote the subgroup of O(@)consisting o h h e elements with determinant 1 . Show that SO(@)has two connected components (and hence that O(0)has four components). (Argue as in (16.1 1.7), using (a) above.) (c) What are the corresponding results when R" is replaced by C" or H" and @ by a Hermitian sesquilinear form? 5.
6.
Show that the groups GL(n, C) and GL(n, H) are connected when n 2 1, and that the same is true of the groups SL(n, R)and SL(n, C) for n 2 1. On the other hand, GL(n, R) has two connected components. (Use the method of (16.11.7).) Let @ be a nondegenerate alternating bilinear form on RZA. Show that the subgroup
Sp(@) of GL(2n, R)which leaves CP invariant is a connected Lie group, and calculate its dimension. (Use the method of (16.11.7).) Consider the same problem with R replaced by C.
7. Let G be a Lie group and suppose that there exists a Lie subgroup H of G and a submanifold L of G such that the mapping (x ,y ) Hxy of L x H into G is a diffeomorphism of L x H onto G. If K is any Lie subgroup of H,show that the manifold G/K is canonically isomorphic to L x (H/K). (Use (16.10.4) and (16.8.8).) Consider in particular the case where G = G L ( h ,R), H = O ( h ,R); then L may be taken to be the manifold consisting of the positive definite matrices (Section 11.5, Problem 15). Deduce that GL(2n, R)/U(n, C) is diffeomorphic to the manifold R"(*"+"x (0(2n, R)/U(n, C)).
12. FIBRATIONS
All the definitions and all the results of Sections 16.12-16.14, with the exceptions of (1 6.12.11) and (16.12.12), remain valid when we replace differential manifolds and C"-mappings by real-analytic (resp. complex-analytic) manifolds and analytic mappings in the statements and proofs. The notions which correspond in this way to those of differential fibrations and (differential) principal bundles are called real-analytic (resp. complex-analytic or
12 FIBRATIONS
holomorphic) fibrations and real-analytic holomorphic) principal bundles.
77
(resp. complex-analytic or
(16.12.1) A differential jibration (or simply a jibration) is by definition a triple A = (X, B, n) in which X and B are differential manifolds and n is a C“-mapping of X into B which is surjective and satisfies the following condition of local triviality:
(LT) For each b E B there exists an open neighborhood U of b in B, a differential manifold F and a diffeomorphism q0:U x F+n-’(U)
such that n(q(y, t)) = y f o r a l l y E U and t
E
F.
The restriction of n to n-’(U) is therefore pr, q-‘, which shows that n is a submersion. The manifold X is called the space of the fibration A, the manifold B its base, and the mapping n its projection. For each b E B, the inverse image xb = n-’(b) is a closed submanifold of X, called thejiber of A over b. By the local triviality condition, there exists a neighborhood U of b such that X,. is diffeornorphic to xb for all b‘ E U. By abuse of language, instead of saying that (X, B, n) is a “fibration” we shall also say that X is a differentialjiber bundle, or simply aJiber bundle with base B and projection n, and that for each x E X the submanifold X,,,, is thejiber through the point x. If all the fibers are diffeomorphic to the same manifold F, then X is said to be a fiber bundle ofjiber-type F. This will always be the case when B is connected, for it follows immediately from the local triviality condition that the set of points b E B such that Xb is diffeomorphic to a given fiber xb, is both open and closed. The tangent space at a point x E X to the fiber X,,,, will be canonically identified with a subspace of T,(X), and the tangent vectors belonging to T,(X,(,,) are called the vertical tangent vectors at x (or the tangent vectors along thejiber at x). They are the elements of the kernel of T,(n). Iff is any mapping of a set E into B, a mappingf’ : E -+ X is called a lifting offif n(f’(z)) =f(z) for all z E E. Let ;1 = (X, B, n) and A‘ = (X’, B’, n’) be two fibrations. A morphism of A into A’ is by definition a pair (f,g ) wheref:B 4B‘ and g : X -+X’ are C“mappings such that 0
(1 6.12.1 .I)
n’og= fox.
The composition of two morphisms ( A g ) and ( f ’ , 9’) is defined to be (f of’, g 0 g’), which is clearly a morphism.
78
DIFFERENTIAL MANIFOLDS
XVI
An isomorphism of 1onto A' is a morphism ( f ,g ) such that f and g are diffeomorphisms. In that case ( f - ' , g - ' ) is an isomorphism of A' onto 1, called the inverse of (f,9). When B = B' and (lB,g ) is a morphism (resp. an isomorphism), g is said to be a B-morphism of 1into A', or, by abuse of language, of X into X' (resp. a B-isomorphism of rl onto A' or of X onto X ) . If g : X -+ X is a B-morphism, then for each b E B the relation (16.12.1 .l) shows that there exists a (?-mapping gb : xb -+ XL such that gb(x) = g(x) for all x E Xb (1 6.8.3.4). (16.12.2) Let 1= ( X , B, n) and 1' = ( X ' , B', n') be two jibrations and let (f,g ) be a morphism of 1 into 1' such that f is a difleomorphism of B onto B'.
In order that ( f , g) should be an isomorphism it is necessary and suficient that gb : X b + x;(b)should be an isomorphism for each b E B.
The condition is clearly necessary. To prove that it is sufficient, we remark first that it implies that g is bijective, so that it is enough to show that g is a local diffeomorphism (16.5.6). By virtue of the condition (LT), we may therefore assume that B' = B,f = I,, X = B x F , and X' = B x F ' ; hence we may write g(b, z) = (b, u(b, z)) for (b, z) E X . The following lemma will then complete the proof: I f u : B x F -+ F' is a mapping of class C" such that for each b E B the partial mapping u(b, . ) : F -+ F' is a difleomorphism (resp. a submersion), then g : (b, z) M (b, u(b, z)) is a diffeomorphism (resp. a submersion).
(16.12.2.1)
For by (1 6.6.5) and (1 6.6.6) we have T(b,z)(g). (hb', hi) = (hd, T b ( U ( .
9
Z))
*
hd + Tz(u(b, . 1) * h,"),
which shows that the linear mapping T(b,z)(g)is bijective (resp. surjective); hence the result, by (16.5.6). Examples (16.12.3) If B and F are two differential manifolds, the triple (B x F, B, prl) is a fibration called the trivial fibration; its fibers are canonically diffeomorphic to F. A fibration 1 = ( X , B , n) is said to be trivializable if there exists a Bisomorphism of 1 onto a trivial fibration ( B x F, B, prJ. Such an isomorphism is called a triuialization of 1. It is important to realize that when a fibration 1 = (X, B, n) is trivializable, there is not in general a distinguished, uniquely determined trivialization of 1. If g1 and g2 are two trivializations of 1,then we have g2(z) = u(g,(z)), where
12 FIBBATIONS
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v : (b, x ) ~ ( bu(b, , x)) is a B-automorphism of the trivial bundle B x F; that is to say (16.12.2.1) u is a C"-mapping such that u(b, is a diffeomorphism of F onto itself for each b E B. In other words, the distinction between a trivial fibration and a trivializable one is that for the former any two fibers are canonically diffeomorphic, whereas for the latter they are diffeomorphic but there exists in general no distinguished diffeomorphism of one onto the other. a)
(16.12.4) A fiber bundle with base B whose fibers are discrete is called a covering (or covering space) of B. From the definition it follows immediately that the projection n : X + B is a surjective local dfleomorphism (16.5.6). Conversely, however, iff : X + Y is a surjective local diffeomorphism, it does
not necessarily follow that (X, Y,f) is a covering of Y. For example, consider a covering (X,, B, n) of B such that n is not injective; if x,, E X, is such that the fiber n-'(n(x,)) has at least two points, consider the space X = X, - {xo} and the restriction f of A to X; it is clear that f is a surjective local diffeomorphism, but (X, Y, f ) is not a covering (cf. (20.18.8)). (Cf. Problem 1.) To say that a triple (X, B, n) is a covering of the differential manifold B is equivalent to saying that X is a differential manifold, ~ta surjective C"-mapping, and that the following condition is satisfied: (16.12.4.1)
(R) For each b E B, there exists an open neighborhood U of b in B such that ~ - ' ( uis) the union of a(finite or infinite) sequence (V,) ofpairwise disjoint open subsets of X, with the property that for each n the restriction n, : V, U of K to V, is a difleomorphism of V, onto U. If B is connected, the condition (R) by itself implies that n is surjective. For n(X) is open in B, and if b is in the closure of n(X),then there exists an open neighborhood U of b in B which satisfies (R) and meets n(X). This implies that the sets V, are not empty, hence that U c n(X); in other words, n(X) is both open and closed in B, hence is the whole of B because B is connected. (16.12.4.2) For example, the Riemann surface Y of the logarithmic function (16.8.11) is a covering of C* = C - (0)of fiber-type Z, the projection n being the restriction of pr, to Y. For each point zo = roeieoE C* (where ro > 0 and 0, E R) has an open neighborhood U in C*, namely the image under the bijection (r, 0)- reie of the open set
V = ( ( r ,e) : r > 0,eo - n
-= 8 < eo in} c R*.
It is clear that the mapping cp : (re", k)wlog r
+ itl + 2kni
80
XVI DIFFERENTIAL MANIFOLDS
is a diffeomorphism of U x Z onto n-l(U) satisfying the local triviality condition (LT). This covering of C* is not triuializable, because it is connected: two points in the same fiber belong to the image in Y of R under a continuous mapping of the form t H (reit,log r + i t ) (3.19).
A covering whose fiber-type is a finite set of n points is called an n-sheeted covering. (16.12.5) I f A = (X, B, n) and A' = (X', B', n') are two fibrations, then it is immediate that (X x X', B x B', n x n') is a fibration, called the product of the fibrations 1and A' and written 1x 1'.For each point (b, b') E B x B', we have (x x X)(b,w) = Xb x .
x;,
A section of a fibration (X, B, n) (or a section of thefiber bundle X) is by definition any mapping s : B 4 X, not necessarily continuous, such that n 0 s = I, (in other words, it is a lfiing of 1),. A section is necessarily an injective mapping. A C"-section of X may be considered as a B-morphism of the trivial bundle (B, B, I,), identified with B, into (X,B, n). It is clear that any trivializable fibration has at least one C"-section, but conversely the existence of such a section does not necessarily imply that the fibration is trivializable (Section 16.16, Problem 1). The sections (resp. sections of class Cr)of a trivial bundle (B x F, B, prl) are the mappings b H ( b , f ( b ) ) ,where f is a mapping (resp. a mapping of class C') of B into F; hence they are in one-to-one correspondence with such mappings. If g is a B-morphism of (X, B, n) into (X', B, n'), then for each Cr-section s of (X, B, z) (where 0 5 r a), the mappingg 0 s : B + X' is a C'-section of (X', B, n'). On the other hand, it should be noted that if (f,g) : (X, B, n) -+ (X', B', n') is a morphism of fibrations with direrent bases, it is not in general possible to define the image of a section of (X, B, n) under such a morphism, because a point of B' may be the image of several distinct points of B. However, when f is a diyeornorphism of B onto B', the image of a section s of (X, B, n) under the morphism (f,g) is defined to be the section b ' H g ( s ( f -'(b'))) of (X', B', n'). (16.12.6)
(16.12.7) (i) A C"-section of afibration ( X , B, n) is an embedding of B into X whose image is closed in X. (ii) A continuous section of a covering (X, B, n) is a diffeomorphism of B onto an open and closed submanvold of X.
12 FIBRATIONS
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(i) Let s be a Cm-section.Since s is injective, the question is local with respect to B, so that we may assume the fibration to be trivial; but in this case the result is immediate, since s(B) is the graph of the mapping b w q(b,s(b))of B into F, in the notation used at the beginning of (16.12). (ii) If U is a connected open neighborhood of a point b E B over which the covering is trivializable, then s(U) must be one of the connected components V, of n-'(U) (16.12.4.1), and slU is the inverse of n, = nIV,. Since n, is a diffeomorphism, slU is the inverse diffeomorphism.Hence the section s is a local diffeomorphism, and the result now follows from (i) and (16.7.5). (16.12.8) Let 1= (X, B, n) be a jibration, B' a differential manifold, and f : B' -+ B a mapping of class C"'.
(i) The set B' x X of points (b', X ) E B' x X such that f(b') = n(x) is a closed submanifold of B' x X . (ii) If n' is the restriction to B' x X of pr, , then 1' = (B' x X, B', z') is afibration such that for eachpoint b' E B' thejiber (B' x x ) b # is canonically difeomorphic to Xf(b'). I f f ' is the restriction of pr, to B' x X,then ( f , f ' )is a morphism of 1' into 1. (iii) Let p' = (Y',B', p') be ajibration with base B' and let g : Y' + X be a C"-mapping such that ( f , g ) is a morphism of p' into 1. Then there exists a unique B'-morphisrn u : Y' -+ B' x X such that g =y0 u. (i) Put h(b', x ) = (f(b'), n(x)),so that h is a C"-mapping of B' x X into B x B (16.6.5). If A is the diagonal of B x B, we have B' x B X = h-'(A). Since the question is local with respect to B, we may assume that there exists a submersion $ : B x B + R" such that A = $-'(O) ((16.8.3) and (16.8.13)). Putting 8 = I) h, we have B' x X = O-'(O), and we have only to show that 8 is a submersion at every point of B' x X;for there will then exist a neighborhood of B' x X in which 8 is a submersion, and we can apply (16.8.8). So let (b', x) be a point of B' x B X, and let b =f(b') = n(x). Since z is a submersion, the image under T(bt, x)(h)of T(b',x)(B' x X) in T(b,b)(B x B) = Tb(B) x Tb(B) contains (0)x Tb(B), which iS a supplement of the diagonal in this product; but this diagonal, when identified with T(b,b)(A),is the kernel of T(b,b)($). Since $ is a submersion, the result now follows from (16.5.4). (ii) Let b&be a point of B', and let b, =f(bb). By hypothesis, there exists an open neighborhood U of b, in B , a differential manifold F, and a diffeomorphism cp : U x F -+ n-'(U) such that n(rp(b,t)) = b for all b E U and t E F. Consider now the open neighborhoodf-'(U) of bb in B'; the mapping 0
a2
XVI DIFFERENTIAL MANIFOLDS
is a bijection off-’(U) x F onto n’-’df-’(U)) such that n’((~’(b’,t)) = b’. For t ) belongs to B’ x X and we have n(cp(f(b’), 2)) =f(b‘); hence (~’(b’, n’((P’(b), t ) ) = b’;
conversely, if (b‘, x) E n‘-‘(f-’(U)), then b =f(b‘) = n(x) E U; hence there exists a unique t E F such that q(b, t ) = x. Finally, the fact that cp’ is a diffeomorphism follows from (16.12.2.1) and the fact that cp is a diffeomorphism. This completes the proof of (ii). so that the point (iii) For each y’ E Y’ we have n(g(y’)) =f@’(y’)), u(y’) = (p’(y‘),g(y‘))belongs to B’ x X,and it is clear that u is the unique B-morphism with the required properties. The fibration L’ is called the inverse image of 1underS, and is denoted by f*(1). If I is trivial, so isf*(A). For each section s : B + X of 1,the mapping s’ :b’ H (b’,s(f(6’)))
is a section off*@), called the inverse image of s underfand denoted byf*(s). If s is continuous (resp. of class C’), then the same is true off*(s). If (XI, B, nl)and (X,, B, n,) are two fibrations with base B and if g is a B-morphism of XI into X, , then it is immediate that the mapping g’:B‘xBX,+B’ x B X ~ defined by g’(b’, xl)= (b‘,g(xi)) is a B’-morphism; it is denoted by f *(g). (16.12.9) Let 1= (X,B, n) be a fibration, B’ a submanifold of B, and j : B’ + B the canonical injection. The set B’ x X is then the image under the mapping (x, b‘) + (b’, x ) of the graph of the restriction of n to n-’(B’), and the mapping x ~ ( n ( x )x ,) is therefore a diffeomorphism of the submanifold n-’(B‘) of X (16.8.12)onto B‘ x X. If we identify B’ x B X with n-’(B’) by means of this diffeomorphism,then A’ is identified with the restriction of n to n-’(B’). In future we shall always make this identification, and we shall say that the fibrationj*(A) = (n-’(B’), B’, x’) is induced by 1on B’.
The inverse imagej*(s) of a section s of I is then the restriction of s to B’. A section of the induced fibrationj*(A) is also called a section of I over B’. The set of Coo-sectionsof 1 over B’ is denoted by T(B‘, X). More generally, for any subset A of B, a section of A (or of X) over A is by definition any mapping s : A + X such that n 0 s = 1, (in other words, such that s(b) E X, for all b E A). The sections of 1over B are sometimes called global sections of 1(or of X). The condition (LT) may be stated in the form that each point b E B has an open neighborhood U in B such that the fibration induced by 1 o n U is trivializable, or (as we shall sometimes say) that I is trivializable ouer U.
12 FIBRATIONS
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(16.12.10) Let A = (X,B, n) and 1'= (X', B, x') be two fibrations with the same base B. We may form the product fibration
1 x A' = (X x X', B x B', n x A'). Let 6 be the diagonal mapping B -+ B x B, and let 1" be the inverse image 6*(1 x A'). The space of this fibration is by definition the submanifold of B x X x X' consisting of all (b, x , x') such that b = n(x) = n'(x'). Now, by (16.12.8(i)), the set X x X of points (x, x ' ) B X x X' such that n(x) = n'(x') is a submanifold of X x X', and up to a canonical symmetry (of B x X x X onto X x X' x B) the space X" of the fibration A" is therefore the graph of the restriction of n pr, (or A' pr2) to the submanifold X x ,,XI of X x X. Hence it is canonically diffeomorphic to this submanifold (16.8.13), and we shall identifv x" with X x BX . Next, if x" is the projection of A", and if U is an open subset of B over which X and X' are trivializable, we have diffeomorphisms cp : U x F 3 n-'(U), cp' : U x F' --t d-'(U), and it is immediately verified that (b, (t, t')>++(b,cp@, 0, cp'(b, t')) is a diffeomorphism of U x (F x F') onto n"-'(U). In particular, over each point b E B, the fiber of A" is diffeomorphic to X, x XL . The fibration A" (or the fiber bundle X x X ) is called thejiberproduct of 1and 1' (or of X and X') over B. 0
0
The following proposition is the analog, for sections of fiber bundles, of the Tietze-Urysohn theorem (4.5.1) : (16.12.11) Let A = (X, B, n) be ajibration withjibers difeomorphic to RN,let S be a closed subset of B, and let g : S 3 X be a section of X over S such that for each b E S there exists an open neighborhood Vb of b in B and a Cr-section (1 S r 5 a)s b Of A over V, which agrees with g on Vb n S . Then there exists a C-sectionf of 1 over B which agrees with g on S . For brevity we shall call g a C'-section of I (or X) ouer S. Let (A,) be a denumerable locally finite covering of B by connected open sets such that 1is trivializable over each A, (12.6.1). Let (B,) be another open covering of B such that B, c A, for each n (12.6.2). Let U, be the union of the A, for 1 5 k g n, and let W, be the union of the Bk for 1 5 k n, so that is the union of the B, for 1 5 k 6 n. By induction on n we shall define a C-sectionf, of I over W,, such that: (l)f, agrees with g on n S, and the and to g section of n-'(U,) over Wnu (U, n S) which is equal t o f , on W,, on U, n S, is of class C'; (2)f,+l agrees withf, on . Since B is the union of the W, ,the section f of 1over B which is equal tof, on W, for all n will have the required properties.
w,
w,,
w,,
84
XVI DIFFERENTIAL MANIFOLDS
Suppose then that fn has been defined. We shall show that there exists a Cr-section h,,,, of 1over A,+, which agrees with f,,on A,,+1 n and with g on A,+, n S. Then the sectionf,,, which is equal to f, on and to hn+, on Bn+, will satisfy the conditions (1) and (2). For at a point x E u (U, n S) which does not belong to B,,,, there exists by hypothesis a neighborhood T c U, of x which does not intersect B,,+l, and a C'-section of A over T which is equal tof, (hence also tof,,,) on T n W,, =T n and is equal to g on T n S; and at a point of B,,+l or of A,,,, n S, the section h,,,, over the neighborhood A,,+1 of this point agrees withf,,, on A,+, n and with g on A,+, n S. It remains to define h,,,,. By virtue of the choice of the A,, and the hypothesis on the fibers of 1,we maylimit ourselves to the casewhere n-'(A,,+,) = A,+1 x RN,so that sections over A,,+, may be identified with mappings of A,+, into RN.Consider now the function u,,+,, defined on
w, w,, w,
wn+l,
w,,+,
iAn+1 n w n )
u (An+, n S),
w,
which is equal to f,,on A,+, n and to g on A,+, n S. This function is of class C' on this closed subset of A,,+,. Indeed, this is obvious at a point of A,,+, n S which does not belong to ,and at a point x E An+, n R,,,there exists by hypothesis a neighborhood T c U,, n A,+, of x and a function of class C' on T which agrees withf, on T n W,, and with g on T n S, and hence agrees with u,,,, on T n (W,, u S). We can now apply (16.4.3) to the N components of u,+, and hence extend u,+, to a Cr-function h,,+l on A,+,. The proof is now complete.
w,,
In particular: (1 6.12.12) If' the fibers of I = (X, B, n) are difleomorphic fo RN,then there exists a Cm-sectionof 1over X.
We have only to apply (16.12.11) with S = @. Remark (16.12.13) A real-analytic fibration can also be regarded as a differential fibration: the differential fibration so obtained is said to underlie the given real-analytic fibration. I n the same way we define the real-analytic fibration underlying a given complex-analytic fibration. One point that should be emphasized is that the words " trivializable "and "trivialization "signify different things for a real-analytic (resp. complex-analytic)fibration and the underlying differential (resp. real-analytic) fibration.
12 FIBRATIONS
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PROBLEMS
1. Let X and Y be two connected differential manifolds of the same dimension and let f:X +Y be a local diffeomorphism. Show that the following properties are equivalent: (a) f is proper (Section 12.7, Problem 2). (b) f i s a closed mapping (that is, the image under f o f any closed subset of X is a closed subset of Y). (c) For each y E Y, the fiber f - ' ( y ) is a finite set, whose number of elements is independent of y. (d) (X, Y , f )is a covering of Y, all of whose fibers are finite. (To prove that (a) implies (b) and that (b) implies (c), argue by contradiction.) Deduce that for a Cm-mappingf: R"+R" to be a diffeomorphism of R"onto itself it is necessary and sufficient thatfshould be a local diffeomorphism and that Ilf(x)II 3 co as llxll+ co,where llxll is any norm on R". (Use the fact that R" is simply connected (16.28.3).)
2. Let X and Y be two pure complex-analytic manifolds of dimensions m and n, respectively, and letf: X x Y --f C"be a holomorphic mapping. Let
20 = {(x, y ) € x x Y :f(x, y) = O}
and let S be the set of points (x.y)eZo at which the tangent linear mapping T,f(x, . ) : Ty(Y)-+ C" is not bijective (i.e., S is the set of "singular points" of Zo). (a) Show that Z = Zo - S is a submanifold of X x Y of dimension m (if not empty), that the restriction p of prl to Z is a local dXeomorphism and that the fiber p-'(x) is discrete for each x E X. (b) Suppose that X = Cmand Y = C, so thatfis an entire function on Cm+'.For each point (xi, yl)E Z, let u1 be the unique holomorphic function on a neighborhood of x1 such that f(x, ul(x)) = 0 for all x in this neighborhood, and ul(xl)= y1 If Z; is the analytic manifold defined by u1 (Section 16.8 Problem 12) and p 1 :Z; +Cm is the canonical mapping, show that there exists a unique isomorphism h of Z; onto the connected component Z1of Z containing (xi,yl), such that p 0 h =p l . (Lift up to Z1 a path in X which is the projection of a path in Z; .) (c) Suppose that X = Y = C and takef(x, y) = xy - sin y. Then pr,(S) is not closed, prl(Z) = X, and (Z, X,p ) is not a covering. (d) Suppose again that X = Y = C, and takef(x, y) = x - 2ey ezy.Then prl(S) = (1) and prl(Z) = X - {I}, but (Z, prl(Z), p ) is not a covering. (e) Suppose that Y = C" and that the restriction of prl to Zo is a proper mapping (Section 12.7, Problem 2). Let T be the open set X - prl(S), and let 2, be a connected ,p(Zl), p ) is a covering of ~ ( 2 ,with ) a finite component of p-'Q in Z. Show that (Zl number of sheets. (Use Problem 1.) (f) Suppose that X is connected, Y = C, and takef(x, y ) to be a polynomial in y,
.
+
f(x, Y ) = y'
+ gl(x)Y-l + ...+ d x ) ,
with coefficients which are holomorphic functions on X. Suppose also that the discriminant of this polynomial does not vanish identically on X. Then prl(S) is closed,
86
XVI DIFFERENTIAL MANIFOLDS
and the open set T = X - prl(S) is dense and connected; hence the results of (e) are applicable, and moreover we have p(Z,) = T.
s'
(g) Suppose that X = Y = C, and take f ( x , y ) = x - exp(t') dt. Then S is empty, prl(Z) = X, but (Z, X,p) is not a covering. (Use Picard's theorem (Section 10.3, Problem 8(b)), and consider the values of y E C of the form e'"/'t, where t E R.) 3. Let (X, Y, p) be a covering, in which the differential manifold Y is compact. Show that of Y such that, for each i, each connected there exists a finite open covering (Ul)i51bm
component of p-'(U') meets at most one connected component of p-'(U,) for each
j # i. (Use (3.16.61.)
4.
Let A = (X,B, m) be a fibration and let f:B1+B and g :Bz+ Bl be two mappings of class c". =fine a B,-isomorphism of (fa g)*(A) onto g*(.f*(h)).
5. Let h = (X, B, a)and h'
= (X, B,
a') be two fibrations with base B, and
A"= (X X g X , B, w") their fiber product over B. (a) Consider the fibration a*@'), with base X and projection p, and the fibration n'*(h) with base X and projection p'. The space of each fibration is X x X . Show that ma = a op = r ' op'. (b) Let p = (Y,B, mi be a fibration with base B, and let f:Y +X and f:Y +X' be B-morphisms. Show that there exists a unique B-morphismf" :Y -+ X x X such that p of" = f a n d p' of" =f'. We havef," = (&,A) for all b E B.
6. Let X, Y be two pure differential manifolds, of dimensions p, q respectively. Let (U, q,p) and (V, #, q) be charts on X and Y,respectively. For each integer r 0 show that the mapping
JXfl
Jh&)
0
JWI
0
J&{x)(v-')
is a bijection j&.#of J'(U, V) onto J'(dU). $(V)). Show that J'(YP(U), ~V))may be canonically identified with dU)x #o x L;.q (Section 16.5, Problem 9) and is hence canonically endowed with a structure of differential manifold, induced by that of RP x R9 x L;. *.Show that the charts (J'(U, V), j& *, N)(where N =p q dim(L;, J ) form an atlas on J'(X, Y) which defines a structure of differential manifold. If a (resp. a') is the mapping which to each jet associates its source (resp. its target), then
+ +
(J'Or, Y), X,a),
(J'KY ) .Y,4,
(J'(X, Y ) ,X
X
Y,(a,a'))
are fibrations in which the fibers are diffeomorphic to Jb(RP9 Y ) ,
J'KR'90 v
Li.99
J'(n
respectively. If f:X -+Y is a C"-mapping, the mapping : x I-+ J:(n is a C"section of J'(X, y)considered as a fiber bundle over X. If s 4r, the canonical mapping J'(X, Y)-+ J*(X,Y) defined in Section 16.9, Problem 1 is a morphism for the fibrations over X, Y and X x Y.Iff: X + Y is a C"-mapping, the jet of order r - s of the mapping x H J:(f) depends only on J'(f). Hence we
13 DEFINITION OF FIBRATIONS BY MEANS OF CHARTS
87
have a mapping u H J'-"(u) of J'(X, Y) into J'-"(X, Js(X, Y)). Show that this mapping is an embedding and a morphism for the fibrations over X. Is it a diffeomorphism? Let X, Y, X', Y' be pure differential manifolds and let u : X + X , v : Y + Y be C"-mappings. For x E X and y' E Y , put x' = u(x) and y = uw),and define a mapping of J:,(X, Y ) into J:(X, Y) by w c-t J:.(v) 0 w 0 J:(u). This gives rise to a C"-mapping j ' ( ~ V, ) : u*(J'(X', Y))= X
Y ) -+ Jr(X,Y ) .
X x* J'(X',
If X = X (resp. Y = Y), the mapping y(lx,v ) (resp. y(u, ly)) fibrations over X (resp. Y).
is a morphism for the
13. DEFINITION OF FIBRATIONS BY MEANS OF CHARTS
(16.13.1) Let 1= (X, B, n) be a fibration. By hypothesis, there exists an open covering (U,) of B such that the fibrations induced by 1 on each U, (16.12.9) are trivializable. This property is then a fortiori true for each open covering which isjiner than (U,) (12.6). For each a, let Fa be the fiber at an (arbitrarily chosen) point of U, . Then by hypothesis there exists a diffeomorphism pa : U, x Fa + n-'(U.)
satisfying (LT) (16.12.1). For each pair of indices (a, /I) we,denote by 9,. the restriction (U, n U,) x Fa + n-'(U, n U,) of p a . Then we have a diffeomorphism (called a "transition function ")
which is of the form
(b, t> H(by e,a(b, t>>, where 8,, is a C"-mapping. Moreover, it follows directly from this definition that if u, B, y are any three indices and if we denote by $ J a , I&, and $fa the restrictions of $, t,bys, and t,bya to (Van U,n U,) x Fa, (U,n U , n U,) x F,,
and (Uan U , n U,) x Fa,
respectively, then we have (1 6.13.1 .I)
= J/;,
0
Now consider two fibrations A = (X, B, n) and I' = (X', B, n') with the same base, and a B-morphism (resp. a B-isomorphism) g of X into X' (resp. onto X'). Then there exists an open covering (U,) of B such that for (16.13.2)
88
XVI
DIFFERENTIAL MANIFOLDS
each a the fibrations induced on U, by both 1 and 1' are trivializable, so that we have diffeomorphisms cp, : U, x F A + n'-'(U,)
cp, : U, x F, + z-'(U,),
satisfying (LT). The composite mapping
is then of the form
(b, f)H(b,o,(b, t)), where o, is a mapping of class C" (resp. a mapping of class C" such that o,(b, * ) is a diffeomorphism of F, onto FA for each b E U,). The mapping g, is called the local expression of g corresponding to cp, and cp; . Also, with the notation of (16.13.1) and analogous notation for the fibration A', if we put g,, = g. I ((U.n U,) x F,), the diagram
(u, n u,)
x F , ~ ( u ,n u,) x F:
is commutative for each pair of indices (a, fi). (16.1 3.3) Conversely, consider a differential manifold B and an open covering (U,) of B; suppose that for each index a we are given a differential manifold F, , and for each pair of indices (a, fi) a mapping $pa:
(Uun
Up) x
Fa
+
(Uu X Up) x Fp
of the form
(b, 0- (b, 0,,(b9 t)), where 0,, is of class C". Suppose also that: (1) for each b E U, n U,, the mapping 0,,(b, * ) : Fa + F, is a diffeomorphism (which implies (1 6.12.2.1) that $, is a diffeomorphism); (2) the "patching condition" (16.13.1 .I)(with the notation used there) is satisfied for each triple of indices (a, fi, 7).
This latter condition, together with the facts that the are homeomorphisms and (U, n U,) x F, is open in U, x F,, allows us to define first of all a topologicalspaceX bypatching together the topological spaces U, x F,
13 DEFINITION OF FIBRATIONS BY MEANS OF CHARTS
89
along the open sets ( V a n U,) x F, by means of the homeomorphisms $,a (12.2). Hence (loc. cit.) we have homeomorphisms cp, : U, x F, + X u , where the X, are open subsets of X which cover X, such that if q,, is the restriction of cp, to (U,n U,) x F a , we have (P,a((Ua
n U,) x F a ) = X a n X,,
$pa
=C P;' 0 Vpa *
Let us first show that X is metrizable, separable, and locally compact. There exists (12.6.1) a denumerable open covering (A,,) of B which is finer than the covering (U,); hence (12.6.2) a denumerable open covering (B,,) of B such that S,,c A,, for all n. For each n, let a(n) be an index such that A,, c U,,,,), and put Y,, = rp,(,,)(B,, x Fa(,,))c X,,,,,_ Since the interior q,,of Y,, in X contains cp,(,,)(B,, x Fa,,,), the open sets y,,cover X. By (12.4.7), it is enough to show that the sets Y,, are closed in X, and for this it is enough (12.2.2) to show that Y,, n X, is closed in X, ,for each index j?.This is evident if X,,,, n X, = 0, and if XUcn,meets X,, then Y, n X, is the image under cpac,,, of the set
,
(Bn
n Up) x F,
9
which is closed in U, x F,. Next we define a mapping n : X + B as follows. Each x E X belongs to some Xu, hence is of the form cp,(b,, t,) with (b,, t,) E U, x Fa; we define n(x) = b,, and from the hypotheses it is immediate that this definition is independent of the choice of the index a. Finally, we transport to X, by means of cp, the structure of (product) differential manifold on U, x Fa; the fact that the I), are diffeomorphisms ensures that the structures induced on X,n X, by those on X, and X, are the same. Hence we have defined a structure of diferentialmanifold on X (16.2.5). It is now clear that I = (X, B, n) is a fibration; it is said to be obtained by patching together the trivialfibrations (U, x Fa, U, ,pr,) by means of the $,. Keeping the hypotheses and notation of (16.13.3), consider another open covering (U;) of B which isfiner than the covering (U,). For each , put F; = F a ( y ) .For each index y let a(y) be an index such that U; c U a ( y )and pair of indices (y, d), let
(16.13.4)
$;,:(U;nU;)xF;+(Uj,nU;)
xF;
denote the restriction of I).(a),a(y)to ( U b n U;) x Fb. It is clear that the $,; satisfy the same conditions as the $, and therefore define a fibration I' = (X', B, n') by patching together the trivial fibrations (Ui x F;, U;, prl) by means of the $ i y .This fibration A' is B-isomorphic to A. For if x' E X', then (with the obvious notation) we have x' = cp;(b, t ) with b E U; and t E F; for some index y ; to x' corresponds the point x = cp,(,,(b, t ) of X, and it is immediately seen that this point x does not depend on the choice of y, and that
90
XVI
DIFFERENTIAL MANIFOLDS
in this way we have defined a B-morphism g : X --* X. Conversely, for each point x E X we have x = cp,(b, t) with b E U, and t E F,, for some index a; there exists an index y such that b E U;, and to x corresponds the point x' = cp;(b, t). Once again, this point x' E X does not depend on the choices of a and y, and thus we have defined a B-morphism h : X + X'. Finally, it is straightforward to check that g 0 h and h o g are the identity mappings, and so our assertion is proved. (16.13.5) Still keeping the hypotheses and notation of (16.13.3), suppose that we are given, for each index a, a differential manifold FA, and for each pair of indices (u, /3), a mapping
+,;
: (U, n U,) x F: + (U, n Us) x F;
such that the conditions of (16.13.3) are satisfied by these mappings. Let 1' = (X', B, n') be the corresponding fibration. Suppose further that we are given, for each a,a mapping of class C" : c a : U ax F,+F:,
and that, if g, : U, x Fa + U, x F: is the mapping defined by g,(b, t ) = (b, c,(b, t)), the diagrams (1 6.13.2.1) are commutative. Then there exists a unique B-morphism g : X + X' such that g , = cp:-' 0 g cp, for each a.For if x is any point of X, there exists an index a such that x = cp,(b, t ) with b E U, and t E Fa; we put g(x) = cp:(g,(b, t)) and the commutativity of the diagrams (16.13.2.1) guarantees that this point does not depend on the choice of index a.The fact that g is a B-morphism is clear. In particular, if c,(b, * ) is a diffeomorphism for each a and each b E U,, then g is a B-isomorphism. Another particular case in which the preceding method may be applied is the definition of a Cm-sectionof the fibration 1:for such a section may be regarded as a B-morphism of the trivial fibration ( B , B, lB) into 1. 0
14. PRINCIPAL FIBER BUNDLES
We recall that a group G is said to act freely (or without fixed points) on a set E (cf. (12.10)) if for each x E E the stabilizer S, of x consists only of the idenfity element of G: in other words, if for each x E E the canonical mapping SHS * x of G into the orbit G . x is bijective. The group G then acts faithfully on E. (16.14.1) Let X be a diferential manifold and G a Lie group acting differentiably and freely on X; suppose that the orbit manifold X/G exists (16.10.3), and let n : X -P X/G be the canonical submersion. Then:
14 PRINCIPAL FIBER BUNDLES
91
(i) (X, X/G, a ) is a jibration. More precisely, each point of X/G has an open neighborhood U for which there exists a C"-mapping Q : U + X such that n(a(u)) = u for all u E U and such that the mapping (u, s ) w s a(u) is a d f e o morphism of U x G onto a-'(U). (ii) Let R c X x X be the set ofpairs (x,y ) such that xand y belong to the same orbit. For each (x, y ) E R,let ~ ( xy ), be the unique element of G such that y = z(x, y ) * x. Then z is a submersion of the submanifold R (16.10.3) into G.
-
(i) Since n is a submersion, it follows from (16.8.3) that every point of X/G admits an open neighborhood U for which there exists a mapping Q : U + X of class C" such that, for each u E U , we have n(a(u))= u and Tu(u)(~(U)) is a supplement of Tu(u)(a-l(u))in Tu(u)(X).Since by hypothesis the mapping cp : U x G + a - l ( U ) defined by cp(u, s) = s * Q(U) is bijective, it is enough to show that rp is a submersion (16.8.8(iv)). This is a consequence of the following more general result: (16.14.1.1) Let X be a diferential manifold and G a Lie group which acts differentiably on X such that the orbit manifold X/G exists. Let 71: X + X/G be the canonical submersion, and suppose that there exists a C"-mapping Q : X/G + X such that n 0 Q = IX,G. Then Q is an immersion, and the mapping cp : (X/G) x G + X defined by cp(u, s) = s * Q ( U ) is a surjective submersion. The fact that
Q
is an immersion follows from the relation
TU(Ud71) Tu(Q) = IT,(X,G). Next, we shall show that cp is a submersion at a point of the form (uo , e). Put xo = o(u0), so that n-'(u0) is the orbit G . x o . By virtue of (16.10.7), the canonical mapping G -P G . xo is a submersion of G onto the submanifold n-'(u0) of X, and we can apply (16.6.6) and (16.8.8). If now ( u o ,so) is any point of (X/G) x G, we remark that cp is the composition of the three mappings X H S O . x, (u, t)l+ t . a@), (u, S ) H ( U , SOIS), the first of which is a diffeomorphism of X onto itself (16.10), the third a diffeomorphismof (X/G) x G onto itself, and the second a submersion at the point ( u o , e). From this it follows that cp is a submersion at the point ( u o , so). O
(ii) Since the question is local with respect to B = X/G, we may assume that there exists a C"-section Q : B + X and that cp : (b, S ) H S * o(b) is a diffeomorphism of B x G onto X. Then the mapping p : xHpr2(cp-'(x)) is of class C", and hence the mapping 2, which is the restriction to R of ( x , Y)-P(Y)P(x)-'9
92
XVI
DIFFERENTIAL MANIFOLDS
is of class C". Moreover, for each x E X, the restriction of z to
{ x } x (G . X ) c R is a diffeomorphism of this submanifold onto G (16.10.7). Hence z is a submersion of R into G.
Examples Let G be a Lie group and H a Lie subgroup of G. It is clear that H acts freely on G (on the right) by the action (s, X ) H X S ; hence it follows from (16.10.6) and (16.14.1) that (G, G/H, n), where n : G + G/H is the canonical mapping, is a fibration. As another example, we have seen (16.1 1.I1) that the group G consisting of the identity mapping and the symmetry x H -x acts freely on S, and has the projective space P,(R) as orbit manifold; since S, is connected and G discrete, the fibration so defined is not triuializable. Again, with the notation of (1 6.11 .lo), the group GL(p, R) acts freely on the right on the space L,,p , and hence defines a fibration of L,, whose base is the Grassmannian Gn,p . It should be remarked that it can happen that a Lie group acts freely on a manifold X but that the orbit space X/G is not a manifold, even if G is discrete (cf. (16.10.3.4)). (16.14.2)
When the conditions of (16.14.1) are fulfilled, the manifold X endowed with the action of G is said to be a differentialprincipalfiberbundle (or simply a principal bundle) with structure group G ; the manifold B = X/G is the base of the bundle, and the fibers are the orbits of the points of x , and are diffeomorphic to G. Usually we shall regard the structure group of a principal bundle as acting on the right. The Riemann surface of the logarithm (16.12.4) is a principal bundle, with base C* and structure group Z. (16.14.3) Let X, X' be two principal bundles, B, B' their bases, n, n' their projections, and G, G their structure groups. A morphism of X into X' is by definition a pair (u, p), where u : X + X' is a C"-mapping and p : G -+ G' is a Lie group homomorphism, such that
u(x . s)
(1 6.14.3.1)
= u(x) * p(s)
for all s E G and x E X. The image under u of an orbit x * G is therefore contained in the orbit u(x) * G'; in other words, there exists a mapping u : B --* B' such that A' u = u n, and it follows from (16.10.4) that u is of class C". The mapping u is said to be associated with the morphism (u, p ) ; it is clear that (u, u ) is a morphism of fibrations (16.12). When p is an isomorphism of 0
0
93
14 PRINCIPAL FIBER BUNDLES
G onto G , it follows from (16.14.3) that the restriction of u to an orbit x . G is a diffeomorphismof x . G onto u(x) G'. If moreover v is a diffeomorphism of B onto B', then (u, u) is an isomorphism of fibrations (16.12.2). In these conditions, (u, p ) is said to be an isomorphism of the principal bundle X onto the principal bundle X . When G = G and p = lG,we shall say simply that u is a morphism of X into X'.
-
Example (16.14.4) Given a differential manifold B and a Lie group G, we define a right action of G on B x G by the rule
(b, t ) * s = (6, ts).
Since the orbits of this action are the sets pr;'(t) for t E G, and since pr, is a submersion, if follows that the orbit manifold exists and may be identified with B (16.10.3). Moreover, it is clear that G acts freely on B x G ; hence, with the above action, B x G is a principal bundle, called a trivial principal bundle. A principal bundle X with structure group G is said to be trivializable if it is isomorphic to a principal bundle of the form B x G. An isomorphism of X onto B x G is called a triuialization of X. (16.14.5) A (differentiable) principal bundle is trivializable if and only if it admits a COD-section.In particular, a principal bundle whose structure group is diiffeomorphic to RN is trivializable (1 6.12.1 1).
The condition is clearly necessary. Conversely, if a principal bundle X with structure group G and base B = X/G admits a Cm-section : B X, then it follows from (16.14.1.1) that the mapping (b, s ) H Q ( ~.)s is a bijective submersion, hence a diffeomorphism (16.8.8(iv)) and consequently an isomorphism of the principal bundle B x G onto X. (16.14.6) Let X be a principal bundle with structure group G, base B = X/G andprojection n : X + B. Let B' be a differential manifold and let f:B' + B be a C"-mapping. The group G acts differentiably and freely on the manifold X' = B' x X (16.12.8) by the rule (b', x ) * s = (b', x * s). With respect to this action, X' is a principal bundle with structure group G, and thejbration of X' may be identijied with the inverse image underfof the3bration 1 = ( X , B, n). Furthermore, if Y' is a principal bundle with structure group G and base B', and if u : Y' + X is a morphism for which f is the associated mapping, then there exists a unique B'-isomorphism w : Y' + B' x X such that u =f' w, where f ' : B' x X + X is the restriction of pr, . 0
94
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DIFFERENTIAL MANIFOLDS
The first assertion is obvious; with the notation of (16.12.8) the orbits of G in X are the fibers n'-'(b') of the fibration A' = (B' x X, B', x') =f*(3,). Since x' is a submersion, the orbit manifold X / G exists and the corresponding fibration of X may be identified with 2,by virtue of (16.10.3). The last assertion follows from (16.12.8). The principal bundle X' defined in (16.14.6) is called the inverse image of X by$ In particular, if B' is a submanifold of B and i f j : B' B is the canonical injection, then the inverse image of X by j may be identified with the submanifold x-'(B') of X, the action of G on this submanifold being the restriction of the action of G on X. This principal bundle is also called the bundle induced by X over B'. In this terminology, (16.14.1(i)) states that each point of B admits an open neighborhood U over which the induced principal bundle is trivializable (16.14.4). (16.14.7) Let X be a principal bundle with structure group G (acting on the right); let B = G\X be the base and n : X -,B the projection. Also let F be a diferential manifold on which G acts diferentiably (on the left). Then G acts diferentiably andfreely on the right on the product X x F by the rule
( x ,y ) * s = (x * s, s-l * y). For this action: (i) The orbit-manifold G\(X x F ) exists. We denote it by X x F , and the projection X x F + X x F by (x,y ) x * ~y. (ii) For each orbit z E G\(X x F), let xp(z) be the element of B which is equal to n(x)for all (x, y) e z. Then ( X x F , B, nF) is ajibration in which all the$bers are difeomorphic to F . More precisely, ifU is an open set in B such that n-'(U) is triviaIizable and ifa : U -+ n-'(U) is a Cm-section of n-'(U), then the mapping (b,y ) ~ a ( b* )y is a U-isomorphism of U x F onto n,'(U) (which is therefore trivializable). (i) Let R be the set of points ( x , x', y , y') E X x X x F x F (identified with the product manifold (X x F) x (X x F)) such that ( x ,y ) and (x', y') belong to the same orbit. With the notation of (16.14.1), R is identified with the set of points (r, y , o(r) * y ) E R x F x F, that is to say with the graph of the mapping (r,y ) H r ( r )* y of R x F into F. By (16.14.1(ii)) and (16.8.13), this is a closed submanifold of R x F x F, hence also of X x X x F x F. By virtue of (16.10.3), this establishes (i). (ii) It is sufficientto prove the second assertion, for which we may assume that U = B and that X is trivial. Then it follows from (16.10.4) that zFis a surjective mapping of class C". Next, for each x E X, put
s(x) = o(x, o(x(x))) E G
14 PRINCIPAL FIBER BUNDLES
95
(withthenotation of (16.14.1)), so that o(n(x)) = x * s(x). Iff: X x F + B x F is the mapping (x, y) H (n(x), s(x)-' * y), we havef(x . t, t-' y ) = f ( x , y ) for all t E G, because s(x * t ) = t - ' s ( x ) by definition. Since f is of class C" (16.14.1(ii)) there exists by virtue of (16.10.4) a mapping g : X x F -,B x F of class C" such that f ( x , y) = g(x .y ) , and it is immediately verified that g is the inverse of the mapping (b, y) Ho(b)* y . When X = B x G is trivial, so that we may take a(b) = b . e and identify X x F with B x F by means of g, we have (b, s) y = (b, s y ) . (16.14.7.1) With the same notation as above, every section cp of X x F over U may be uniquely expressed in the form cp : b H a ( b ) . $(b), where $ is a mapping of U into F. The section cp is of class C' (r an integer or + m) if and only if I) is of class C'. Since a is a diffeomorphism of U onto a submanifold a(U) of X, the inverse of c being the restriction of IC, we may also write $(b) = cP(o(b)),where 0 = $ 0 (n1 a(U)); the mapping @ is of class C' if and only if $ is of class C'. Moreover, by taking U to be sufficiently small, we may suppose that (D is defined on a neighborhood of a(U) in X, and is of class C' in this neighborhood if cp is of class C' (16.4.3). (16.14.7.2) For x E X, y E F, and t E G we have ( x * t) * y = x * (t .y). The relation x y = x * y' signifies that x = x * t and y' = t-' .y for some t E G ; hence y' = y , so that y Hx .y is a dzfeomorphism of F onto the fiber IC;'(K(X)). It should be noted carefully that the group G does not act canonically on a fiber ICE'@) of X x F: we can make G act on this fiber by choosing a point xo in z-'(b) and putting t (xo y) = xo (t . y); but this action depends in general on the choice of x,; for if xh = xo . t o , we have x; .y = xo * (to * y), so that on replacing xo by x; the new action of G on n;'(b) is 1
(4 xo * Y ) H X o
*
((to
- A,
which is not the same as the previous action unless the commutator subgroup of G acts trivially on F. This condition will be satisfied in particular if G is commutative. As in (16.10), putting m(x, y ) = x * y , we denote the tangent linear mappings T , b ( . r)) and T,(m(x, . )) by ?
(16.14.7.3)
h,Hh;y,
kYt-+x* k,,
respectively. Then we have (16.14.7.4)
h,
*
( t * y ) = (h,
*
t) * y ,
x .( t
. k,)
= (X
. t)
*
k,
96
XVI
DIFFERENTIAL MANIFOLDS
-
for all t E G , and the mapping k y H x k, is bijective. It follows (16.6.6) that
which implies that m is a submersion. The space X x F is called the bundle of fiber-type F associated with X and the action of G on F. This notion will be especially useful in Chapter XX. At this point, we shall make use of it to prove the following proposition: (16.14.8) Let X be a principal bundle with structure group G, and let H be a Lie subgroup of G. Then H acts on X (on the right) by restricting the action of G. The orbit-manifold H\X exists, so that X is a principal bundle with base H\X and group H (16.14.1). Also if x : H\X + G\X is the mapping which associates with each H-orbit the unique G-orbit containing it, then (H\X, G\X, n ) is a fibration whose fibers are difeomorphic to the homogeneous space G/H.
If R c X x X (resp. R c X x X)is the set of pairs ( x , y ) which belong to the same G-orbit (resp. the same H-orbit), then in the notation of (16.14.1) we have R = z-'(H), which shows that R is a closed submanifold, because T is a submersion ((16.14.1) and (16.8.12)). Next we remark that G acts differentiably on the left on G/H, so that we can define the associated bundle X x (G/H) over G\X. Let no :
xx
(G/H) -+ G\X
be the projection. We shall define a diffeomorphism
# : x x~(G/H)+H\x such that the diagram
(1 6.14.8.1)
G\X is commutative; this will prove the proposition. Let cp : G + G/H and p : X + H\X be the canonical projections. Let f : X x G + H\X be the composite mapping (x,s ) ~ p (.xs). For each t E H, we have&, st) = f ( x , s), so that we may writef ( x , s) = g(x, cp(s)), where g : X x (G/H) --+ H\X is a mapping of class C" (16.10.4). Further, for s' E G, we have
g(x . s', s'-' . cp(s))
=f ( x
.s', s'-'s)
=f ( x , s) = g(x, cp(s))
14 PRINCIPAL FIBER BUNDLES
97
so that we may write g(x, q(s)) = u(x * q(s)), where u : X x (G/H) + H\X is a C"-mapping (16.10.4). Next, for each x E X, put f'(x) = x q(e), which defines a Cm-mappingf' : X --* X x (G/H). For each t E H, we have
-
f'(x
*
- -
t ) = ( x t ) q(e) = x (t p(e)) = x * q(e) =f'(x)
because t * q(e) = p ( t ) = p(e) since t E H. Hence we may writef'(x) = u'(p(x)) with u' a mapping of class C" (16.10.4). It remains to verify that u and u' are inverses of each other and that the diagram (16.14.8.1) is commutative, which is straightforward. In particular, and changing the notation : (1 6.14.9) Let G be a Lie group and H, K be two Lie subgroups of G such that K c H. Let n : G/K + G/H be the mapping which associates with each left coset of K the left coset of H which contains it. Then (G/K, G/H, n) is a Jibration withjbers diffeomorphic to the homogeneous space H/K. If K is a normal subgroup of H, then G/K is a principal bundle over G/H with structure group H/K.
The last assertion follows from the fact that H/K acts freely on G/K on the right, because xKt = xtK for all x E G and t E H. Examples (16.14.10) It follows in particular from (16.14.9) and from (16.11.4) and (16.11.6) that for p = 2 , . . . , n the Stiefel manifold S,,,(R) (resp. S,,,(C), resp. S,,,(H)) isjiberedover S,,,p-l(R) (resp. S,,,p-l(C), resp. S,,,,-,(H)) with fibers diffeomorphic to the sphere Sn-, (resp. S 2 ( n - p ) + lresp. , S4(n-p)+3). Again, by virtue of (1 6.11.9), S,,,,(R) (resp. S,,,,(C), resp. S,,,,(H)) is a principal bundle over the Grassmannian G,,,,(R) (resp. G,,,,(C), resp. G,,,,(H)) with structure group O(p,R) (resp. U(p, C), resp. U ( p , H)). In particular, the sphere S,, (resp. S2n+l,resp. S4n+3)is a principal bundle over the projective space P,,(R) (resp. P,,(C), resp. P,,(H)) with structure group {- 1, l} (resp. U(1, C), which is isomorphic to the multiplicative group U of complex numbers of absolute value I, hence also isomorphic to T, resp. U(1, H), which is
+
isomorphic to the multiplicative group of quaternions of norm 1).
More particularly, if we take n = 1 (having regard to (16.11.12)) we obtain a fibration of S1over S, with fiber-type { - 1, l}; a fibration of S, over S, with fiber-type S,; and a fibration of S7 over S, with fiber-type S, . Since S, is connected, it is clear that the first of these three fibrations is not trivializable, and it can be shown that the same is true of the other two (" Hopf fibrations").
98
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DIFFERENTIAL MANIFOLDS
+
If F is any differential manifold on which the group G = { - 1, 1) acts, we obtain from the first of these three principal bundles an associated bundle with fiber-type F and base U = S1. Taking F = R, the action of - 1 on R being fH -t, we obtain the orbit-manifold (U x R)/G, where - 1 acts by (z, t ) ~ ( - z ,- t ) . This manifold is called the Mobius strip. Taking F = U, with - 1 acting by ZH -2, we obtain the orbit-manifold (U x U)/G, where - I acts by (z, z')H(-z, -2'). This manifold is called the twisted torus. Finally, taking F = U again, with - 1 now acting by complex conjugation ZH Z, we obtain the orbit-manifold (U x U)/G, with - 1 acting by (2,2') H (- 2, 2').
This manifold is called the Klein bottle.
PROBLEMS
1. Let X be a principal bundle with structure group G , base B = X/G, and projection n, and let (U.) be an open covering of B such that for each index a there exists a section a . of X over U. for which ya : (b, s) Hurn@) . s is a diffeomorphism of U. x G onto n-'(UJ. For each pair of indices (a, let cp,. denote the restrictionof cpa to U. n u@ , and put !IJ,= = c p ~ l c p ~ which ~, is a transition diffeomorphism of the form
m,
0
where e p , : U. n u@ G is a Cm-mapping. Further, for each triple of indices (a,8 and each point b E U, n Up n U, we have the " cocycle condition " --f
,~)
Conversely, let B be a differential manifold and (U.) an open covering of B; and suppose that we are given a C"-mapping : U, n UB+G for each pair of indices a,p, these mappings satisfying the cocycle condition (2). Show that there exists a principal bundle X with structure group G and base B, and for each index a a section a, of X over U. such that the transition diffeomorphisms are of the form (1). If for the same covering (U,) of B we are given another family of mappings & :U, n u@ + G satisfying the condition (2), and hence defining a principal bundle X' over B with structure group G, show that for X and X to be isomorphic it is necessary and sufficient that there should exist for each index a a C"-mapping pa : U., + G such that, for each pair of indices (a,B>
e,,
for all b E U. n u@
14 PRINCIPAL FIBER BUNDLES
99
Describe the relations between two families (8~8) and (8$) defining the same principal bundle, corresponding, respectively, to an open covering (U,) and a finer open covering (Ul). When G is commutative, given two principal bundles X, X over the same base B and with G as structure group, we can define (up to isomorphism) their composition X . X' as follows: for a given open covering (U.) of B, suppose that X (resp. X ) is satisfying the cocycle condition (2); then the defined by the family (88.d (resp. family (8,.8;,) also satisfies (2) and hence defines a principal bundle with structure group G, denoted by X . X'. Verify that up to isomorphism this bundle is independent of the choice of families (OD,) and (&) defining X and X . In this way the set of isomorphism classes of principal bundles with base B and structure group G is endowed with a commutative group structure. 2. Let X be a connected complex-analyticmanifold. Show that the ring O(X) of (complexvalued) holomorphic functions on Xis an integral domain (Section 16.3, Problem 3(a)). Let Ro(X) denote the field of fractions of O(X). If u,v E O w ) and v # 0, the function x Hu(x)/v(x) is defined and holomorphic on a dense open subset of X. If u/v = u l / q in the field Ro(X) (i.e., if the holomorphic function uvl - ulv is identically zero), then
the functions x Hu(x)/v(x) and x H ul(x)/vl(x) are defined and equal on a dense open subset of X. Hence, for eachfE Ro(X), there is a largest dense open set 6 ( f ) in X with the property that for each point xo E 6 ( f ) there exist two elements u, v in O(X) such that u/v = f and v(xo) # 0, so that u(x)/v(x) is defined and holomorphic in a neighborhood of xo . Putf(xo) = u(xo)/v(xo);then the complex numberf(xo) depends only on f (and xo). Hence we have defined a holomorphic mappingf: 6 ( f ) + C, and the mappingf-fis bijective. Usually therefore we shall identify fandA and say that f i s an elementary meromorphic function on X (by abuse of language), whose domain of definition is 6 ( f ) . I f f an d g are two elements of Ro(X), thenf+ g,fg, and l/f(iff# 0) are defined as elements of Ro(X); but all that can be said about their domains of definition is that S ( f + g) and &fg) contain 6 ( f ) n 6 ( g ) , and in general neither of the sets 6 0 , 6(l/f) is contained in the other. A meromorphic function on X is by definition a function f which is defined and holomorphic on a dense open subset U of X, such that for each x E X there exists a connected open neighborhood V, of x and an elementary meromorphic function fx E ROWs)such that 6(f,) = U n V, and such that fx agrees with f on U n V,; then there exists no holomorphic function on an open set U strictly containing U, which extends f. Show that the set R(X) of meromorphic functions on X Can be endowed with a field structure which induces the field structure of each Ro(V,). If X is compact and connected, then O(X) = Ro(X) = C (Section 16.3, Problem 3(b)). If X = P,(C) (which is compact and connected), show that, for each pair P, Q of nonzero homogeneous polynomials of the same degree on Cn*', there exists a meromorphic function f on X such that, if m : C"+'- {0}+ P.(C) is the canonical mapping, we have P(z)/Q(z) =f(m(z)) at all points z # 0 in C"+'such that m(z) E 6 0 . When n = 1, we obtain all meromorphic functions on P,(C) in this way (use Liouville's theorem (9.11 .I)).
3. Let X be. a connected complex-analytic manifold. Apredivisor on X is a pair consisting of a covering (U,) of X by connected open sets and a family (f.), where f . is meromorphic on U. (Problem 2) and not identically zero, such that for each pair of indices a,p, there exists a holomorphic function go. :U. n U b + C* such that
fa(x)= g8a(x)h(x)
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at all points x E UQn U, at which f, and f p are defined. Two predivisors (0, (f.)) and ((Ui), (fi)) are said to be. equivalent if, for each x f X, there exists an open neighborhood V, of x contained in some U. and in some U; ,and a holomorphic function hz :V, +C*, such that f . ( y ) = h,(y)f;(y) at all points y E V, at which f, and f i are defined. A divisor on X is an equivalence class of predivisors. If D, D' are two divisors, then there exist two predivisors belonging to D and D , respectively, and corresponding to the same open covering (UJ. If ((Uo), cfo)) and ((Urn), (f:)) are two such predivisors, then we denote by D D' the divisor containing Show that D D' does not depend on the choice of the predivisor ((Ud, predivisors in D and D'. The mapping (D, D') HD D' defines a commutative group structure on the set Div(X) of divisors on X. The neutral element of this group (denoted by 0) is the divisor containing the predivisor consisting of X and the constant function 1. A principal divisor is a divisor containing a predivisor of the form (X, f), where f is a meromorphic function on X, not identically zero. The divisor containing this predivisor is called the divisor off and is denoted by Div(f). Two meromorphic functionsJ g. neither of which is identically zero,have the same divisor if and only if there exists a holomorphic function u on X which does not vanish at anypoint of X, such that f(x) = u(x)g(x) at all x E X, wherefand g are both defined. The principal divisors form a subgroup &c(X) of Diva, isomorphic to R*(X)/O*(X), where R*(X) is the multiplicative group of the field R(X) and O*(X) is the group of invertible elements of the ring 0 0 .
uQfL)).
+
+
+
4. We retain the hypotheses and notation of Problem 3. If (WE),(f.))is a predivisor on X, the functions gp. define (Problem 1) a principal bundle over X with structure group C*, and equivalent predivisors give rise in this way to isomorphic principal bundles, so that to each divisor D on X there corresponds, up to isomorphism, a principal bundle P@). The bundle P(D) is trivializable (as a holomorphic fiber bundle) if and only if D is principal. Two bundles P(D) and P(D') are X-isomorphic if and only if D D = Div(f), wherefis a meromorphic function on X. In this way we obtain an isomorphism of the quotient group Div(X)/Princ(X) onto a subgroup of the multiplicative group of isomorphism classes of principal bundles over X with structure group C* (Problem 1).
-
5.
L e t r b e thecanonicalmappingCn+l-{O}-+P.(C),andforj=O, 1, ...,nletU ,be the image in P.(C) of the open set consisting of the points z = (zo, zl, .. ,z") f C"+' such that z' # 0. The U, are connected open sets which cover P.(C). For each j , let f i be the holomorphic function on U, whose value at r ( z ) E U, is zo/z'. Then ((U,), V;)) is a predivisor, and the corresponding divisor D1or D,(C) is called the fundamental divisor on P,(C). Show that when n 2 1, this is not a principal divisor (Section 16.3, Problem 3(b)) and that its class in Div(X)/hinc(X) generates a subgroup isomorphic to Z. It follows that there are infinitely many nonisomorphic holomorphic principal bundles over P.(C) with structure group C*. Show that the principal bundle P(D,(C)) is that defined by the action of the multiplicative group C* = GL(1, C) on the space L.+', '(C) = C"+' - {O}, (16.11.10). Consider the analogs for real-analytic manifolds of the definitions and results of hoblems 1-5. In analogous notation, show that the fundamental divisor D,@)
.
on P.(R) is such that 2D,@) is principal. (Consider the function R"+' - {Oh)
14
PRINCIPAL FIBER BUNDLES
101
6. Let B be a differential manifold, G an at most denumerable discrete group, and U, V two open subsets of B. Describe all the isomorphism classes of principal bundles over B with structure group G, such that the induced principal bundles over U and V are trivializable (cf. Problem 1). In particular, if U n V has exactly two connected components, then the isomorphism classes in question are in bijective correspondence with the conjugacy classes in G. 7. Define the product of two principal bundles X, X' with structure groups G, G' and bases B, B', respectively; also define the fiber product of two principal bundles X, X over the same base, with structure groups G, G', respectively. Show that if X is a principal bundle over B whose structure group G is the product of two subgroups G', G", then X is canonically isomorphic to the fiber product over B of two principal bundles X',X" over B with G',G" as respective structure groups; and that X is trivializable if and only if X' and X" are trivializable. 8.
Let X be a principal bundle with base B, structure group G, and projection T, and let E = X X F be a fiber bundle with fiber-type F and projection wF, associated with X. For each C"-sectionfof E over B, there exists a unique mapping rp, : X + F such that x . p,(x) = f ( ~ ( x ) )this ; mapping is of class C" and satisfies the relation
p,(x . s) = s-' . p,(x) for all x E X and all s E G. Show thatf Hp, is a bijection of the set of C"-sections of E over B onto the set of C"-mappings p : X + F such that p(x . s) = s-I . p(x) for all x E X and s E G. In particular, if there exists yo E F such that s . yo = yo for all s E G, then the bundle E admits a section over B; if G acts trivially on F, the bundle E is trivializable. 9. Let X, X' be two principal bundles, with bases B, B , structure groups G, G', and projections w , T' respectively; let E = X x F, E = X' x c' F' be the fiber bundles with fiber-types F, F associated with X, X', respectively, and let T F , wk, be their projections. Let (u,p) be a morphism of X into X (16.14.3) and let u : B + B' be the mapping associated with u. Show that for each mapping f: F + F such that f(s . y ) = p(s) .f(y) for all y E F and s E G, there exists a unique mapping w, :E + E such that the pair (u, w,) is a morphism of the fibration (E, B, wF)into (E, B', T;.), and that w,(x . y ) = U ( X ) . f ( y ) for all x E X and y E F. Such a morphism of fibrations is called a (u,p)-morphism. If G = G', X = X , and if u = lx ,p = l o , then w, is said to be an X-morphism.
10. Let X be a principal bundle with base B and structure group G , and let E = X x F be a fiber bundle associated with X, with fiber-type F. Let u : B' + B be a Cm-mapping, and let X = v*(X) = B x X be the inverse image of X by u. Show that there exists a unique mapping f:E = X' x F + u*(E) = B' x E such that f((b', x ) . y ) = (b', x * y ) forb' E B', x E X, and y E F, and that /is a B'-isomorphism of fiber bundles. What factorization property analogous to (16.1 2.8(iii)) can be stated in this context? = (E, B, T ) be a fibration in which B is a connected differential manifold. In order that should be B-isomorphic to a fibration (X X F, B, wF) associated with a principal bundle X, it is necessary and sufficient that there should exist: (1) a Lie group G acting differentiably and faithfully on a differential manifold F diffeomorphic to
11. Let
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DIFFERENTIAL MANIFOLDS
the fibers of E; (2) an open covering (U,) of B and for each a a diffeomorphism vm:U,, x F 'IT-~(U=) such that P ( ~ ~ (yX) ) ,= x for all (x, y) E U, x F;(3) for each pair of indices (a, a C"-mapping g,= : U. n UB--f G such that if rpaais the restriction of 'p. to (U. n U,) x F, then the transition diffeomorphism a/~,* = cpG' vPD is of the form (x, y) H(x, g,.(x) . y) (cf. Problem 1). 0
12. Let X, Y be pure differential manifolds, of dimensions p and q, respectively. Let R'(X) (resp. R:(X)) denote the set of invertible jets of order r from X to Rp (resp. invertible jets of order r from X to RP,with source x ) (Section 16.9, Problem 1). We can define on R'(X) a structure of differential manifold as in Section 16.12, Problem 6; with respect to this structure, the group G'(p) (Section 16.9, Problem 1) acts differen-
tiably on the right on R'(X) and defines on R'(X) a structure of principal bundle over X. Show that the fibration (J'(X, Y), X x Y, (n,d))(Section 16.12, Problem 6) is isomorphic to a fibration associated with the principal bundle R'(X) x R'(Y), with fiber-type L;, q ; likewise that the fibration (J'(X, Y ) ,X, 'IT) (resp. (J'(X, Y). Y, 'IT')) is isomorphic to a fibration associated with the principal bundle Rr(X) (resp. R'(Y)), with fiber-type Ji)(Rp, Y) (resp. J'(X, Rq)o). 13. Show that the twisted torus (16.14.10) considered as a fiber bundle with base S , and fiber-type S, , i s trivializable.
14. With the hypotheses of (16.14.8), suppose in addition that H is a normal subgroup of G. Show that the quotient group G/H acts differentiably and freely on the right on the manifold H\X, and that the orbit-manifold may be canonically identified with G\X, so that H\X is a principal bundle with base G\X and group G/H.
15. Let (X, B, n) be a principal bundle with structure group G, and let H be a Lie group acting differentiably on the right on X. Suppose that (x .s ) . t = (x . t ) . s for all x E X, s E G, and t E H;this implies in particular that for each b E B the set w-*(b). t is a fiber n-'(b') for some b' E B, so that if we write b' = b . t, then H acts differentiably on B, and equivariantly (16.10.10) on X and B. Suppose further that H actsfreely on B and that the orbit-manifold H\B exists, so that B is a principal bundle over H\B with structure group H (16.14.1).
(a) Show that H acts freely on X, that the orbit manifold H\X exists, and that if 'IT' : H\X H\B is the unique mapping which makes the diagram --f
X
& H\X
commutative, then (H\X, H\B, n') is a principal bundle with structure group G. (Reduce to the case where (B, H\B, q) is trivial.) (b) Let F be a differential manifold on which G acts differentiably on the left. Show that there exists a unique differentiable right action of H on X x G F such that (x . y) . t = (x . t ) . y for t E H, x E X, and y E F. Furthermore, show that the orbit manifold H\(X x F) exists and is canonically diffeomorphic to (H\X) x F. (c) For each pair (x, x') of elements of X such that ~ ( x=) 'IT(x'),let T ( X . x') denote
14 PRINCIPAL FIBER BUNDLES
103
the element of G such that x = x’ . T(X, x’). Suppose that there exists a Cm-section u of X over B and a Cm-mapping p : H + G such that u(b . t) . t-’ = u(b) . p ( t ) for all b E B and all t E H. (a) Show that /?is a homomorphism of H into G. (#?) For each x E X, put f ( x ) (n(x), T(u(T(x)), x)), so that f is a Cm-mappingof X into B x G. Show that the unique mapping g which makes the diagram
X
-
H\X
commutative (where H acts on the left on G by the rule (I, s) Hs/?(t -I)) is an isomorphism of principal bundles with base H\B and structure group G. 16. Let (X, B, v ) and (X’, B , n‘) be two principal bundles with structure groups G, G’, re-
spectively. Suppose that G acts differentiably on the left on X and that s.(x’.t’)=(s.x’)*t‘
(which we shall denote by s . x’ . t‘) for all s E G, x’ E X’, and t‘ E G’; then G also acts differentiably on B , and equivariantly on X and B’. Show that the unique mapping p :X x X X x B’, which makes the diagram --f
XXX
-
XXGX
commutative, is such that (X x X , X x B , p ) is a principal bundle with structure group G’, the action of G’ being such that ( x . x’) . f’ = x . (x’ . f ’ ) for all x E X, x’ E X , t’ E G’. (Reduce to the case where X is trivial.) Furthermore, the composite mapping p’:
1 X X d
X x X - X x B P x X ~ B
is such that (X x X , X x B , p’) is a principal bundle with structure group G x G’ (the action of G x G’ on X x X being defined by ( x , x’) . (s, t’) = (x . s, s-’ . x’ . t’)). Let F be a differential manifold on which G’ acts differentiably on the left; then G acts differentiably on the left on the fiber bundle X x G’ F associated with X with fiber-type F’, by the rule s . (x‘ . z’) = (s . x’) . z’. Show that there exists a unique diffeomorphism (called canonical) : (X x X‘) x G’ F’+X x
(X’ x O’ F’)
for which the diagram
XXXXF’
(X
XG
X)
XG’
F‘ 7 x X G (X’
XG’
F)
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XVI DIFFERENTIAL MANIFOLDS
is commutative, where f and g are the canonical mappings. Hence on the space Y = (X x X') x G' F' we have two structures of fiber bundle. one with base X x B' and fiber-type F', and the other with base B and fiber-type X' x G' F'. 17. Let (X, B, a)be a principal bundle with structure group G, and let p : G +G' be a homomorphism of the Lie group G into a Lie group G . Then G acts differentiably on the left on G by the rule (s, t') H p ( s ) t ' , and we may consider the fiber bundle X x G associated with X by this action. Show that G' acts on the right on X x G' by the rule ( x . s') * t' = x . (s't') and that with respect to this action (X x G', B, a') (where a' is the canonical mapping) is a principal bundle with base B and group G'; this bundle is called the p-extension of X. If X is a principal bundle with base B and structure group G', and if u : X + X is such that (u, p) is a morphism (16.14.3) of X into X , show that there exists a unique isomorphism u : X x G + X such that u = u 0 q, where :X + X X G is the canonical mapping x H X * e' (e' being the
neutral element of G).
18. Let (ek)l S k S Z n be the canonical basis of C2n,and identify R2" with the real vector subspace of C""spanned by the ek; then C2"= R2"@ iR2", that is to say every vector z E Cz" can be written uniquely in the form z = x iy with x, y in RZn.Put x = Wz. Let B(z, w) be the symmetric bilinear form on C2" for which B(e,,, eJ = ,a (Kronecker delta). Let V be a totally isotropic subspace of C2"of maximum (complex) dimension n, relative to the form B. Then the mapping z HWz is an R-linear bijection of V onto R"". Let fv be the inverse bijection, and put j v ( x ) = W(ifv(x)) for x E RZn, so that jvis an R-linear bijection of R"" onto itself.
+
(a) Show that j<(x) = -x, B(jv(x), jv(y)) = B(x, y), and that B(x, jv(x)) = 0 for all x, y E RZn. (b) The orthogonal group of the restriction of B to R2"may be identified with O(2n). It acts transitively on the set of mappings jvsatisfying the relations in (a) by the rule jv Hjscv, = sjVs-'(O(2n)may be considered as a subgroup of O(B)).The stabilizer of jv(or of the corresponding subspace V) may be identified with the unitary group U(Sv), where SV@,Y)= B(x, Y)
+ iB(x,jv(y))
is a positive definite Hermitian form relative to the complex vector space structure on Rz" defined by ( a pi) * x = ax Pjv(x). The set r. of maximal totally isotropic subspaces of Cz"relative to B is therefore in canonical bijective correspondence with the homogeneous space 0(2n)/U(n, C), and the structure of real-analytic manifold of this homogeneous space may therefore be transported to I?. by means of this correspondence. This manifold has two connected components, one of which corresponds to SO(2n)/U(n,C ) and is denoted by TL. (c) Show that r.+ is endowed with a fibration with base S2"-" and fiber diffeomorphic to F$- . (Consider all the maximal totally isotropic subspaces V containing a given isotropic vector whose projection on R2" is e, .) (d) Show that the real Stiefel manifold SZm,2n-z(R)is endowed with a fibration whose base is diffeomorphic to r$ and whose fibers are diffeomorphic to the complex Stiefel manifold Sn,n-l(C). (Observe that a totally isotropic subspace of C2"of dimension n - 1 is contained in a unique subspace belonging to r.' .)
+
+
15 VECTOR BUNDLES
105
15. VECTOR BUNDLES
(16.15.1) Let (E, B, n) be a differential fibration such that for each b E B the fiber E, = n-'(b) is endowed with a structure of a finite-dimensional real (resp. complex) vector space. Then E , endowed with the structure defined by the fibration (E, B, n) and the vector space structures on the fibers Eb, is said to be a real (resp. complex) vector bundle if the following condition is satisfied:
(VB) For each b E By there exists an open neighborhood U of b in B, a finite-dimensional real (resp. complex) vector space F , and a dizeomorphism cp : U x F -+n-'(U)
such that n(cp(y,t)) = y for all y E U and t E F , and such that for each point y E U the partial mapping cp(y, * ) is an R-linear (resp. C-linear) bijection of the vector space F onto the vector space E, . This condition is equivalent to the following: (VB') For each b E B , there exists an open neighborhood U of b, an integer n (depending on b) and n mappings si : U + E of class C" such that n si = 1, for each i and such that the mapping 0
r~ : (Y, 5 ' 3
* *
3
F>H t1s1(y) +
* * *
+ Fsn(Y)
is a diffeomorphism of U x R" (resp. U x C") onto n-'(U).
For if (VB) is satisfied and if is a basis of F over R (resp. C), then the mappings y ~ s , ( y=) cp(y, a,) (1 S i 5 n) satisfy (VB'). Conversely, it is clear that the mapping cp defined in (VB') satisfies (VB) with F = R" (resp. C"). If V is an open subset of B, the induced fibration (n-'(V), V, n In-'(V)) (16.12.9) and the vector space structures on the fibers E, for b E V clearly define a structure of a real (resp. complex) vector bundle on n-'(V). This vector bundle is said to be induced by E on V and is sometimes denoted by E IV. A mapping cp satisfying (VB) is called a framing over U, and the sections si (1 5 i 5 n) over U are said to form a frame for the vector bundle E over U. Sometimes we shall call a basis of E, over R (resp. C ) a frame for E (or E,) at the point 6. For n sections si (1 5 i 5 n) o f class C" of E over U to form a frame, it is necessary and sufficient that, for each point y E U, the vectors si(y) should be linearly independent over R (resp. C ) in E,; this follows from (16.12.2.1).
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DIFFERENTIAL MANIFOLDS
Let cp : U x F + n-'(U) and cp' :U' x F -+ n-'(U') be two framings. Then the restrictions $ and $' of cp and cp' to (U n U') x F are framings over U n U', and the transition diffeomorphism $ $'-' : (U n U') x F + ( U n U') x F 0
is of the form
where XI+&) is a Coo-mappingof U n U' into GL(F). We denote by 0, the zero element of the vector space El, for each b E B. The support of a section s of E over an open set U is by definition the closure in U of the set of points y E U at which s(y) # 0,. The mapping b H o b is a Coo-sectionof E, by virtue of (VB),and it is the only section with empty support. It is called the zero section of E and is denoted by 0, or 0. Every C'-section of a vector bundle E over a closed subset S of B can be extended to a C'-section over the whole of B.For each b E B and each ub E Eb, there exists a C'O-sections of E over B such that s(b) = ub. (16.15.1.2)
The first assertion is a particular case of (16.12.11). The second follows from the first, because by virtue of (VB)there exists a Coo-sectionof E over U taking an arbitrary value at the point b. Suppose that U and cp satisfy the conditions of (VB), and let (V,$, m) be a chart on B at the point b such that V c U; also let p be an isomorphism of F onto R" (resp. C"). Then it is clear that (x-'(V), ($ x p ) q-', m + n) (resp. (x-'(V), ($ x p) cp-', m + 2n)) is a chart on E at each point of the fiber x-'(b). A chart on E obtained in this fashion is called a$bered chart. The local expression (16.3) of a section s of E over V relative to this chart and the chart (V,$, m) on B is a mapping of the form 0
0
of $(V) into $(V) x R" (resp. $(V) x C"). The mapping f : $(V) + R" (resp. f : $0 + C") is sometimes called the vector part of s (relative to the charts under consideration). The support of s is equal to the support of f, and s is of class C' if and only iff is of class C'. The dimension of the vector space Eb over R (resp. over C) is called the rank of E at b and is denoted by rkb(E). By (16.8.8), for each x E Ebwe have
15 VECTOR BUNDLES
(16.15.1.4){
dim,(E) = dimb(B)+ rk,(E) dim,(E) = dimb(B)+ 2 rkb(E)
107
if E is a real vector bundle, if E is a complex vector bundle.
It follows from (VB) that the rank rkb(E)is locally constun? on B, hence constant on each connected component of B. When rkb(E) is constant its value is called the rank of E. A vector bundle E of rank 1 is called a line-bundle. When E is a complex vector bundle over B, it is clear that the fibration of E, together with the real vector space structures of the fibers Eb (underlying their complex vector space structures) define on E a structure of a real vector bundle. This real vector bundle Eo is called the real vector bundle underlying E; we have rkb(E0) = 2 rkb(E)for each b E B. (16.15.2) If E and E' are two real (resp. complex) vector bundles, (E, B, n) and (El, B', n') the corresponding fibrations, then a vector bundle morphism (or simply a morphism) of E into E' is by definition a morphism ( f , g ) of (E, B, n) into (E', B', R') (16.12) such that for each b E B the restriction gb of g to E, is an R-linear (resp. C-linear) mapping of Eb into E>(b).When B = B' andf= l,, we say that g is a (linear) B-morphism of vector bundles. The set of linear B-morphisms of E into E' is then denoted by Mor(E, E') (cf. (1 6.1 5.8)). An isomorphism (f,g) of E onto E' is a morphism of vector bundles which is an isomorphism for the corresponding fibrations (16.12). For (f,g) to be an isomorphism it is sufficient thatfshould be a diffeomorphism and gb bijective for each point b E B (16.12.2). The local expression of a morphism of E into E' relative to the fibered charts corresponding to charts (U, $, rn) and (U', $', m') on B and B', respectively (16.15.1), is therefore of the form
where F is a C"-mapping of +(U) into $'(U') and X H A ( X )is a C"-mapping of $(U) into the set Hom(R", R"')(resp. Hom(C", C"')),(identified, if we prefer, with the set of matrices of type (n', n) over R (resp. C)). Example (16.15.3) Let B be a differential manifold and F a real (resp. complex) vector space of dimension n, and consider the trivial fibration (B x F, B, pr,) (16.12.3). By transporting the vector space structure of F onto the set
{b} x F = pr;'(b)
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XVI DIFFERENTIAL MANIFOLDS
by means of the mapping t w ( b , t), we obtain on B x F a structure of a real (resp. complex) vector bundle of rank n corresponding to the trivial fibration, because the condition (VB) is satisfied by taking U = B and cp to be the identity mapping. Such a vector bundle is said to be trivial. A vector bundle E over B is said to be trivializable if there exists a B-isomorphism of E onto a trivial vector bundle; such an isomorphism is called a trivialization of E, and it is precisely the inverse of a framing over B (16.15.1). (16.15.4)
The tangent bundle of a digerential manifold.
Let M be a differential manifold and let T(M) be the union of the (pairwise disjoint) tangent spaces T,(M) as x runs through Mt. Let oM :T(M) -+ M be the mapping which associates with each tangent vector h, E T,(M) the point x E M (the " origin " of h,). We shall show that there exists a unique structure of diyerential manifold on T(M) such that z = (T(M), M, oM) is a jibration (the tangent spaces T,(M) being the fibers) and such that the following condition is satisfied:
(TB) For each chart c = (U, cp, n) on M, the mapping (cf. (16.5.3))
of U x R" onto oi'(U) is a digeomorphism. An equivalent condition is that, for each h E R",the mapping x~(d,cp)-' * h is a Cm-section of T(M) over U. For if this is so, and if for each vector e, of
the canonical basis of R" we put Xi(x) = (d, q)-' the mapping (x, t',
.,., 5") H c n
i= 1
*
ei, and h =
c tiei,then n
i= 1
('Xi@)is a diffeomorphism, by virtue of
(I 6. I2.2), hence (TB) is satisfied. The existence of these sections shows at the same time that the vector space structures of the T,(M) and the fibration z define on T(M) a structure of a real vector bundle; this vector bundle is called the tangent vector bundle, or more briefly the tangent bundle, of the differential manifold M (cf. Section 16.12, Problem 6).
t By definition (16.5), an element of TJM) is an equivalence class of mappings of R into M, hence a subset of the set S(R,M) of all mappings of R into M; consequently T,(M) M)), and the sets {x} x T,(M) are pairwise disjoint in the set is a subset of %3(9(R, M x @(F(R, M)). As a set, T(M) is defined to be their union.
15 VECTOR BUNDLES
109
To establish the existence of the differential manifold T(M), consider an open covering (U,) of M such that for each a there exists a chart (U, , cp, ,nu) on M. Let a, f? be two indices such that U,n U, # @, which implies that nu = n, = n say; let .cp, and cpa, be the restrictions of cp, and cps , respectively, to U,n U,, and let fs, = qsU cpGi be the transition diffeomorphism. Then (16.5.7) we have 0
Dfs.((P,(x)) = (4cp,3 O (dx),cp. for x E U, n U, , and the mapping $su:
-
(U,n U,) x Rn+(U, n U,) x Rn
defined by
$&,
h) = (X, D f s u ( ' P p ( 4 )
*
h)
is evidently of class C", hence a diffeomorphism by (16.12.2.1). Further, if for each triple of indices (a, f?, y) we denote by f;,, f;,, andf: the restrictions of the mappings f,,,f y B, and j;,to (P,(Uu n up n U J , cp,(uu n u, n UJ,
and
cp,(U, n u,
UY),
respectively, then we have&{ =f,ys o f i , and by (8.2.1) the patching condition (16.1 3.1 .I) is satisfied. This establishesthe existence ofthe fibration z (16.13.3); moreover, it follows from the construction in (16.1 3.3) that this fibration satisfies (TB) and is the unique fibration with this property; for the condition (TB) implies that each mapping (16.15.4.1) is aframing of T(M) over U. This framing 1,9= is said to be associated with the chart c. (16.15.4.2) A section of T(M) over a subset A of M (16.12.9) is called a tangent vector JieId (or simply a vector field) over A. With the preceding (16.5.1) for notation, the mappings XHX,(X) = (dXq)-'* e, = O;?Je,) 1 i 5 n are C"-vector fields on U which form a frame of T(M) over U. These vector fields are called the vector fields associated with the chart c, and the frame they form is the frame ussociuted with c. Every vector field on U is uniquely expressible in the form (16.15.4.3)
X H X(x)
c ai(x) n
=
*
Xi(X),
i= 1
where the u' are n scalar-valued functions on U. For X to be of class C', it is necessary and sufficient that the ai should be of class C'. (16.15.4.4)
With the notation of (16.15.4.1), we see that
(oii(W, (cp x 1m) 0 $,
',2n)
110
XVI DIFFERENTIAL MANIFOLDS
is aJibered chart on T(M) (16.15.1), called the fibered chart associated with the chart c = (U, cp, n) on M. If (U, cp', n) is another chart on M with the same domain of definition, and if u = cp cp'-' : cp'(U) + cp(U) is the transition diffeomorphism, then the transition diffeomorphismfor the associated fibered charts on T(M) is the mapping 0
of cp'(U) x R" onto q(U) x R". If a vector field X on U is given by (16.1 5.4.3), its local expression relative to the fibered chart associated with (U, cp, n) (16.15.1.3) is the mapping (16.15.4.6) i= 1
of cp(U) into cp(U) x R". (16.15.5) When M is an open subset of R",the inverse of the framing associated with the chart (M, 1, ,n) is a trivialization of T(M), called the cunonical trivialization and given by h x H( x , zx(hx)) (16.5.2). Usually we shall identify T(M) with M x R" by means of this trivialization. A vector field on M is then of the form X H ( x , f(x)), where f is a mapping of M into R". (16.15.6) Let M, N be two differential manifolds and let f:M -+ N be a Then the mapping mapping of class C' (where r is an integer 2 1, or + a). (16.15.6.1)
(which we shall write in the more legible form
is a mapping of class C'-' (with the convention that r - 1 = 00 if r = co) of T(M) into T(N). For if (U, cp, m) and (V, $, n) are charts on M and N , respectively, such thatf(U) c v, and if F is the local expression offrelative to these charts (16.3), then the local expression of T(f) relative to the associated fibered charts (16.15.4) is the mapping (16.15.6.3)
-
(x, h)H(F(x), F'(x) h)
of cp(U) x R" into $(V) x R".Iffis of class C", then (f, T(f)) is a morphism of oector bundles (16.15.2). It is clear that
15 VECTOR BUNDLES
Ill
Hence iffis a subimmersion (resp. an immersion, resp. a submersion), so also is T(f). Also it is clear that T(1M) = lT(M), and that if g : N + P is another C'mapping, we have T(g of) = T(g) T(f). Iff is a diffeomorphism, then so is TCf), and T(f-') = T(f)-'. If M and N are two differential manifolds and pr, ,pr, are the projections of M x N onto M and N, respectively, then the mapping (T(pr,), T(pr,)) is a canonical isomorphism of T(M x N) onto T(M) x T(N) (16.6.2); usually we shall identify these two vector bundles over M x N. As an application of these results, we remark that if a Lie group G acts on a differential manifold X, then the mapping (16.10.1.2) 0
(hs, k,)Hs- k,+ h;x
of T(G) x T(X) into T(X) is of class C". Likewise, if (X, B, n) is a principal bundle with structure group G and if E = X x F is the associated fiber bundle with fiber-type F, then the mapping (16.14.7.2)
of T(X) x T(F) into T(E) is of class C". (16.15.7)
The tangent bundle of a vector bundle.
If 1 = (X, B, n) is a fibration, then (T(X), T(B), T(n))is also a Jibrution. For if U is an open subset of B such that there exists a diffeomorphism rp : U x F + II-'(U) with II o cp = pr, , then it follows that T(q) is a diffeomorphism of T(U) x T(F) onto T(n-'(U)) such that T(n) T(cp) = pr, . We recall (16.12.1) that the tangent vectors h, E T,(X) such that T(a) * h, = 0 in Tn(.JB) are said to be trertical, or along thefiber. Now let E be a vector bundle with base B and projection II.Then T(E) is also a vector bundle with base T(B) and projection T(II). Since the question is local with respect to B, we may assume that E is trivializable, and hence we reduce to the case where E = B x F, with F a vector space of dimension n and B an open set in R".Then (16.15.6) T(E) may be identified with T(B) x T(F), and the canonical trivializations of T(B) and T(F) (16.15.5) therefore finally identify T(E) with (B x R") x (F x R").Since F x R" is canonically endowed with the structure of a vector space of dimension 2n, it follows that T(E) is endowed with a vector bundle structure over T(B); but it has to be shown that this structiire is independent of the trivialization of E from which we started. 0
112
XVI
DIFFERENTIAL MANIFOLDS
Now a transition diffeomorphism from one trivialization to another is of the form
-
$ : (b, Y) I--r (b, &) Y), where A : B + GL(F) c End(F) is a C"-mapping; and then T($) is the diffeomorphism (16.15.7.1)
-
((b, h), (Y, k))H((b, h), (A(W * y, 44 k + (A'(b) * h) * y))
((8.1.3) and (8.9.1)), and our assertion follows from the fact that the mapping
+
(y, k)HA(b) * k (A'@) * h) . y is linear. Hence we have two vector bundle structures on T(E): one with base E and projection oE,the other with base T(B) and projection T(n); moreover we have oE T(n) = n oE. Furthermore, if w = oB T(n) = n 0 oE, then (T(E), B, m) is ajbration over B; but although, for a given trivialization of E, each fiber of this fibration can be endowed with the structure of a vector space of dimension 2n m, yet it is not possible to define in this way a vector bundle structure on the fibration. For the preceding calculation shows that the righthand side of (16.15.7.1) is not linear in (h, y, k), although it is linear in (y, k) and in (h, k), which correspond to the two vector bundle structures on T(E) previously described. In the particular case where E = T(M), the tangent bundle of a differential manifold M, the two vector bundle structures on T(T(M)) both have the same base T(M), but are quite distinct from each other. 0
0
0
+
(16.15.8) Let E be a real (resp. complex) vector bundle with base B and projection n, and let &(B; R) (resp. B(B; C)) be the set of C"-mappings of B into R (resp. into C), which is an R-algebra (resp. a C-algebra). Consider also the set Mor(E, E') of B-morphisms of E into another real (resp. complex) vector bundle E' over B. If u', u" E Mor(E, E'), we define u' U" to be the mapping of E into E' such that (u' + u")b = u; + u; for all b E B, and it follows from (VB) that u' + u" is a B-morphism. Again, for each function f E B(B; R) (resp.fE B(B; C)) and each u E Mor(E, E'), we define a mappingf. u : E + E' by the rule (f. u)b =f(b)ub, and f * u is again a B-morphism. Hence we have defined on the set Mor(E, E') a structure of 8(B; R)-module (resp. &(B; C)module). In particular, the set T(B, E) or T(E) of all Cm-sectionsof E may be considered as the set of B-morphisms of the trivial bundle (B x {0}, B, pr,) into E, and therefore T(E) is an B(B; R)-module (resp. an &(B; C)-module). If there exists a frame over B (in other words, if E is trivializable), then this module is free, and every frame over B is a basis of it.
+
15 VECTOR BUNDLES
113
By applying these remarks t o the vector bundle induced on n-'(U), where
U is open in B, we define a structure of &(U; R)-module (resp. $(U; C)module) on the set T(U, E) of Cm-sectionsofE over U. For each u E Mor(E, E'), the mapping S H U o s of T(U; E) into T(U; E') is &(U;R)-linear (resp. &(U;C)-linear). Remark (16.15.9) In exactly the same way we can define the notion of a real (resp. complex) vector bundle over a real-analytic manijold B; such a bundle is called a real- (resp. complex-) analytic vector bundle over B. Everything goes through as before, with the exception of (16.15.1.2). When the base B is a complexanalytic manifold, thecorresponding notion of vector bundle is of interest only when the fibers are complex vector spaces; such vector bundles are called holomorphic vector bundles. When M is a real- (resp. complex-) analytic manifold, the tangent bundle T(M) is a real-analytic (resp. holornorphic) vector bundle over M. All the developments of Sections 16. I6 and 16. I 7 (with the exception of (I 6.17.3)) extend immediately to real-analytic and holomorphic vector bundles.
PROBLEMS
1. A differential manifold M is said to be paralfelizabfe if the tangent bundle T(M) is trivializable. Show that the differential manifold underlying a Lie group is parallelizable. In particular, the spheres S1 and Snare parallelizable. 2.
Let m be a bilinear mapping of R' x
R" into R" such that
Ilm(r, x) II = l l II ~. Ilx II
(the norms being Euclidean norms).
', we have (m(y, x) I m(y', x)) = (y 1 y') IIx ]I2, and for all x, x' in R" (a) For all y, y' in R we have (m(y, x) I m(y, x')) = Ilyllz(xI x'). For each y f 0, the mapping x H m(y, x) therefore belongs to O(n,R). If (ef)lslsk is the canonical basis of Rk and if we put v(x) = m(ek,x), then the mapping (x, y) H mo(y, x) = m(y, v-'x) has the same property as m, and we have mo(ek,x) = x. In order that there should exist such a bilinear mapping m, ,it is necessary and sufficient that there should exist k - 1 elements u l ~ O ( n , R( )I s i i k - 1 ) such that u 2 = - 1 , u i u , t u , u , = O f o r 1si,j6 n and i # j (in the algebra EndCR")). For each x E S.-l, the k - 1 vectors (x, uf(x))E Tx(Sn-i) (where Tx(Sn-l)is identified with a subspace of T,(R")) are then linearly independent, and so we have k - 1 vector fields on S.- I which are linearly independent a t each point of
114
XVI DIFFERENTIAL MANIFOLDS
(b) We shall assume the existence of an (associative) algebra C, over R, of dimension 2", generated by the unit element 1 and m elements c, (1 i 6 m) such that c f = - 1, CICJ cj CI= 0 for i # j , so that the element 1 and the products el ct2 * * * ctpfor
+
1 6il < it
< ... < ipz m
form a basis of C, . The algebra C, is called the Cliffordalgebra of index m. In order that there should exist a mapping m with the property considered in (a) above, it is necessary and sufficient that there should exist a homomorphism of Cr-, into End(R"). (Observe that if r is the (finite) group generated by the imam u1of the c1 in GL(n, R), then the bilinear form (s . x 1s * y) is positive-definite on R"and invariant under I?.) aar
(c) It can be shown that for 0 5 m following table: m=O
1 2
3
4
5 7 the Clifford algebras C,,, are given by 5
6
the
7
R C H H x H M 2 0 M*(C) Me.@) Ma@)xMa@) and that Cm+8is isomorphic to C, &M16(R). Deduce that if each integer n 2 1 is expressed in the form n' * 2c(n)16d(") with 0 <= c(n) 5 3, d(n) 2 0, and n' odd, then there are p(n) - 1 = 2c(n) W(n)- 1 vector fields on S.-l which are linearly independent at each point of Sm-l. (Usethe fact that, if K is a field or a division ring, the simple m(K)-modules are of dimension n over K.) In particular, S , is a parallelkable manifold (Problem 1). It can be shown that there do not exist p(n) vector fields of class Coon Sn-* which are linearly independent at each point; in particular, the only parallelizable Spheres are S1,Sj,and S,.
+
3. The notion of a bundle of R- (resp. C-) algebras over B is defined as in (1 6.1 5.1), replacing F by a finite-dimensional R-(resp. C-) algebra, and the linear bijections in the condition (VB) by algebra isomorphisms. If X is any differential manifold, define a canonical algebra bundle structure on P(X)= J'(X, R) (Section 16S, Problem 9). 4.
Let B be a pure differentialmanifold and A = (X, B, p ) a differential fibration over B, such that X is a pure manifold. For each b E B let PI,(B, A) denote the subset of Jg(B, X) consisting of jets of sections of X over a neighborhood of 6, and let P(B, X ) denote the union of the PI,@, A). Show that P(B,A) is a closed submanifold of J'(B, h) (Section 16.12, Problem 6),and that the restrictionsto P(B, A) of the projections of the fibrations of J'(B, X) define fibrations on P(B, h) over the graph of p in X x B, B, and X, respectively. If s r, the canonical mapping of J'(B, X) into J'-$(B, Ja(B,X)) ( W o n 16.12, Problem 6), restricted to P(B, A), is a B-morphism of P(B, A) into P-'(B, p), where p is the fibration (B, P(B, A), 1~). If x'= QC', B, p') is another fibration over B, and if g :X' +X is a B-morphism, then g defines canonically a B-morphism P(g) : P(B, A) +P(B, A'). If E is a vector bundle over B and h = (E, B, p ) is the correspondingfibration, we write P(B, E) in place of P(B, h). Then P(B, E) is canonically endowed with a vector bundle structure over B, and also with a module bundle structure over the algebra bundle P(B)(Problem 3 ;show how to define a module bundle over an algebra bundle). If E = B x F is trivial, P(B, E) is canonically identified with J'(B, F).
5.
Let G be a Lie group acting differentiably on a differential manifold M. Show that if the orbit manifold M/G exists, then so does the orbit manifold T(M)/G (cf. Section 16.19, Problem 5).
16 OPERATIONS ON VECTOR BUNDLES
115
16. O P E R A T I O N S ON VECTOR B U N D L E S
In this section and the following one we shall consider only real vector bundles. The extension of the definitions and results to complex vector bundles is left to the reader. (16.16.1) Let E', E" be two vector bundles over the same base B, with projections d,n". Let E @ E" (resp. E @ E") be the disjoint union of the sets EL @ EL (resp. EL @ EL) as b runs through B, and let c (resp. p) denote the mapping E @ E" + B (resp. E @ E" + B) which sends each element of EL@ E; (resp. ELBE;) to b. If U is open in B and if s', s" are sections of E , E", respectively, over U, let s' @so (resp. s' @ s") denote the mapping b ~ s ' ( b@s"(b) ) (resp. b-s'(b) @ s"(b)) of U into E' @ E" (resp. E' @ E"). We shall show that E @ E" (resp. E' 8 E") carries a unique vector bundle structure with base B and projection c (resp. p), satisfying the following condition: for each open set U in B and each pair of Cm-sectionss', s", of E and E", respectively, over U, s' 8 s" (resp. s' @ s") is a Cm-sectionof E' @ E" (resp. E @ E") over U. The bundle E' @ E" is called the sum (or Whifneysum) of E' and E", and E' @ E" is called the tensor product of E' and E". Let us prove for example the existence of E @ E". Consider an open covering (U,) of B such that for each u there exists a finite-dimensionalvector space F; (resp. Ff)and a diffeomorphism cp: : U, x F; +n'-'(U,)
(resp. cpz : U, x Ff -+n"-l(U,))
satisfying the condition (VB). For each pair of indices a, B we have transition diffeomorphisms (1 6.13.1)
I&,:(U, n U,) x F:
(1 6.16.1.I)
I&,:(U,
n U,) x F:
-+
(U, n Up) x F; ,
-,(U,n Up) x Fi ,
which are of the form (b, t') + (b,fim(b, t')) and (b, t")w(b,f;,(b, t")), respectively, where for each b E U, n U, the mappings&',(b, . ) andf;,(b, . ) are linear. Now consider the mappings (1 6.16.1.2)
+,
: (U, n U,) x (F; @ FE)+ (U, n U,) x (Fb @ Fi)
defined by
(b, t)
(by&a(b, t)),
-
where, for each b E U, n U, ,&,(b, * ) =f;,(b, ) @f;,(b, * ) (tensor product of linear mappings). It is immediately seen (by taking bases in FL and Ff)that are diffeomorphisms and satisfy the patching condition (16.13.1 .I). the
116
XVI
DIFFERENTIAL MANIFOLDS
They therefore define (16.13.3) a fibration ( E @ E , B, p), and it is clear that E' €3 E is thus endowed with a vector bundle structure satisfying the condition on the sections stated above. Conversely, suppose that E 63 E" is endowed with a vector bundle structure satisfying this condition. With the notation used above, let (eAi) be a basis of FA and (eb) a basis of Ff. Then the sections bt+ &(b, eLi)(resp. bt+ q: (b, eij))form aframe (&) (resp. (sij))of E (resp. E ) over U,, and the condition on the sections shows that the sLi @ sij form a frame for E' €3 E" over U, . Let q, : U, x (FA €3 Ff) + p-'(U,) be the diffeomorphism corresponding to this frame. It is then immediately verified that for any two indices a, the transition diffeomorphisms
(U, n Up) x (FA €3
+
(U, n Up) x (Fb €3 Pi)
correspondingto the q. are precisely the mappings (16.1 6.1.2). This establishes the uniqueness of the vector bundle structure on E' €3 E". It follows moreover that if U is any open set in B such that n'-'(U) and n"-'(U) are trivializable, then T(U, E @ E") is an &(U; R) module isomorphic to E) E"). If F', F" are two vector bundles over B and if u' : E' --f F' and u" : E" -+ F" are two B-morphisms, then one shows in the same way that there exists a unique B-morphism u' €3 u" : E' €3 E" -+ F' €3 F" such that, if s' and s" are sections of E', E", respectively, over an open set U in B, and if u;, u: and (u' €3 u"), are the restrictions of u', u" and u' €3 u" to n'-'(U), n"-'(U), and p-'(U), respectively, then we have
w,
w,
(u' @ 24")"
0
(s' 0 2") = (24;
0
s') @ (246 s"). 0
The restriction of u' @ u" to a fiber (E' @ = EL @ EL is the tensor product u; @ ug of the linear mappings uL : Ei -+ FL and u; : E, + F; . If u' and u" are B-isomorphisms, then so is u' @ u". The proofs of the corresponding assertions for E' 0 E" are analogous and simpler. For each open U c B such that n'-'(U) and n"-'(U) are trivializable, T(U, E' 0 E") is an &(U; R)-module isomorphic to
r(u, E') 0 r(u, E"). There are canonical injective B-morphismsj' : E' + E' 0 E", j" : E" -+ E' 0 E" and canonical surjective B-morphisms p' : E' 0 E" -+ E', p" : E' 0 E" E" such that, with the same notation as above, -+
j;
p;
0
0
s' = s' 0 OE,, ,
(s' 0 s")
= s',
j;
p';
o
S" =
0,. 0 S",
(S' 0 s") = sir.
Furthermore, if F is a vector bundle over B and if u' : E' F and u" : E" F are B-morphisms, then there exists a unique B-morphism u : E' 0 E -+ F -+
-+
16 OPERATIONS ON VECTOR BUNDLES
117
such that uu (5’ 0 s ” ) = u; s‘ 4-u; 5“. We write u = U’ + u”. If G’, G” are vector bundles over B and if u’ : G’ -+ E’ and u” : G” E” are B-morphisms, we denote by u‘ 0 u” the morphism ( j ’ u‘) ( j ” 0 u”) of G‘0 G” into E’ 0 E”. For each b E B we have (u’ + u ” ) ~= u; + u:, (u’ 0 u”), = u; 0 u; . Finally we remark that the fibration (E’ 0 E”, B, a) is isomorphic to the fiber product E‘ x E” defined in (16.12.10). 0
0
0
+
0
(1 6.16.2) The definitions of Whitney sum and tensor product generalize immediately to the case of any finite number of vector bundles over B. In particular, for each integer m > 0, we define a multiple mE (resp. a tensor power Esm) of a vector bundle E as the sum (resp. tensor product) of m copies of E. We use an analogous notation for B-morphisms. m
A E is defined: this m space has for its underlying set the disjoint union of the sets A E, as b runs m m through B, and its projection 1 : A E B sends each element of A Eb to the Likewise, for each integer m > 0, the exterior power
-+
point b E B. For each sequence (sj), of m sections of E over an open set U in B, we denote by s1 A s2 A . . * A s, the mapping bHS,(b) A
S2(b) A
* ‘ *
A
S,(b);
the vector bundle structure on A E is defined by the condition that, for each sequence (sj), of m sections of class Cmof E over an open subset U of B, m
m
the mapping s1 A s2 A ... AS,,, is a Cm-section of /”\ E. If n is the projection E -+ B, and if we are given an open covering (U,) of B together with diffeomorphisms rp, : U, x F, -+n-’(U,) satisfying (VB), then the fiber bundle
...
A E may be constructed as follows: we take the transition homomorphisms
(b, t)t+(b,fS,(b, t)) corresponding to the rp,,
and we form the mappings m
(b, t
) (b,~ gs.(b, t>>, where for
each b E U, n U,, gp,(b,
-1 = A f&b, *I.
ai,A ai2A . - .A aim
of
A E such that i, < i, m
c
m
< i, form a frame of A E over U.
If u : E -+ F is a B-morphism of vector bundles, there exists a unique B-
morphism we have
A u : A E A F such that, if s, , . . . ,s, m
m
m
-+
m
( A U )
0
(sl A S2
A
. ‘ A- Sm)
= (U
0
S,) A (U
0
S,)
are sections of E over U,
A
. * *
A(U
0
Sm).
118
XVI DIFFERENTIAL MANIFOLDS
For each b E B we have m
(A U)b = A U b . If u is an isomorphism, then so is m
A Conventionally E@O,or
m
0
A E, or I, denotes the trivial line-bundle B x R.
With the notation of (1 6.1 6.1), let Hom(E', E ) denote the disjoint union of the sets Hom(E; , EL) (the set of all linear mappings of E; into EL) as b runs through B, and let q : Hom(E', E ) + B be the mapping which sends each element of Hom(E;, EL) to b. If s' (resp. u) is a section of E (resp. Hom(E, E))over an open subset U of B, let u(s') or u * s' denote the sec) ) E" over U. Then there exists on Hom(E, E") a tion b ~ ( u ( b ) ) ( s ' ( b of unique structure of a vector bundle over B with projection q, satisfying the following condition: for each open subset U of B and each pair of sections s', u of E and Hom(E, E"), respectively, over U, if s' and u(s') are Coo-sections of E and E", respectively, over U, then u is a Coo-sectionof Hom(E', E") over U. The proof is analogous to that given in (16.16.1) and we shall merely sketch it. With the same notation as before, consider this time the mappings (1 6.1 6.3)
m,, : (U, n U,) x Hom(F: ,F:) --., (U, n U,) x Hom(Fi ,Fj) of the form (byu ) H ( ~ ,h,,(b, u)), where for each b E U, n Us we have h,,(b, u) =f[,(b, 0 u of&(b, - ) - I . We leave it to the reader to check the details, which are analogous to those in (16.16.1). Consider in particular the trivial bundle B x R (16.15.3). If E is any vector bundle over B, the bundle Hom(E, B x R) is called the dual of the bundle E and is denoted by E*. If s and s* are sectionsof E and E*, respectively,over an open U c Bywe write (s, s*) or (s*, s) in place of s*(s). If ( Q ~ is )a frame of E over an open set U c B (16.15.1), there exists a unique frame i s n of E* over U such that (a?,a j ) = 0 if i # j and (a:, ai>= 1 for all i. The frame (a:) is called the dual of (ai). If u' : F' --+ E' and u" : E" * F" are two B-morphisms, there exists a unique B-morphism w : Hom(E, E") --+ Hom(F, F"), denoted by Hom(u', u"), such that for each section f of Hom(E', E") over an open U c B and each section s' of F' over U, we have (in the notation of (16.16.1)) (wu f)(s') = u; f(u& s'). If u' and u" are isomorphisms, then so is Hom(u', u"). In particular, for each B-morphism u : E --., F of vector bundles, we define its transpose 'u : F* --+ E* by the condition 0
)
0
0
0
('u"
0
s*, r ) = (s*, uu 0 r )
for any two sections r of E and s* of F* over an open subset U of B.
~
~
~
16 OPERATIONS ON VECTOR BUNDLES
For each b E B we have Hom(u', u y b = Horn(#:, u:) : f in particular (%)b = '(ub).
b H
ui
0
fb
0
119
ub , and
(16.16.4) With the notation of(16.16.3), for each open U c B and each C"section u ofHom(E, E") over U, let ti denote the mapping (by t')H(b, ( W w " ofx'-'(U) into n"-'(U). Then is a U-morphism ofn'-'(U) into n"-'(U), and the mapping U H ~is an isomorphism ofthe 8(U; R)-module T(U, Hom(E, E")) onto the 8(U; R)-module Mor(n'-'(U), n"-'(U)).
Since the question is local with respect to B, we reduce immediately to the situation in which U = B and E , E" are trivial; in which case Hom(E', E") is also trivial, and then the result is obvious. (16.1 6.5) We denote by T:(E) the vector bundle ((E*)@4)6 (E@3. This bundle is called the tensor bundle of type ( p , q) over E ( p is the contravariant index and q the covariant index). Conventionally we put TA(E) = E, TY(E) = E*, Tg(E) = B x R . A section of T;(E) over a subset A of B is sometimes called a tensor$eld of type ( p , q ) on A (relative to the bundle E). If (ai)' is a frame of E over the open set U c B , and (a:), - the dual frame (1 6.16.3), then the np+4tensor fields a:,
6a:2 €9
* *.
0 ak*,6a j , 6 aj26 *
-
*
6ajp
form a frame of T:(E) over U, called the frame induced by (a,). If u : E + F is a B-isomorphism, we define T:(u) to be the B-isomorphism of T:(E) onto T:(F) induced from u by transport of structure, that is to say the tensor product (('u-')@~) 0 (uBP).(If p > 0 and q > 0, we cannot define Ti(#) for an arbitrary B-morphism u, because 'u is a B-morphism of F* into E*, not of E* into F*.) Remark
A
(1 6.1 6.6). It is also necessary to consider morphisms such as u' 0 u", u, Hom(u', u") in the context of isomorphisms of vector bundles over different base manifolds. For example, let E , E" be two vector bundles over B, , and F', F" two vector bundles over B , ; let (f,u') be an isomorphism of E' onto F', and (f,u") an isomorphism of E onto F", corresponding to the same diffeomorphism f of B, onto B , . Then (f,Hom(u', u")) will be the isomorphism of Hom(E, E ) onto Hom(F', F") which sends hb E Hom(E;, E") to the element g,,,, : y;(b)++u:(hb(u;-'(y>(b)))) of Hom(F;, E). We leave it to the reader to write down the definitions in the other cases.
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PROBLEMS 1. For each differential manifold B, let I or I(B) denote the trivial real line-bundle B x
R;
it is an identity element for the tensor product of vector bundles over B. Likewise Ic or Ic(B) denotes the trivial complex line-bundle B x C. The sum ml (resp. mIC) of m copies of I (resp. Ic) may be identified with the trivial bundle B x Rm(resp. B x Cm).
,= G,,,,(R) and the subset Un,= Urn,,(R) of the (a) Consider the Grassmannian Gm, trivial bundle nI(G., p ) = Gn, x Rn consisting of pairs (V,x ) where V is a vector subspace of dimensionp in R", and x E V. Show that U , is a vector subbundle of nI(G., p), isomorphic to the bundle associated to the principal bundle L,,, (16.1 4.2) with structure group GL(p, R), with fiber-type Rp (for the canonical action of GL(p, R) on Rp on the left). Define in the same way the complex vector bundle Urn..(C) over G-,p(C).The bundle U., ,(R) (resp. Urn. ,(C)) is called the canonical (or tautological) vector bundle over Gm,,(R) (resp. Gm, ,(C)). In particular, when p = 1 (so that Gm,,(R) = Pn-l(R). Gn,,(C) = Pn-,(C)), we write Ln-,(I) or Ln-,,&) in place of U,,,,(R), and Ln-,,&) in place of U,,.,(C). The principal bundle P(D,(R)) (resp. P(D,(C))) (Section 16.14, Problem 5) may be identified with the complement of the zero section in Lm-,(I) (resp. Lm-I.c(l)). We denote by L.-,(k) (resp. Lm-,.c(k)), for each integer k > 0, the tensor product of k copies of Lm-l(l) (resp. Lm-l.c(l)). (b) If n 2 1, the bundles L.(2) are trivializable as real-analytic bundles, but L.(1) is not trivializable as a differential bundle. (Argue by contradiction, by lifting a section which is # O at each point of the space Rn+'- {O}; show that, for each x E P.(R), there exists a real-analytic section of L.(1) over the whole of P,(R) which is # O at the point x.) On the other hand, the holomorphic bundles L.,,(k) for k 2 1 are pairwise nonisomorphic and admit no holomorphic section over P,,(C) other than the zero section (same method, using Section 9.10, Problem 5). However, for each x E P.(C) there exists a real-analytic section of Ln.&) over P.(C) which does not vanish at x. (c) If we endow R" with the Euclidean scalar product, then the mapping which sends each p-dimensional subspace V c R" to its orthogonal supplement VL, of dimension n - p , is a real-analytic isomorphism w of Gnv onto G,,." - ~Show . that the direct sum Un. @ w*(U., "-,,) is a trivializable bundle over G,,,p . Define in the same way a holomorphic isomorphism w of (2". ,(C) onto (2". "- p(C), but show that the analogous assertion about the canonical bundles is false (consider the isotropic vector subspaces of C"). On the other hand, there exists a real-analytic isomorphism w o of Gnvp(C) onto Gn,"-,,(C) such that U,,, ,(C) @ wt(U., n-p(C))is trivializable as a real-analytic bundle.
,
2.
(a) Let L be a real or complex line-bundle over a differential manifold B. Show that the tensor product L 0L* is trivializable (cf. (16.1 8.3.5)). Likewise for real- or complexanalytic line-bundles over a real-analytic manifold, and for holomorphic line-bundles over a complex manifold. By reason of this fact, vector bundles of rank 1 are also called invertible vector bundles; we write L""" = L*, L @ ( - k =)(L*)@' for all k > 0, with the convention that Leo = I. (b) The tensor product of k copies of L.(I)* (resp. L.,c(l)*) (Problem 1) is denoted by L.(-k) (resp. Ln,&-k)). Show that for each k 2 I and each x E P.(C) there exists a holomorphic section of Ln,,=(--k)over PJC) which does not vanish at x.
17 EXACT SEQUENCES, SUBBUNDLES, QUOTIENT BUNDLES
121
(c) Let L be a (differential) real line-bundle over a differential manifold B. Show that L 0 L is trivializable. (Observe that if (f,),is, is a finite sequence of sections of L over B, t h e n z f , Of, is a section of L 0 L over B which vanishes only at the points where J
all thef, vanish.) What is the analogous result for complex line-bundles?
17. E X ACT SEQUENCES, SUBBUNDLES, A N D Q U O T I E N T BUNDLES
All vector bundles considered in this section have the same base B, and all morphisms are B-morphisms. (16.17.1) Let E be a vector bundle, E' a subset of E. For each b E B let E, = E' n E, . Then E' is said to be a subbundle of E if the following two conditions are satisfied :
(I) For each b E B, EL is a vector subspace of E, . (2) For each b E B, there exists an open neighborhood U of 6, a frame (sl, . . ., s,) of E over U and an integer m S n such that, for each y E U, the vectors sl(y), . . . , s,,,(y) form a basis of EI, . Under these conditions: (i) E' is a closed submanifold of E. (ii) If x' is the restriction to E' of the projection x : E + B, then the space E together with the fibration (E', B, x ' ) and the vector space structures on the fibers EL is a vector bundle over B (which justifies the terminology introduced above). It is enough to prove these assertions when E is replaced by E n x-'(U), where U is as in (2) above; but then assertion (i) follows from (16.8.7(ii)), and (ii) is obvious, having regard to the condition (VB'). It is clear that the canonical injection j : E' bundles.
+E
is a morphism of vector
Example (16.17.1.1) Let (X, B, 71) be a fibration, and consider the tangent bundle E = T(X) of X. For each x E X, let V, c E, = T,(X) be the subspace of oertical tangent vectors at the point x (16.12.1). Then the union V(X) of the V, is a subbundle of T(X). For, since the question is local with respect to X, we may assume that X = B x F ; if x = (b, z), let U be an open neighborhood of b in B, let W be an open neighborhood of z in F, and consider tangent vector fields Yi(1 5 i S n ) on U (resp. Z j ( I C J 5 m) on W) forming a frame
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DIFFERENTIAL MANIFOLDS
of T(B) (resp. T(F)) over U (resp. W).Identifying T(X) with T(B) x T(F), we have a frame of T(X) over U x W, obtained by taking the vector fields (b', 2')- (Y,(b'),0) and (b', 2')- (0, Zj(z')). For each x' = (b', 2') E U x W, the vectors (0, Zj(z')) form a basis of V ,, which proves our assertion (cf. Section 16.19, Problem 4). (16.17.2) With the hypotheses and notation of (16.17.1), let EL = &,/EL for all b E B, and let E" be the disjoint union of the sets E; as b runs through B. Let A" : E" + B be the mapping which sends each element of EL to 6, and let p : E + E" be the mapping whose restriction to Eb is the canonical mapping Eb + E,",for each b E B.Then there exists on E" a unique structure of a vector bundle over B with projection A", such that p is a morphism of vector bundles. For if U is an open set in B satisfying condition (2) of (16.17.1), and if F,, is ...,s,(y), for each y E U, then the vector subspace of E,, spanned by ~,,,+~(y), it is clear that the union F of the spaces F,, for y E U is a subbundle of n-'(U), and the restriction q of p to F must be an isomorphism of F onto n"-'(U), in view of (16.12.2.1). From this follows the uniqueness of the bundle structure on E". On the other hand, if we put sib) = p(s,(y)) for m + 1 g k n and all y E U, it follows from above that (s:+~ , . . ., s,") must be a frame of E" over U. The existence of a vector bundle structure on E" possessing these frames is then verified by the same method as in (16.16.1), and we leave the details to the reader.
The vector bundle E" thus constructed is denoted by E/E' and is called the quotient of E by the vector subbundle E . It is clear that the canonical morphism p : E -P E" is a submersion (16.12.2.1). With the notation and hypotheses of (16.17.1) and (16.17.2), there r : E" 4E such that p r = I,,,. Consider a locally finite denumerable open covering (U,) of B such that each U, satisfies condition (2) of (16.17.1). Let E, = n-'(U,), E L = n"-'(U,), and let p, : E, Ef be the restriction of p; then it is immediately clear that there exists a morphism r, : E; + E, such that p , 0 r, = 1,:. Let (f.)be a partition of unity subordinate to the covering (U,) and consisting of C"-functions (16.4.1). For each index a,let r; : E" -+ E be the morphism whose restriction to E L isf,(b)(r, I Ei) when b E U,, and 0 when b $ U, . Then the morphism r = C r; has the required Propmy(16.17.3)
exists a morphism
0
-+
(I
It follows immediately that the morphism j + r : E' @ E" -+ E is an isomorphism; F = r(E") is a subbundle of E, such that EL and Fb are supplementary subspaces of Eb for each b E B. Every vector subbundle of E with this property is called a supplement of E in E.
17 EXACT SEQUENCES, SUBBUNDLES, QUOTIENT BUNDLES
123
(16.17.4) Let E, F be two vector bundles over B, and u : E -+ F a B-morphism. For each b E B let UI, : E, + F, denote the linear mapping which is the restriction of u to E,. The rank rk(ub)is called the rank of u at b. Put Nb = Ker(Ub), 1, = Im(#b) and
N=
U NbCE,
beB
I=
U IbCF.
boB
With this notation we have: (16.17.5) (i) The mapping bwrk(ub) of B into the discrete space N is lower semicontinuous. (ii) The following conditions are equivalent:
(a) The function bwrk(ub) is continuous (and therefore constant on each connected component of B). (b) N is a vector subbundle of E. (c) I is a vector subbundle of F. Since the question is local with respect to B, we may assume that E and F are trivial, of the form B x P and B x Q, respectively, where P and Q are vector spaces. Then u is of the form (b, t)H(b, f(b,t)) where, for each b E B, f(b, ) is a linear mapping of P into Q, and moreover the elements of the matrix of f(b, * ) relative to bases of P and Q are C"-functions on B. Since rk(ub)is the rank of f(b, * ), it is the largest integer p such that there exists at least one nonzero p x p minor in the matrix of f(b, * ). Assertion (i) now follows immediately. Since rk(ub) = dim(1,) = codim(Nb), it is clear that the conditions (b) and (c) in (ii) each imply (a). Conversely, suppose that (a) is satisfied; since the question is again local with respect to B, we may assume in addition that rk(ub)= p is constant on B, and that there exists a basis (ei)l of P, and a basis (e;),sjg,,, of Q , such that for each b E B, 1, is a supplement of the subspace of {b} x Q generated by the (b, e))for p + 1 5 j 5 m, and such that the vectors (b, f(b, ei))(1 5 i 5 p ) form a basis of I b . If f(b,e i )=
m
C
j= 1
aij(b)ej
for
1 S i s n,
this signifies that the p x p minor A(b) in the matrix (aij(b)),formed by the elements with both indices S p , is nonzero. Then the fact that I is a vector subbundle of F follows because the sections b w ( b , f(b, ei)) b- (b, e;)
5 i 5 P), (p+ 1 5 j S m ) (1
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DIFFERENTIAL MANIFOLDS
form a frame of F satisfying condition (2) of (16.17.1) relative to I. On the other hand, Cramer's formulas show that f(b, e k ) =
P
(P + 1 5 k 5 n),
C Pki(b)f(b,ei) i= 1
where the Pki are of class C"; and the fact that N is a vector subbundle of E follows because the sections
(p
Pki(b)ei)
+ 1 5 k In),
(1 5 i s p )
form a frame of E satisfying condition (2) of (16.17.1) relative to N. When the conditions of (16.17.5(ii)) are satisfied, the vector bundles N and I are called, respectively, the kernel and the image of u and are written Ker(u) and Im(u). If p : E -,E/Ker(u) and j : Im(u) + F are the canonical morphisms, then the unique mapping v : E/Ker(u) + Im(u) such that u = j o v o p is an isomorphism of vector bundles (16.15.2). (16.17.6) Let E 5 F G be a sequence of two morphisms of vector bundles over B. The sequence is said to be exact if, for each b E B, the sequence
Eb-
Ub
Fb+
ub
Gb
of linear mappings is exact. A finite sequence UI
E,+E,
-+*.*-+
- -
E,-
5 E,
of morphisms of vector bundles over B is said to be exact if each of the Uk + 1 Uk+2 sequences Ek Ek+l Ek+z(0 5 k 5 n - 2) is exact. If we denote by 0 the trivial bundle B x {0}, then a morphism u : E -,F of vector bundles over B is injective (resp. surjective) if and only if the sequence 0 -,E F (resp. E 5 F + 0) is exact. For each vector subbundle E' of E, the
:
sequence
o -,E! L E Z E / E ~-,o (with the notation of (16.17.1) and (16.17.2)) is exact. (16.17.7) If E 5 F G is an exact sequence of morphisms of vector bundles over B, then u and v satisfy the equivalent conditions of (16.17.5(ii)), and the bundles Im(u) and Ker(v) are therefore deJined and equal.
18 CANONICAL MORPHISMS OF VECTOR BUNDLES
125
It is enough to observe that rk(ub) + rk(ub) = dim F, is a continuous function of 6 ; since rk(u,) and rk(ub)are lower semicontinuous as functions of b, it follows that they are continuous. In particular, for each injective (resp. surjectiue) morphism u : E + F, Im(u) (resp. Ker(u)) is defined, and E is isomorphic to Im(u) (resp. F is isomorphic to E/Ker(u)). (16.17.8) If E‘, E” are two vector bundles over B, then E’ (resp. E”) may be identified with a subbundle of E‘0 E” by means of the canonical injection j’ (resp.j”)(16.1 6.1), and also with a quotient bundle of E @ E by means of the canonical surjection p” (resp. p’). (16.17.9) The properties of tensor products and spaces of linear mappings relative to exact sequences in the category of vector spaces give rise to analogous properties of vector bundles: if E’ -+ E + E” is an exact sequence of vector bundles over B and if F is a vector bundle over B, then the sequences
E’ @ F + E @ F + E @ F, Hom(E, F) + Hom(E, F) -+ Hom(E’, F), Hom(F, E‘) + Hom(F, E) + Hom(F, E”) are exact. Examples (16.17.10) Let E be a vector bundle over B and let m be an integer > O . For each b E B, let S,(E,) (resp. A,(Eb)) be the subspace of EFm consisting of symmetric (resp. antisymmetric) tensors. By considering a frame of E over an open subset U of B, and the corresponding frame of E@“, it is immediately verified that the union S,(E) (resp. A,(E)) of the S,(E,) (resp. the A,(Eb)) is a vector subbundle of Earn,called the bundle of symmetric (resp. antisymmetric) (contrauariant) tensors of index m.
IS. CANONICAL MORPHISMS OF VECTOR BUNDLES
Except in (16.18.5), we shall again restrict our discussion to real vector bundles, and leave it to the reader to develop the corresponding results for complex vector bundles.
126
XVI DIFFERENTIAL MANIFOLDS
Let E , E , F be three vector bundles over the same base manifold B, and let n', z", IL be their projections onto B. Consider the vector bundle E' 63 E" and its projection cr. A B-morphism u of thefibration ( E @ E,B, a) into thefibration ( F , B, n) is said to be bilinear if, for each b E B, the restriction ub : Ei @ EL + Fb is a bilinear mapping. For each open subset U of B, the mapping (s', s") I+ uu 0 (s' 0 s") is an b ( U ;R)-bilinear mapping of T(U, E ) x T(U, E") into T(U, F). Multilinear B-morphisms are defined in the same way. In particular, there exists a unique bilinear B-morphism
(16.18.1)
m : E'@ E" +E' @ E"
such that, for each pair of sections s', S" of E and E" over an open set U c B, we have m(s' @ 2") = s' @ s". This follows immediately from the definitions of the vector bundles E @ E and E' @ E (1 6.16.1) and the local definition of a morphism of fibrations (16.13.5). The morphism m is called canonical. Furthermore, given any bilinear B-morphism u : E' @ E" + F, there exists a unique linear B-morphism v : E' @ E" + F such that u = v m. Here again, by virtue of (16.1 3.5), the proof reduces to the case of trivial bundles and rests in the last analysis on the corresponding algebraic proposition and the fact that polynomials are functions of class C" in R". 0
(16.18.2) The preceding argument applies in the same way to all the canonical linear or multilinear mappings defined in algebra, and provides canonical linear or multilinear B-morphisms correspondingly. Moreover, whenever a canonical linear mapping defined in algebra is bijective for finitedimensional vector spaces, the corresponding B-morphism is an isomorphism (1 6.15.2).
We shall restrict the followingdiscussion to defining the most important of these canonical B-morphkms by characterizing their effect on sections. First of all, we have the isomorphisms of associativity and distributivity:
which maps a section s1 €3 (s2 €3 s3) over an open set U to the section ($1
(16.8.2.3)
@ s2) €3 s3 ;
Hom(E @ F, G) + Hom(E, Hom(F, G))
18 CANONICAL MORPHISMS OF VECTOR BUNDLES
127
such that if s‘, s”, u are sections of E, F, Hom(E 0 F, G), respectively, over an open set U, then the image of u is the section v of Hom(E, Hom(F, G)) such that (v(s”))(s’) = u(s’ 8 s”) ;
Hom(E‘0 E”, F) + Hom(E, F) 0 Hom(E“, F),
(1 6.18.2.4)
such that if s’, s”, u are sections of E’, E”, Hom(E’ 0 E”, F), respectively, over an open set U, then the image of u is the section v‘ 0 v”, where v‘(s‘) = u(s‘), v”(s”) = u(s”);
(16.18.2.5)
Hom(E, F’) 0 Hom(E, F”)
-+
Hom(E, F’ 0 F“)
such that if s, v’, v” are sections of E, Hom(E, F’), and Hom(E, F”), respectively, over an open set U, then the image of v’ 0 v” is the section v such that v(s) = v‘(s) 0 v“(s); (16.18.2.6) Hom(E’, F’) C3 Hom(E”, F”)
+ Hom(E’ 8 E”,
F‘ 0 F”)
such that if s’, s”, u’, u“ are sections of E , E“, Hom(E, F’), and Hom(E”, F”), respectively, over an open set U, then to u’ 8 u” there corresponds the section u such that u(s’ 8 s”) = u’(s’) 8 ~ “ ( s ” ) . (16.18.3) (16.18.3.1)
Second, we have the isomorphisms related to duality: E + E**
such that, if s, s* are sections of E, E*, respectively, over an open set U, then to s there corresponds the section a of E** such that (s*, s) = (3, s*); (1 6.18.3.2)
Hom(E, F) + Hom(F*, E*)
such that, if r , s*, u are sections of E, F*, and Hom(E, F), respectively, then to u there corresponds the section ‘u such that (u(r), s*) = ( r , ‘u(s*)); (16.18.3.3)
(E 8 F)* + E* 8 F*,
which is a particular case of (16.18.2.6); (16.18.3.4)
E* 0 F + Hom(E, F)
128
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DIFFERENTIAL MANIFOLDS
such that, if r, r*, and s are sections of E, E*, and F,respectively, then r* @ s is mapped to the section u of Hom(E, F) such that u(r) = ( r , r*)s. In particular, we have a canonical isomorphism of End(E) on E* @ E, under which the morphism lE corresponds to a section of E* @ E = Ti(E), canonically associated with E, and called the Kronecker tensor field; if (aJi is a frame of E over an open set U, and (a:) the dual frame (1 6.16.2), then the Kronecker tensor field is
C 6; a: @ aj , i. i
where S i = 0 if i # j and 6; = 1 for all i. Finally, there is a canonical B-morphism (16.18.3.5)
E*@E+B x R
such that if s*, s are sections of E*, E, respectively, then s* Q s is mapped to the section (s*, s) of the trivial bundle. These morphisms may be combined with each other and with the preceding ones to give canonical morphisms between tensor bundles over a vector bundle E (16.16.5). First of all we have tensor multiplication, which is a bilinear morphism
which maps a pair of sections
to the section
This bilinear morphism corresponds canonically (16.18.1) to the associativity isomorphism (1 6.18.2.1) :
Next, we have a canonical isomorphism
which is obtained fiom (16.18.3.4) by using the canonical isomorphism
18 CANONICAL MORPHISMS OF VECTOR BUNDLES
(Tp))*
(16.18.3.9)
+
129
T:(E)
which in turn comes from (16.18.3.3), (16.18.3.1), and the associativity isomorphisms. From all this it follows that a bilinear B-morphism of into T:(E) corresponds canonically to a global Cm-sectionof the tensor bundle T;: 2 3 9 . Letp,qL 1 andletj,kbesuchthat 1 SjSp, 1 $ k s q . T h e n t h e r e i s a canonical linear morphism which generalizes (1 6.18.3.5), called contraction of the indices j and k: c i : TXE) + T:I:(E)
(1 6.18.3.10)
- €3 t: €3 t , €3 €3 t pto the section €3 t:- 1 €3 t:+ €3 - €3 t,*€3 tl €3 - €3 tj- €3 tj+ 63
which maps a section t: €3 *
* * *
1
* *
*
*
1
1
*
.
' ' €3 t p
In particular, the bundle End(E) = Hom(E, E) is identified with T:(E) = E* €3 E, and a B-morphism u : E +E is identified with a global section of T:(E) (16.16.4). The contraction c:(u), which is a section of the trivial linebundle B x R,is called the trace of u and is written Tr(u); its value at b E B is the number Tr(ub). Likewise, the section u(s) of E, where s is a given section of E, is identified with c:(u @ s) (u @IS being a section of T:(E)). (16.18.4) We shall now consider B-morphisms connected with the exterior algebra. First of all, there is a canonical B-morphism
A : E@"+~"\E
(1 6.18.4.1)
such that, if sl, . . .,,s are sections of E over an open set U c B, the section s1 @I s2 €3 * * * @ s, is mapped to the section s1 A s2 A * * A sm, so that 1 is surjective. There exists a canonical B-morphism u : E@"'+ E@'",whose image is a supplement of Ker(1) (16.17.3), which is such that u 0 u = u and which maps a section s1 €3 . €3,s of E@"'to its antisymmetrization
-
-
(here 6, denotes the symmetric group of all permutations of the set { 1,2,
m
. ..,m}). This B-morphism enables us to identify canonically A E with
the subbundle A,(E) of E@"'(16.17.10).
130
XVI DIFFERENTIAL MANIFOLDS P
4
Next, let s, t be sections of AE, A E over U (where p
2 1, q 2 1). We
denote by s A t the section of A E over U such that (s A t)(b) = s(b) A t(b) for all b E U; s A t is called the exterior product of s and t . We then have a canonical B-morphism P+4
(1 6.18.4.2)
A E the direct sum of the bundles P A E for 0 5 p 5 rk(E), then r(U, A E) is the direct sum of the T(U, A E)
which maps s @ t to s A t . If we denote by P
and is endowed by the B-morphisms (16.18.4.2) with a structure of an anticommutative graded &(U;R)-algebra. If E is trivializable over U, so that T(U, E) is a free 8(U; R)-module, then T(U, A E) is canonically identified with the exterior algebra A (T(U, E)) of this module. Next, there is a canonical isomorphism
6:
(1 6.18.4.3)
i;\(~*)+ (AE)*
such that, if sl, ...,,s are sections of E and sr, . ..,:s are sections of E* over an open set U c B, we have (sl A s2 A
(16.18.4.4)
A
*
s, 6(s:
A
s;
m
A*
A)):s
= det((s, ,s;).
m
We shall henceforth identify A(E*) with (A E)* by means of 6, and we shall m
write them both as A E*. P Finally, if s is a section of E and z* a section of A E* over U (wherep 2 l), we denote by i(s)r* or i, * z*, the section of
P- 1
A E* whose value at each b E U
is the interior product s(b) J z*(b) in A EE; this section is called the interior product of s and z*. If sl, . . . ,sp-l are sections of E over U, we have P- 1
(16.1 8.4.5)
(i(s)z*, s1 A s2
A
A
sPd1)= (z*, s A s1
and in particular, when p = 1, (1 6.18.4.6)
i(s)s* = (s*, s)
A
.**
A
sP-J
18 CANONICAL MORPHJSMS OF VECTOR BUNDLES
for any section s* of E*. For arbitrary p and sections,:s we have
131
..,,s*,of E* over U,
(with the usual convention that the symbol under the circumflex is to be omitted). Also we have (1 6.1 8.4.8) (16.18.4.9)
i(s) 0 i(s) = 0,
+
i(s)(z,*A 2 ); = (i(s)z,*)A z,* (P
I)PZ,* A (i(s)z,*),
where zf,z$ are sections of A E*, A E*, respectively, over U. This last formula may be expressed by saying that for each section s of E over U, i(s) is an antiderivation of degree - 1 of the b(U;R)-algebra T(U7A E*). We have then, for each integer,p 2 1, a canonical B-morphism (16.18.4.10)
4
LIE*
E @I (AE’)
such that, if s and z* are two sections as above, then s 69 z* is mapped to the section i(s)z*. (1 6.18.5) Finally, we consider B-morphisms connected with extension of scalars (from R to C). The (real) vector bundle E may be canonically identified with the tensor product E @ (B x R); it follows immediately from the definitions that the tensor product E @I (B x C) (wherein B 6C is considered as a real vector bundle) admits a canonical structure of a complex vector bundle of the same rank as E at each point of B. This complex vector bundle is called the complexijication of E and is written E(c,. We have a canonical injective B-morphism (16.18.5.1)
E --* Em
7
under which a section s of E over U is mapped to the section s @I 1 of E(c,. We remark that the real vector bundle underlying E(c, is the sum E @I i E of its two real subbundles E and iE. Let E, E’ be two real vector bundles over B. Then we have canonical isomorphisms
132
XVI DIFFERENTIAL MANIFOLDS
(16.18.5.3) (16.18.5.4)
(I6.18.5.5)
(E*)(C, -.+ (E(C))*,
(b)
+
it\(E(C)),
(C)
which on the sections reduce to the usual linear mappings. For example, if u is a section of Hom(E, E ) over U, then to u @ 1 there corresponds under (1 6.18.5.3) the section v of Hom(E(,, , E;,)) defined as follows: if s is a section of E and 5 a section of B x C over U, then v(s @I 5) = u(s) €35. In particular, if s* is a section of E*, s a section of E, 1and p two sections of B x C, the identification of s* @ 1with a section of (Ecc,)* identifies (s* 8 I , s @ p ) with (s*, s ) l p .
19. INVERSE IMAGE O F A VECTOR BUNDLE
Let E be a real (resp. complex) vector bundle over B, and let 1= (E,B, n) be the corresponding fibration. Let B' be a differential manifold, f:B' + B a C"-mapping, and consider the inverse imagef*(I) = (E', B', n') of I underf, and the canonical morphism (Jf')of f*(A) into I (16.12.8). We recall that, for each b' E B', the restriction f;, : Ei, + Ef(b,)o f f ' is a bijection; we can therefore transport to EL, the real (resp. complex) vector space structure on E,(,,,) by means of the inverse bijection&',-'. We shall show that, with these vector space structures and the fibration f *(A), E' is a real (resp. complex) vector bundleover B'. We reduce immediatelytothesituation where E = B x F is trivial, and then it is immediate that E = G x F, where G c B' x B is the graph off. If p is the restriction to G of the projection pr, : B' x B -.+ B , then the mapping p - l x 1, : B' x F -.+ G x F = E is a diffeomorphism which evidently satisfies the condition (VB) of (16.15); this proves our assertion. The bundle E is called the inverse image of the vector bundle E by the mapping f and is denoted by f *(E). It has the following " universal property " relative to$ (16.19.1) Let El be a real (resp. complex) vector bundle over B', and let g : El + E be a C"-mapping such that (f,g ) is a morphism of vector bundles (1 6.15.2). Then there exists a unique B'-morphism u : El +f *(E) such that g =f' o u. If%, : (El)b, --f is bijective for all b' E B', then u is a B'-isomorphimt.
19 INVERSE IMAGE OF A VECTOR BUNDLE
133
This follows immediately from (16.12.8(iii)) and the definition of the vector space structures on the fibers off*(E). If sl,. . .,s, are sections of E over U which form a frame (16.15.1) over U, then their inverse images s; ,. ..,s; under f (16.12.8) form a frame of f*(E) overf-'(U). If E, F are two vector bundles over B and if v : E + F is a B-morphism (resp. a B-isomorphism) of vector bundles, then f*(u) : f*(E) +f*(F) (1 6.12.8) is a B'-morphism (resp. a B'4somorphism) of vector bundles. A particular case is that in which B' is a submanifald of B and f is the canonical injection; in that casef*(E) is said to be the vector bundle induced by E on B'. Examples (1 6.19.2) Let X be a differential manifold, Y a submanifold of X, j : Y -,X the canonical injection, and consider the morphism (j,TCj)) : T(Y) + T(X) of vector bundles (16.15.5), which is an immersion; hence there exists a unique Y-morphism u : T(Y) +j*(T(X)) such that T(j) =j' u, where 0
j' : j*(T(X)) + T(X)
is the canonical mapping. It is clear that the mapping ywrk(u,) = dim,(Y) is locally constant, so that u is an isomorphism of T(Y) onto a vector subbundle ofJ*(T(X)). Usually we shall identify T(Y) with its image inj*(T(X)); the quotient bundlej*(T(X))/T(Y) is called the normal bundle of Y in X (this name will be justified in Chapter XX) (cf. Problem 5). (16.19.3) Let El , E, be two vector bundles over B. With the notation introduced at the beginning of this section, consider the canonical injections j , : El + El 0 E, , j , : E, + El 0 E2 (16.16.1). To them correspond injections f*(jl) :f*(El) -+f*(El@E,) and f * ( j d :f*(E,) -+f*(El @ E2), and hence we have a B'-morphism
f * ( d+f*(i2) :f*&) Of*@,) --+f*(El8 E2)It is immediately verified (by reducing to the case where El and E, are trivial bundles) that this is a B'-isomorphism. We shall usually identifyf*(E,) @f*(E,) withf*(El 0 E,) by means of this isomorphism. If m : El @ E, +El @I E, is the canonical bilinear B-morphism (16.18.1), then f * ( m ) :f*(El) @f*(E,) +f*(El 8 E,) is a bilinear B-morphism, and therefore factorizes into f*(El) @f *(Ed %*(El)
sf *(Ed -9*(E1€3
Ez)9
134
XVI
DIFFERENTIAL MANIFOLDS
where u is a linear B'-morphism. It is immediately checked that D is a B'isomorphism, which permits us to idenrifyf*(E,) @f*(E,) withf*(El €3 EJ. Under this identification, if s1 ,sz are sections of El , Ez , respectively, over an open subset U of B,f*(s,) @f*(sz) is identified withf*(s, @ sz) (sections overf-'(u)). Likewise, we have a canonical B'-isomorphism
if u is any section of Hom(E, , E2)over U, this isomorphism identifiesf*(u) with the section v of Hom(f*(E1),f*(E,)) such that v(f*(sl)) =f*(u(sl)) for all sections s1 of El over U. In particular, f*(E*) is thus identified with the dual bundle (f*(E))*, for any vector bundle E over B. Finally, with the same notation, for each integer p 2 1 we have a B'P
P
E) + Af*(E), which identifiesf*@, A sz isomorphism f*(l\ f*(sl) A .. * A f*(sp)(sl , . . . , sp being sections of E over U).
A
*
A
sp) with
PROBLEMS
(a) Let U:, be the subset of the trivial bundle nl over the Grassmannian G,,, consisting of all pairs (V,x), where V is ap-dimensional subspace of R"and x is orthogonal to V (with respect to the Euclidean scalar product on R").Show that U:, is a vector subbundle of nI and that UA, ,,@Urn,= nI, in the notation of Section 16.16, Problem I . (b) Show that the inverse image ~ J * ( U ~ (Section , ~ - ~ ) f6.16, Problem I(c)) is isomorphic to U;. p .
,
(a) Define a canonical injection of Gnv into Gn+,,,, and show that the inverse image of Urn+,,,.under this injection is isomorphic to LJ". c , ,+, and show that the inverse (b) Define a canonical injection of Gm, into Gn+mv p + n under this injection is isomorphic to Urn,@ m l . image of Urn+,,,, (a) Show that the normal bundle of S. (considered as a submanifold of R"+')is trivializable. (b) Define a canonical injection of S. into Sn+pand show that the normal bundle of S. in &+,, is trivializable.
Let (E, B, n-), (E', B', 7')be two real (resp. complex) vector bundles and let (f,g) be a morphism of the first into the second (16.1 5.2). Suppose that the kernel of& : Eb+ E;(b) has locally constant dimension. Show that the union of the Ker(gb) for 6 E B is a vector subbundle of E (factorize g by using (16.19.1)). This bundle is called the kernelof the morphism (f,g). In particular. let M,M be differential manifolds, and f: M + M' a submersion of M into M'. Then the morphism (f,T(f)) of vector bundles (16.15.6) has a kernel.
19 INVERSE IMAGE OF A VECTOR BUNDLE
135
More particularly, let (X, B, a ) be a fibration. Then the kernel of the morphism (a, T(a)) is the bundle V(X) of vertical tangent vectors (16.17.1.1), and the quotient bundle T(X)/V(X) over X is isomorphic to the inverse image T(a)*(T(B)). 5. (a) Let M be a submanifold of R", so that for each x E M the tangent space T,(M) is
contained in T,(R"). Let N,(M) be the orthogonal supplement of T,(M) in T,W) (with respect to the scalar product obtained by transporting the Euclidean scalar product on R" by means of the canonical isomorphism T;~ (16.5.2)). Show that the N,(M) are the fibers of a vector subbundle (over M) ofj*(T(R")) (where j : M -+R"is the canonical injection), isomorphic to the normal bundle of M in R". (b) Suppose that M is a fiber bundle over B. For each b E B and x E M, , let P,(M) be the orthogonal supplement in T,(M) of the subspace T,(M,) tangent to the fiber M b(with respect to the scalar product on T,(R") defined in (a)). Show that the Px(M) are the fibers of a subbundle P(M) of T(M), supplementary to the subbundle of vertical tangent vectors V(M) (16.17.1.1).
6. Let M be a submanifold of R",and suppose that M is endowed with a fibration (M, B, T ) making it a principal bundle for a compact group G; suppose also that G is a subgroup of the orthogonal group O(n)and that, for each x E M and s E G, x . s is the image of x E R" under the orthogonal transformation s-'. Then the group G acts freely on T(M), N(M), V(M), and P(M), in the notation of Problem 5, and orbit mani-
folds exist for all these actions, and are canonically endowed with vector bundle structures over B (cf. Section 20.1 and Section 16.14, Problem 15(b)). Show that G\P(M) may be canonically identified with T(B), and that
G\V(M) @NN(M)) may be canonically identified with M x R". Deduce that T(B) 0 (G\V(M)) 8(G\N(M)) is isomorphic to M x G R". 7. Consider the principal bundle (S., P,,(R), .rr) with structure group 2/22, identified with the subgroup of O(n 1) consisting of the identity and the symmetry X- - x . Show that the associated vector bundle S. x '/"R is isomorphic to the canonical vector bundle L,,(I) on P,,(R) (Section 16.16, Problem 1). (Consider the mapping (x, t ) - ( v ( x ) , t x ) of S. x R into P,(R) x R"+l.)Deduce that T(P.(R)) @ 1 (notation of Section 16.16, Problem 1) is isomorphic to (n l)L.(l) (use Problem 6). State and prove the analogous result for P.(C).
+
+
8. Let E be a real vector bundle of rank k , with base B and projection a.
(a) Show that there exists a canonical one-to-one correspondence between C"mappings u : E +Rm whose restriction to each fiber Eb is linear and injective (such mappings are called Gaussian) and B-isomorphisms of E into f*(U,,,,,J, where f:B-tG,,t is a C"-mapping (the notation is that of Section 16.16, Problem 1). (Definef(6) to be u(Eb).) (b) Deduce from (a) that if V is any relatively compact open subset of B, there exists an integer N such that the bundle w l ( V ) is B-isomorphic to an inverse imagef*(UNv3. (Cover V by a finite number of open sets over which E is trivializable, and use a partition of unity.)
XVI DIFFERENTIAL MANIFOLDS
136
9. Let E be a real vector bundle of rank n over a differential manifold B, with projection
The multiplicative group R* acts differentiably on the open subset E -O(B) of E by the rule (c, ub)H cub. Show that the orbit manifold (E -O(B))/R* exists. We denote it by P(E). If p : P(E) + B is the mapping obtained from w by passing to the quotient, show that (P(E), B,p) is a fibration whose fibers are diffeomorphic to Pm-l(R); also that the transition diffeomorphisms are diffeomorphisms of V x P.- ,(R) onto itself (where V is an open set in B) of the form (x, z) t+ (x, u(x, z)), where u(x, .) belongs to the projective group PGL(n, R)= GL(n, R)/R* (R* being identified with the center of GL(n, R)). P(E) is called the projectiue bundle defined by E. The canonical line bundle over P(E) is the subbundle LE(1) of p*(E) consisting of all pairs (z, u) f P(E) x E such that p(z) = P(U) and such that either u = 0 or else u has z as its orbit under the action of R* on E -O(B). Show that if we identify one fiber P(E), with Pn-l(R), the bundle induced by LE(l) on P(E)b is isomorphic to the canonical line-bundle Ln-l(i) (Section 16.16, Problem 1). P.
@x'
10. Let E be a vector bundle of rank n, L a subbundle of E of rank 1, and F a supplement
of L in E. Show that jt\E is canonically isomorphic to
( hF) @ (L
F).
11. Let E be a vector bundle with base B and projection rr. For each b E B and each u1,E E, , let -rub : Tub(Eb)+Eb denote the canonical bijection (16.5.2) of the space of vertical tangent vectors at the point ub onto the fiber Eb. Show that for each pair of vectors u b ,vb in Eb there exists a unique vertical tangent vector h(ub,vb) E Tub(&,)such that TU,(h(ub, vb)) = vb. Considering w*(E) = E x E as a vector bundle over E, show that h : E X E +T(E) is an injective E-morphism of vector bundles whose image is the bundle V(E) of vertical tangent vectors (16.17.1.1). For each tangent vector kUbE TU,(E),let p(kuddenote thepoint(u,, T(P) . kub)of
,
,,
E x T(B). Show that if we consider E x T(B) = a*(T(B)) as a vector bundle over E, then p :T(E) -+ E x ,,T(B) is an E-morphism of vector bundles and that the sequence
o +E x ,,E LT(E) 2E x
T(B) +o
is exact. 12. (a) Let (X, B, P), ( X , B, w') be two fibrations with the same base B, and letf: X + X be a B-morphism which is an embedding of X into X'. Show that the inverse image f*(V(X')) of the bundle of vertical tangent vectors to X is isomorphic to V(X) 0 N(X), where N(X) is the normal bundle of X in X . (b) Suppose moreover that ( X , B, P') is a vector bundle over B. Show that V(W 0 N(X) is isomorphic to P*(X) (use Problem I I). 13. The group G = (1, - I } acts analytically on C", the action of the element Z w - Z.
(a) Consider the holomorphic mappingf : C"+ CN,where N = *n(n
f(cl, cz, .. . .51) = ( ~ [ J ) l s r 5 J s nShow . tlrat ffactorizes into C"5 C"/G
2CN,
-I
being
+ l), defined by
20 DIFFERENTIAL FORMS
137
where n is the canonical mapping onto the orbit space (12.10.6) and j is a homeomorphism onto a closed subset V. of CN.The canonical image of V. - {0} in the projective space PN-l(C) is analytically isomorphic to Pnb1(C). (b) The complex manifold obtained by blowing up the point 0 in CN(Section 16.11, Problem 3) is analytically isomorphic to the canonical bundle LN-i,C(I). If 4 :LN-l.dl)+CN
is the canonical projection of this blowing-up (it is a local isomorphism everywhere outside the fiber 9-'(0)), show that 9-'(V,,) is analytically isomorphic to the bundle Lm-1vc(2) over P.-,(C) (Section 16.16, Problem 1).
20. DIFFERENTIAL FORMS
(16.20.1) Let M be a differential manifold. The dual T(M)* of the tangent bundle T(M) is called the cotangent bundle of M. If F is the transition diffeomorphism between two charts on M, then, as we have seen, the transition diffeomorphism between the associated fibered charts of T(M) is (x, h) ++ (W, DFW . h) (16.15.4.5). Hence the transition diffeomorphism between the associated fibered charts of T(M)* is ( x , h*)++(F(x), 'DF(x)-l . h*). We shall write TG(M) in place of T:(T(M)) when there is no risk of confusion; in particular, therefore, TA(M) = T(M) and TY(M) = T(M)*. A section of T,P(M) over a subset A of M is called a tensorfield (or, by abuse of language, a tensor) of type ( p , q) over A. The set T(M, T;(M)) of tensor fields of class C" over M is denoted by Y--qP(M)or F;,R(M); it is a module over the ring b(M) = b(M; R) (also denoted by &R(M)) of real-valued C"-functions on M, and is a free module when T(M) is trivializable (16.15.8).
For each p 2 I , a section over A of the bundle A T(M)* of tangent pcovectors is called a diferential p-form on A (or simply a differential form P
P
when p = 1). The set r(M, AT(M)*) of differential p-forms of class C" on M is denoted by bp(M) or by G?~,~(M). It is a module over b(M), and is free when T(M) is trivializable. We have FY(M) = b,(M), and b,(M) may be identified with the module of antisymmetric p-couariant tensorfields (16.18.4) of class C" over M. Example (16.20.2) Let f be a real-valued function of class C' (r 2 I ) on M, so that for each point x E M the vector d, f E T,(M)* is a tangent covector at x (16.5.7). Then the mapping x ~ df ,is a differentialform of class C-' on M (with the
138
XVI
DIFFERENTIAL MANIFOLDS
convention that r - 1 = a if r = 00); it is denoted by df and is called the diferential off. By means of a chart we reduce immediately to the case where M is an open subset of R",and hence T(M) is identified with M x R"(1 6.1 5.5). If we denote by (e:) the basis dual to the canonical basis (ei) of R",then it follows from (16.5.7.1) that the differential form df is the mapping (1 6.20.2.1)
i=1
which proves that it is of class C'-'. Hence, by (16.5.7)
Iff and g are two real-valued C"-functions on M, then clearly we have
(16.20.3) Let c = (U, cp, n) be a chart on M, where cp = (cpi)lsi.dn. Then it follows from (16.5.7) and (1 6.5.4) that the n differential forms dcp' (1 5 i 5 n) form a frame of T(M)* over U, called the frame nssociated with the chart c. This frame is the dual of the frame (Xi)ldisnof T(M) associated with c ((1 6.1 5.4.2) and (1 6.1 6.3)) ; in other words, (16.20.3.1)
= S{
(Kronecker delta).
It follows that every differential form o on U is uniquely expressible in the form
c n
(16.20.3.2)
x H o(x) =
a&%) * dcp'(x),
i= 1
where the ai are n scalar functions on U. The form w is of class C' if and only if the functions ai are of class c'. The framing defined by the sections dqi is
and (okl(U), (cp x lEn) (+*):I, 2n) is a jfibered chart of T(M)* (16.15.1), associated with the chart c. If a differential form o on U is given by (16.20.3.2), its local expression relative to the fibered chart associated with (U, cp, n) is the mapping 0
20 DIFFERENTIAL FORMS
139
of q(U) into q ( U ) x (R")*. (16.20.4) With the same notation, every tensor field Z of type (p, q ) on U is uniquely expressible in the form (1 6.20.4.1)
the summation running over all sequences ( i k ) l s k s q and ( j h ) l s h s p of indices from 1 to n. Likewise, every differentialp-form a on U is uniquely expressible in the form a=
(1 6.20.4.2)
aHdpH, H
where H = {il , i 2 , . .. ,ip}runs through the set of all subsets of p elements of the set (1,2, .. . ,n} (the sequence (id being strictly increasing), and where
The coefficients in (16.20.4.1) (resp. (16.20.4.2)) are scalar functions on U which are of class C' if and only if Z (resp. a) is of class C'. Identifying P
P
AT(M)* with the dual of the bundle AT(M) (16.18.4), we see that if Yl , .. ., Ypare p vector fields on U, then we have (16-20.4.4)
(a, Y1
A
Y2 A . . *
A
Y,) = C UH * det((dq'",
Yk)),
H
where h and k run from 1 to p in each determinant. (16.20.5) When M is an open set in R",the preceding remarks may be applied to the canonical chart (M, l M ,n), so that q* = pri (1 S i 5 n). By abuse of notation, we denote the coordinates of a point x by t', t2,..., r", and the differential forms d(pri) by dt' (1 S i n), so that a differential pform on M is written
(16.20.5.1)
(t', t2,... ,Y ) H ~u H ( t l ,..., r") dt" A d t i 2 ~ A dtip. *
H
*
140
XVI DIFFERENTIAL MANIFOLDS
For each integer r 2 0 (or r = a), the differentialp-forms on an open subset M of R"
form a basis (called the canonical basis) of the module of differentialp-forms of class C' on M, over the ring &(')(M) of real-valued functions of class C' on M. Example
Let M be any differential manifold. We shall define a differential I-form K M (or K) of class C" on the manifoId T(M)* (the cotangent bundle of M), called the fundamentalform, as follows. Let 02 : T(M)* 4M be the projection map for the bundle T(M)*, and consider the mapping (16.20.6)
T(o6) : T(T(M)*)
T(M)
(16.15.6): for each covector h: E T,(M)* and each tangent vector
kh:
E Thz(T(M)*),
the vector Th:(o$) * kh: is a tangent vector belonging to T,(M). We may there fore consider the linear form k h g w(h:, Th:(o$) . kh;), which is a covector KM(h:) at the point h: E T,(M)*. To show that this does in fact define a differential form of class C" on T(M)*, we may assume that M is an open set in R",so that T(M)* is identified by the canonicaltrivialization with M x (R")*, and T(T(M)*) with T(M) x T((R")*) and therefore with (M x (R")*)x (R"x R")
after permuting the factors. If h: the mapping Th,(oM) is
= (x,
h*) and kh, = ((x, h*), (y, k)), then
((x, h*), (Y, k)) H(x, Y)
and hence KM(h:) is the covector ((x, h*), (Y, k)>H(h*, Y>
on Th,(T(M)*). With the notation of (16.20.5), the fundamental form is
on M x (R")*.
20 DIFFERENTIAL FORMS
141
(16.20.7) Consider now a mapping f : M’ --t M of class C“, and the bundle f*(T(M*)) on M‘, the inverse image under f of the bundle T(M)* (16.19.1). We shall show that there exists a unique M’-morphism of vector bundles
w :f*(T(M)*)
(1 6.20.7.1)
such that, for each x’
E
+ T(M’)*
M’, wx. is the composite linear mapping
v,, being the canonical isomorphism of fibers defined in (16.12.8(ii)). This property defines the mapping w uniquely, and it remains to show that w is of class C“. Now, if (U, cp, n) and (U’, cp’, m) are charts on M and M’, respectively, and if F : cp(U) cp‘(U‘) is the local expression off, then it is easily checked, by virtue of the definition of v,. (16.19.1) and the local expression of T,.(f) (16.15.6.3) that the local expression of w is --f
(16.20.7.2)
(x’, h * ) H (x‘, ‘DF(x’) * h*),
whence the assertion follows. For each differential 1-form w on M, the differential form w of*(w) is called the inverse image of w by f and is denoted by ‘f(o).In view of (16.20.7.2), the form ‘f(o) is equivalently defined by the condition that, for each x’ E M’ and each tangent vector hx.E T,,(M), we have
(16.20.8)
,
hx,) = (w(f(x’)),T , U )
. hx,)
or again
In particular, if g is a real-valued function of class C’ on M, the formula (16.20.8.1) and the definition of dg (16.20.2.2) give (‘f(d9Xx’h hx*) = (dg(f(x”9 = Tf(x,)(g)*
T X W . hx,)
(TAf) - hx,) = T
X
b of)
*
h,,
by (16.5.4). Hence we have the formula
Let c = (U, (o, n) be a chart on M, where cp = (cp’), sisn. If a differential form on U is given by
XVI DIFFERENTIAL MANIFOLDS
142
1 ai dq' n
UJ
=
i= 1
(16.20.3.1), then ' ~ ( u Jonf-'(U) ) will be given by
Consider also a chart (V, $, m) on M' such thatf(U) c V, and let F be the local expression offrelative to these two charts, and XH ( x , w*(x)) the local expression of w (where w*(x) E (Rn)*);then the local expression of ' ~ ( u Jis) x' H (x', 'DF(x') * w*(F(x'))).
(1 6.20.8.4)
In the particular case where M (resp. M') is an open set in R" (resp. R"), then with the notation introduced in (16.20.5), the inverse image of the form w : (51, . . ., g.)H
c n
i= 1
Ui(51,
. . . ,g.)dt'
by the mapping F = (F', . . . , F") of M' into M will be given by
'F(o) * (t",
f
i='
. ..,t ' " ) ~
(i(uj(F'(5'', . . . ,t""),...,F"(T", ...,t""))DiFi(5",...,t'")))
at".
Under the hypotheses of (16.20.7), by replacing each of the linear mappings w,. by a tensor or exterior power, we may define in the same way the inverse image under f of a covariant tensor field Z or of a diFerentialp-form a on M. These inverse images are denoted by 'f(Z) and 'f(a),.respectively. If (U, q, n ) is a chart on M, and if Z and a are given on U by
(16.20.9)
z= (16.20.9.1)
(il,
H
We may also write
i,)
uiri2...'p
uH dq'l
a=
then we have
c....
A
dqil Q dqi2 @ ...C3 dqip,
dqiZA * . . A dqip
(H = {il, i 2 , .. . , i,]),
20 DIFFERENTIAL FORMS
143
(16.20.9.3)
If M and M‘ are pure, this shows in particular that if dim(M) = n and dim(M’) = n’,then for a p-form a on M we have ff(a) = 0 if n’ < p n. Suppose that n’ = n, and consider two charts on M and M’, respectively. If XH (x, V(x))(where V(x) E R) is the local expression of a differential n-form u on M, and if F (a C”-mapping of an open set in R” into an open set in R”) is the local expression off, relative to the two charts under consideration, then the local expression of ‘f(u) is x’ H(x’, J(x’)V(F(x’))),
(1 6.20.9.4)
where J(x‘) = det(DF(x’)) is the Jacobian of F at the point x’ (8.10.1). It is clear that if a is a differential p-form and ja differential q-form, then
Finally, if g : M“
--t
M’ is another C“-mapping, we have
There are analogous formulas for covariant tensor fields. (16.20.10) If M’ is a submanifold of M and j : M‘ + M is the canonical injection, then y(Z) (resp. y(a)) is said to be the covariant tensor field induced by Z on M’ (resp. the differentialp-form induced by a on M’); we have y(a) = 0 ifp > dim(M’). If M is an open set in R”, and M’ an open set in R” (a situation to which we may reduce by means of charts), and if we identify the vectors of the canonical basis of R” with the first m vectors of the canonical basis of R”, then we pass from the p-form a=
H
a“({’,
. .. , t”)d t ”
A
dtiZA * .
-
A
dt’p
to the induced form ?(a) by replacing the Ck and the dtk with k > m by zero. (16.20.11) It should be noted carefully that there is no equivalent of the preceding developments for contravariant tensor fields or mixed tensor jields. For such fields neither inverse images nor direct images relative to a C“-mapping f:M’ + M can be defined. The reasons behind this are that the mappings T,,(f) are not necessarily bijective (which prevents us defining the inverse image of a tangent vector) and that f itself need not be bijective (which
144
XVI
DIFFERENTIAL MANIFOLDS
prevents us defining the direct image of a section (16.12.6)). Of course, iff is a direomorphism of M’ onto M, we can define the imagef(Z) of any tensor field Z on M’, by transport of structure (16.16.6); if u’ is a differential p-form on M’, then f(a’) is the inverse image ‘f-’(a’) of u’ under f - l , as defined above. Examples Let M, M‘ be two differential manifolds. A homogeneous contact transformation from M to M‘ is by definition a diffeomorphismf of an open set in T(M)* onto an open set in T(M’)*, such that the image underfof the form induced by the fundamental form xM (16.20.6) is the form induced by the fundamental form xM,. If g : M + M’ is a diffeomorphism, we obtain canonically from g by transport of structure a contact transformation (1 6.20.12)
T( g)* : T(M)* + T(M’)* : for each x E M and each covector h: E T,(M), we have T(g)* . h:
(16.20.12.1)
= ‘T,..g)-’ *
h,* ,
and it is immediately verified that the image of xY under T(g)* is xM,. However, it is easy to define contact transformations which are not of this type. Take for example M = M’ = R”,so that T(M)* may be identified with R” x (R”)*,and let U be the open set in T(M)* consisting of the ((ti),(fti)) such that qn # 0. Then the mapping
may be verified to be a contact transformation of U onto U; it is a particular case of a Legendre transformation.
(16.20.13) Let G be a Lie group acting differentiably (on the left) on a differential manifold M. We have already seen that there is a canonical differentiable action of G on the tangent bundle T(M) (16.15.6). We shall show that there is also a canonical action of G on the cotangent bundle T(M)*. Since y(s) : XHS x is a diffeomorphism of M, we obtain by transport of structure a diffeomorphism of T(M)*, which transforms a covector h: at the point x E M into the covector ‘T,.(y(s))-’ * h,* at the point s . x . If we denote this covector by s * h: we shall have, for every tangent vector k, . x at the point s .x ,
-
(1 6.20.13.1)
(S
*
h:, k,..J =
. kS.,.)
20 DIFFERENTIAL FORMS
145
and since the right-hand side is a C"-function on the submanifold of
G x T(M) x T(M)* where it is defined, it follows that the action of G on T(M)* so defined is indeed differentiable. If o is a differential form on M, its image s * w or y(s)o under the diffeomorphism y(s) is therefore the form defined by (16.20.13.2)
((s
*
. x), s-' h,).
w)(x), h,) =
*
Remark (16.20.14) The definitions (1 6.20.8.1) and (16.20.9.3) retain their validity under the weaker hypothesis that the mapping f : M' -+ M is of class C' for some r >= 1 ; the formula (16.20.8.4) then shows that the inverse image under f of a form of class Cs is of class inf(r - 1, s). (1 6.20.15 )
Vector-valueddcyerentialforms.
Instead of considering in (16.20.1) the vector bundle T(M*) = Hom(T(M), M x R) or AT(M)*
= Hom(hT(M),
M x R),
we may more generally consider the vector bundle Hom(T(M), M x F) or Hom(A T(M), M x F), where F is a real vector space of finite dimension. A section of Horn(AT(M), M x F) over M is called a uector-ualueddlfferential p-form on M, with values in F. If (a,), is a basis of F, such a p-form is uniquely expressible as a =
r
cli a,,where
the a, are differential p-forms in
i= 1
the sense defined earlier, or (as we shall sometimes call them) scalar-valued differential p-forms. For each X E M, a ( x ) takes its values in the space of P
linear mappings of AT,(M) into F (identified with (M x F)*); if h, , . . . , h, are p tangent vectors belonging to TJM), we denote by a(x) . (h, A . A hp) the value of a(x) at the p-vector h, A h, A .. * A h,, so that we have r
(16.20.15.1) a ( x ) . ( h ,
A *-. A
h,) =
C (cli(x), h, A ... A hp)a,.
i= 1
If X , , . . . , X,, are p vector fields on M, we denote by
146
XVI DIFFERENTIAL MANIFOLDS
(1 6.20.15.2)
~.(X,AX,A*.-AX,)
the function on M, with values in F, x H a ( x ) * (Xl(x)
A
X,(x)
A
A
X,(x)).
In particular, iff = C f iai is a C1-mappingof M into F, then df = C (&)ai i
i
is a differential 1-form with values in F, called the diflerential off; it is precisely the mapping XI--, d, f (16.5.7). For example, if E is a finite-dimensional real vector space, the mapping XHZ, (16.5.2) is the differential d(l,) of the identity mapping. Iff: M’ + M is a C”-mapping, we define the inverse image ‘f(a) of a under f by the same condition as in (16.20.8): for each x’ E M’ and each p-vector h; A h; A . * * A h;, where the hi belong to T,,(M’), we put (16.20.15.3)
‘f(a)(x’)
*
(hi A . * * A hi)
= a(f(x’)) . (T,.(f)
*
hi A * *
A T,.(f)
*
hb).
Equivalently, we have (16.20.1 5.4)
~ ( a =)
f
i= 1
‘f(ai)ai
which brings us back to scalar-valued p-forms. There is no “exterior calculus ” for vector-valued differential p-forms analogous to the exterior algebra of scalar-valued differential forms, but we shall be led to consider in Chapter XX operations analogous to the exterior product in certain particular cases. Consider three finite-dimensional real vector spaces F , F”, F, and let B : F’ x F” + F be a bilinear mapping. If we are given vector-valued differential I-forms a’,a“on M, with values in F and F , respectively, we can associate with them a differential 2-form with values in F, in the following manner: the mapping (16.20.15.5)
(h, , k,)W B ( ~ ’ ( X*) h, , o”(x) * k,) - B(~’(X). k, , o”(x) * h,)
of T,(M) x T,(M) into F is bilinear and alternating, and hence there exists a 2
unique linear mapping of AT,(M) into F, for which the right-hand side of (16.20.15.5) is the value at the bivector h, A k,. We denote this mapping by (a’A a”)(x), and thus we have defined a 2-form a’A a”with values in F. (When there is no risk of confusion we shall write a‘A a“,but it should be remarked that W” A a’in general has no meaning.) If (a& is, is a basis of a basis of F”, and if we put F and (a;)1
20
DIFFERENTIAL FORMS
147
(16.20.15.6)
where the wl and w; are scalar-valued 1-forms, then we have
o’A
(1 6.20.15.7)
(ofA w;)B(ai
0’’=
, a:).
i,j
In the particular case in which F’ = F” = E, o’= o”= o,and the bilinear mapping B is alternating, we do not use the notation o A a.The mapping
is already bilinear and alternating, and we denote by B(o, o), the corresponding linear mapping of A T,(M) into F. If a’,o”are 1-forms with values in E. we have 2
B(o’
(16.20.15.9)
and if we put o = (16.20.15.10)
+ a”,w‘ + a”)= B(o‘, a‘)+ a’A
o”f B(o”, a”),
c o i a i ,where (ai)is a basis of E, then 1 B(o, o) - c (mi oj)B(ai, aj) 2 i
=
i,j
A
(summed over all pairs of indices ( i , j ) ) ,which can also be written in the form (1 6.20.15.11) (1 6.20.16)
B(o, o)=
1(miA oj)B(ai, aj).
i< j
Diflerential forms on complex manifolds.
All the definitions in this section can be transposed to the context of a complex manifold M, by replacing C“-mappings throughout by holomorphic mappings, and real vector bundles by complex vector bundles. The cotangent bundle T(M)* is a holomorphic bundle (16.15.9) of (complex) rank n, if M is a pure manifold of (complex) dimension n. If we denote by MI, the differential manifold of dimension 2n underlying M, then the cotangent bundle T(M,,)* is a real vector bundle of rank 2n over MI,, and its complexification (T(MIR)*)cc) is a complex vector bundle of (complex) rank 2n over MI,. The results of (16.5.13) show that there exists a C-automorphism ‘J of this fiber bundle such that ‘J2 = - I ; the images of the two morphisms p’ = -)(I - i ‘J),p ” = *(I + i‘J) are, respectively, the cotangent bundle T(M)* and its complex conjugate T(M)*, so that (T(M,R)*)(C) = T(W* 8 T(M)*.
148
XVI DIFFERENTIAL MANIFOLDS
To any complex functionfof class C' on MI, we can associate two sections d ' f = p ' 0 df, d"f=p" df of T(M)* and T(M)*, respectively, so that the complex differential form df is equal to d ' f + d"$ The function f is holomorphic if and only if d"f = 0. If (U, cp, n) is a chart on the complex manifold M, where cp = (cp'), s j 6 n , then the dcp' form a frame of T(M)* over U, and the @ a frame of T(M)* over U. 0
PROBLEMS 1. Let A be one of the rings &(M), B(')(M) (where r 2 1) on a differential manifold M. For each x E M, let m, denote the ideal of functions belonging to A which vanish at x.
Consider the mapping f~ d, f of m, into T,(M)*. Show that, if A = &M), the kernel n, of this mapping is equal to m: , but that if A = b(')(M) with r finite, then m', is of infinite codimension in n, . (Observe that in the latter case the product of two elements of m, has a local expression which admits derivatives of order r 1 at the point of the chart corresponding to x.)
+
2.
Let M be a differential manifold. (a) For each real-valued Cm-functionf'on M, let $denote the C"-function on T(M) defined by h,H
U)
+ DF(x). k.
(b) Show that there exists one and only one involutory diffeomorphismj of T(T(M)) satisfying the following conditions for all k E Tu,(T(M)):
6)
OT(M)(j(k))
=
'
k;
(ii) T ( 0 d .i(k) = oT(M)(k); (iii) for all real-valued C"-functionsfon M. (Here h, = T(oM) . k.) (For the proof use (a) and (8.12.2).) The involution j is called the canonical involution on T(T(M)); it is an isomorphism of the vector bundle (T(T(M)), T(M), oT(M)) onto the vector bundle (T(T(M)), T(M), T(oM)). 3. Let M,M' be two open subsets of R".A point of R x T(M)* will be written (z, (xi), (yJ) with 1 5 i 5 n, and similarly for R x T(M')*. A diffeomorphism f of an open subset V of R x T(M)* onto an open subset V' of R x T(M')* is called a nonhomogeneous contact transformation if the image under f of the form dz -
R
y i dx' i=1
where p is a real-valued C"-function which does not vanish in V.
is
20 DIFFERENTIAL FORMS
149
Let N, N' be two open sets in R"+'. A point of T(N)* will be written ((ti), (Ti)) with 0 5 i 5 n, and likewise for T(N')*. Let U (resp. U') be the set of points of T(N)* # 0). Let M (resp. M') be the projection of (resp. T(N')*) such that qo # 0 (resp. U (resp. U') on R" (the subspace spanned by the last n vectors of the canonical basis). Define a C"-mapping g of U into R X T(M)* as follows: g((fl), (Ti))= (z, (xi),( y , ) ) where z = x1= [' for 1 5 i 5 n, and yi = Ti/TO. Define g' : U' +R x T(M')* likewise. Show that, for each nonhomogeneous contact transformationf'of V = R x T(M)* onto V' c R x T(M')*, there exists a unique homogeneous contact transformation F of g-'(V) onto g'-I(v'), for which the diagram
to,
g-'(V)
-
v
9
is commutative. 4.
With the notation of Problem 3, consider the graph ,?J o f f in V x v'. Show that the projection S of this graph on (R x M) x (R x M') contains no nonempty open set.
+
(a) Suppose that S is a submanifold of dimension 2n 1 and that the restrictions (R x M) x (R x M') are submersions. Suppose that S is defined by an equation
p , p' to S of the two projections of
H(z, x ' , . . . ,x", z', x",
(1)
. . . ,x'") = 0,
where H is of class C". Show that in S we have
(3)
aH
-yi+az
(4)
aH ax'
(I
=O
sisn),
and deduce that H determines f' completely, provided that the functional determinant of the left-hand sides of (I)and (4) with respect to z', x", . . .,x'" is not zero. (b) Determinefwhen H has one of the following forms: n
ZZ'
+ iC xixli - 1 =1
(transformation by recigrocalpolars),
"
+ C (x"
(z' - z ) ~
- xi)2 - r z
(dihtation).
1=1
+
(c) Generalize to the case where S has dimension n k (2 f k 5 n + 1). Consider the case where n = 2 and S is one of the submanifolds of dimension 4 defined by the following equations: (XI)*
+
( X y
+ zz =
(X'l)Z
+
(X'y
+
2'2,
zz'
+
X1X'l
+
XZX'Z = 0
150
XVI DIFFERENTIAL MANIFOLDS
(apsidal tramformation), xlz'
+ z + x'l = 0,
x 1 d 2+ x2 - 2' = 0
(Lie's transformation).
(d) Let X be a submanifold of R x M. For each x E X, let TX(X)' be the subspace of T,(R x M)* consisting of the covectors which are orthogonal to T,(X) (relative to the canonical duality between TJR x M) and TJR x M)*).If the projection on R X M' of the image under F (Problem 3) of the union of the T,(X)I is a submanifold X of R x M', then the union of the TX,(X)Iis the image under F of the union of the TJX)'. Then X' is said to be the transform of X by f. 5. Let X be a submanifold of R",of dimension n - p , defined by p equations fi(x) = 0 (1 5 j s p ) , where thefi are of class C" in a neighborhood of X and are such that the differential forms dfJ are linearly independent at each point of X. Let F be a C"-function defined in a neighborhood of X,and let G be its restriction to X. Then in order that dG(x) = 0 at a point x E X, it is necessary and sufficient that dF(x) should be a linear combination of the p covectors dfi(x), or again that dF(x) A dfl(x) A * . . A dfp(x)= 0 (Lagrange's method of undetermined multipliers).
21. ORIENTABLE MANIFOLDS A N D ORIENTATIONS
(16.21 .I) Let X be a pure diflerential manifold of dimension n 2 0 (16.1.5). Then the following properties are equivalent:
(a) There exists a continuous direrential n-form u on X such that u(x) # 0 for all x E X . (b) There exists an atlas '% of X such that, iy(U, cp, n) and (U,cp', n) are and i f we put I) = cp I (U n U'), any two charts of 9.l for which U n U' # 0, = cp'l (U n U') and 8 = I)' : rp(U n U') -,cp'(U n U'), then the Jacobian J(0) of 8 (8.10) is strictly positive at each point of cp(U n U').
+'
0
To show that (a) implies (b), we remark first that there exists an atlas a0 of X all of whose charts have a connected domain of definition. For each such chart (U, cpo ,n) we may write, for each x E U, U(X)
= W ( X ) d,(pA
A
d,cpE
A
A
d,cpi
(16.20.4.2), where w is a continuous mapping of U into R. By hypothesis we have W ( X ) # 0 for all x E U, hence the sign of w(x) is the same for all x E U. Put cp = cpo if w(x) >Ofor X E U, and put cp = (-cpA, cp:, . .., &)if w(x) < 0 for x E U; then it is clear that (U, cp, n ) is a chart of X and that the set '2l of these charts is an atlas. We have to show that '% satisfies condition (b). For each x E U n U' we may write u(x) = w(x)d,cp'
A
d,q2
A
0
.
.
A
d,cp" = w'(x)d,cp"
A
d,cp"
A
* - *
A
d,cp'",
21 ORIENTABLE MANIFOLDS AND ORIENTATIONS
I51
where by hypothesis w ( x ) > O and w'(x) > 0 at all x E U n U ' . Since the matrix of transition from the basis (d,qi) of T,(X)* to the basis ( d , q f i )is the transpose of the Jacobian matrix of 8, by virtue of (16.5.8.4), it follows that
d, q"
A
*. *
A
d, q'#= J(B)(q(x)). d, q1 A * . . A d, q n
and consequently w(x) = J(B)(q(x)). w'(x), which shows that J(e) is positive at 444. Conversely, let us show that (b) implies (a). By considering the restrictions of charts of % to open sets belonging to a denumerable locally finite covering of X which refines the covering formed by the domains of definition of the charts of 2l (12.6.1), we may suppose that 2l is a finite or denumerable set of charts (uk , qk, n), where the uk form a locally finite open covering of X. Let (A) be a C"-partition of unity on X, subordinate to the covering (uk) (16.4.1). Put uk(x) = 0 if X uk, and uk(x) =fk(X) d,V: A * . * A d,& if x E U k ;then uk is a (?-differential n-form on X. Moreover, each x E X has a neighborhood which meets only finitely many of the uk; hence the sum u = uk is defined and is a differential n-form of class C" on X. We have to k
show that u(x) # 0 for all x E X. Let ho, h, , . .., h, be the indices k such that XEUk, andputz = qho(x), $ j = qh,I(Uhj n Uho), ej = $ j o$;'forI 5 j I m ; then, by definition, we have
But the fk(x) are 20, and fh&)
+
m
fh,(x) = 1 ; hence at least one of the
j= 1
fh,(x) for 0 5 j m must be > O ; and since all the J(ej)(z)are >O by hypothesis, we have u(x) # 0.
If X is a pure differential manifold satisfying the two equivalent conditions of (16.21.1), X is said to be orientable. The proof shows that there is then a differential n-form u on X of class C" such that u(x) # 0 for all x E X. Any manifold of dimension 0 is orientable, since the conditions of (16.21.1) are then trivially satisfied. It is clear that every open submanifold of an orientable manifold is orientable, and that a manifold X is orientable if and only if its connected components are orientable. Let X be an orientable pure differential manifold of dimension n, and let u,, be a C" differential n-form on X such that u&) # 0 for all x E X. (16.21.2)
n
Since the vector space /\(T,(X))* is of dimension 1, every differential n-form
152
XVI
DIFFERENTIAL MANIFOLDS
u on X may be written uniquely as u = f -uo ,wherefis a real-valued function on X, which is of class C' if and only if u is of class C'. Suppose moreover that X is connected and u is continuous. In order that u(x) # 0 for all x E X, it is necessary and sufficient thatf(x) # 0 for all x E X, and then we have eitherf(x) > 0 for all x E X, or elsef(x) < 0 for all x E X (3.19.8). Let 0, (resp. 0,) be the set of differential n-forms of class C" on X
which satisfy the first (resp. the second) of these conditions. Then 0,and 0, are called the orientations of the connected orientable manifold X, and the pairs (X, 0,)and (X, 0,) are called the oriented manifolds (with orientations respectively 0,and 0,), having the orientable manifold X as underlying manifold. The definitions of 0,and 0,remain unchanged if we replace uo by any other n-form of class C" belonging to 0,. The orientations 0, and 0,(and the corresponding oriented manifolds) are said to be opposites of each other. If X is an oriented manifold and uo is a differential n-form belonging to its orientation, then for any differential n-form u =f uo on X, we write u(x) > 0 (resp. u(x) < 0) iff@) > 0 (resp. f ( x ) < 0). This relation is independent of the form uo chosen in the orientation of the manifold. Likewise, we say that an n-covector u, at a point x E X is > O (resp. KO) if it is of the form cuo(x) with c > 0 (resp. c < 0). A sequence (2, , . . . ,2,) of n vector fields over X is said to be positive or direct (resp. negative or retrograde) if we have
-
A
- - A Z,(x)> > 0
(resp. < O ) for all x E X. If X is an oriented manifold and U is open in X, then the restriction to U of a differential n-form on X belonging to the orientation of X is # O at all points of U, and therefore defines an orientation on U, called the induced orientation. (1 6.21.3) Let X, X' be two oriented connected differential manifolds of the same dimension n, and letf: X' --* X be a local diffeomorphism (16.5.6). If u is a differential n-form on X belonging to the orientation of X, it is clear that ff(u)(x') # 0 for all x' E X'. We say that fpreserves (resp. reverses) the orientation if ff(u) belongs to the orientation of X' (resp. to the opposite orientation). Let x' E X ' and let (U, cp, n) be a chart of X at the point x = f ( x ' ) , and (U', t,b, n) a chart of X' at the point x', and suppose that the differential nforms dcp' A A dcp" and d$' A . * .A d$" belong, respectively, to the orientations induced on U and U' by the orientations of X and X'. We may also suppose thatf= $ - I F cp, where F is a diffeomorphism of cp(U) onto an open subset of $(U'). Thenfpreserves (resp. reverses) the orientation according as the Jacobian J(F)(x) is > O (resp.
0
21 ORIENTABLE MANIFOLDS A N D ORIENTATIONS
153
Examples (16.21.4)
The spaces R" are orientable, for the canonical n-form dr'
drz
dr is # O at every point of R". The orientation containing this n-form is called the canonical orientation. Whenever we shall consider R* as an oriented manifold it is the canonical orientation that is to be understood, unless the contrary is expressly stated. A
A
A
(16.21.5) Let X, , X, be two orientable pure manifolds; then so is their product X = XI x X,. Let p = dim X , , q = dim X,, so thatp + q = dim X; let u, (resp.u,) be ap-form on X, (resp. a q-form on X,) of class C", such that u,(x,) # 0 for all x, E X , (resp. uz(xz) # 0 for all x2 E X,). Then it is immediately seen that the (p + q)-form u = 'pr,(u,) A 'pr,(u,) on X is such that u(xl , x,) # 0 for all (x, ,x,) E X. The orientation to which this form belongs is called the product of the orientations of X, and X, defined by u, and v, , respectively. We remark that the canonical diffeomorphism
x, x x, -+x,x x, which interchanges the factors does not preserve the product orientation unless either p or q is even. (1 6.214) Let X be an orientable connected differential manifold, f : Y +X an dale morphism (16.5.6). Then Y is orientable. For if v is a differential nform belonging to an orientation of X, it is immediate that 'f(u)(y) # 0 for all
Y E Y ,since f is a local diffeomorphism. The orientation defined by 'f(u) is said to be induced by f from the orientation of X defined by v. (1 6.21.7) Let X, Y be two differential manifoldr;f : X Y a submersion, x a point of X, andy =f ( x ) ;let n = dim,(X), m = dim,(Y), so that dim,( f -'(y)) = n - m (16.8.8). Let j :f -'(y) + X be the canonical injection, u = T,(f), w = T,(j), so that 'u : T,(Y)* -+ T,(X)* is injective and
'w : T,(X)* m
-+
T,(f-'(y))* n-m
A ('w) is surjective. Then,for m each n-covector u, E A (T,(X)*) and each m-covector g, # 0 in A (T,(Y)*), n-m there exists a unique (n - m)-covector a, E A (T,( f -'(y))*) such that
is surjective; consequently A('.)
is injective and
n
(1 6.21.7.1)
0,
=
(K
c'u)(e$)
A 0:
154
XVI
DIFFERENTIAL MANIFOLDS
for every (n - m)-covector
A (T,(X)*)
n-rn
0: E
such that
By virtue of (16.7.4) we reduce immediately to the case where X = R",
Y = R", f is the projection (t', . .. ,t">(t', .. ., t"), and x and y are the origins, so that f -'(y) = R'-". We can then identify T,(X)* (resp. Ty(Y)*) with R" (resp. R"), and T,( f -'(y))* with R"-";'u is the canonical injection ( t l , ..., ( m ) ~ ( ( l , . .., t,, 0,. . .,O), and 'w is the canonical projection
(tl, . .., 5 , ) ~((,,+' , ..., (,,). Moreover, if (e:) is the basis dual to the canonical basis of , ' R we may suppose that [,, = e: A A e: and D,
-
= c er
. * .A e:.
(
n-m
The (n - m)-covectors 0: such that u, c * e:+'
A
= A('u)([,)) A 0: A
A
e:
are then of the form
+ z*,
where z* is a linear combination of (n - m)-covectors, each of which is an exterior product of certain of the e; in which at least one of the factors has indexj 2 m. For all these covectors it is clear that the image same (n - m)-covector c e:+ A * * A e: = a,.
A (%)(a:)
n-m
-
is the
We denote by ux/ly the (n - m)-covector a, whose existence has just been established. The above proof shows that, for fixed y and # 0, we have:
ry
(16.21.8) If ~ H D ( X ) is a differential n-form of c1m.s C' on X, then XHV(X)/C, is a differential (n - m)-form of class C' on f - ' ( y ) .
For, by the use of a chart we may suppose that f coincides in a neighborhood of 0 with the canonical projection R*+ R" considered in the above proof, and that u(x) = h(x) dtl A . - .A dt", where h is of class C'; it follows that u(x)/[, = h(x) d("+' A * .. A d(", h being restricted to R"-". Furthermore, it is clear that the relation D(X) # 0 implies that u(x)/cy # 0, whence : (16.21 -9) I f X is an orientable manifold and f : X + Y a submersion, then for each y Ef(X), thefiber f - ' ( y ) is an orientable submanifold of X.
21 ORIENTABLE MANIFOLDS AND ORIENTATIONS
155
Remarks (16.21.9.1) Under the hypotheses of (16.21.9), suppose that Y also is an orientable manifold, and that CY = C(y) is the value at y of a differential mform C of class C' on Y which is #O at all points of Y. Then the most practical method of calculating the form CJ = v/C, on f -'(y) is often the following: determine an (n - m)-form c0 of class C' on X such that
Y(C) A 0 0 = 0
(16.21.9.2)
(in general there will be infinitely many). Then it follows from (16.21.7) that 0 = 'j(ao). We shall also write v/'f({,) when there may be ambiguity about5 For example, if Y is open in R" and if f = (f',. . . ,f '"), where the f are real-valued functions of class C", to say that f is a submersion signifies that at each point x E X the m covectors d, f j are linearly independent (16.7.1). The form 'f(C) is then (1 6.20.9.2) (u0f)df'
A
df2 A
-.*
A
df",
where a is a real-valued function on Y. To determine r~ locally we may, by restricting ourselves to a neighborhood U of a point of the fiber f - ' ( y ) , complete thef' to a system of local coordinatesf', . . . ,f".Then we have
-
v = b dfl
A
-*.
A
df",
where b is a real-valued function on X, and in U we may take CJ,,
= b(u 0
f ) - ' df '"+' A * * *
A
df".
(16.21.9.3) If X and Y are two oriented manifolds, u an n-form belonging to the orientation of X, and [ an m-form belonging to the orientation of Y, then the orientation off - ' ( y ) determined by of[, is said to be induced by,ffrom the orientations of X and Y. In view of (16.21.5) and the fact that the projections of XI x X, are submersions, it follows from (16.21.9) that a product X, x X, of pure manifolds is orientable ifand only ifeach of the factors X, , X, is orientable. (16.21.10) The spheres S, are orientable.
We may assume that n 2 1, and then the assertion is a n immediate consequence of the last remark, because the open set R"" - {0}is diffeomorphic to the product R*, x S, (16.8.10).
156
XVI
DIFFERENTIAL MANIFOLDS
For later use we shall construct on the sphere S, (where n )= 1) an n-form which is everywhere nonzero, by the method of (16.21.9.1). Consider S, as the submanifold r-'(l), where r : XI+ llxll = ((to)'+ (t')' * (e")')"' is a submersion of Rn+' - (0) onto R*,(16.8.9). On R*, we take the form ( = t-' d t , and on Rn+' - (0)the canonical (n 1)-form
+ -+
+
v = dt0 A dt' A ... A d y , and we shall first of all construct an n-form uo on R"" - ( O ) such that u = 'r([) A a,; then we take u = 'j(uo), where j : S, -rR"+'- (0) is the canonical injection. Now, we have 'r(r)= r-' dr = r-'
immediately checked that the n-form (16.21.10.1)
uo =
n
C (- 1)'t'dto A dt' i-0
A
* *
-
i=O
*.
A
dt'
A
* * *
A
t idt', and it is
dp
(where as usual the caret means that the symbol underneath it is to be omitted) has the required property, and the induced n-form c on s, is just v/[(l). The function r and the form v are invariant under the rotation group
+
SO(n 1, R) acting on R"" - (0). By virtue of the uniqueness of the n-form c (16.21.7), this n-form is also invariant under the action of SO(n 1, R) on S,, and changes sign under an orthogonal transformation of determinant - 1. In particular, if s : x H -x is the symmetry transformation on S, , we have
+
(16.21.10.2)
's(a)(x) = (-
l)nf'O(X).
We shall write u(") in place of u. Conventionally, when n = 0, do)is the 0-form (= function) on So = {- 1, 1) which takes the value 1 at the point 1 and the value - 1 at the point - 1, so that the formula (16.21.10.2) remains valid. When we take on S,, the orientation induced by the function r from the canonical orientations of R"" and R (1 6.21.9.2), the sphere S, is said to be oriented toward the outside. With the opposite orientation, S, is said to be oriented toward the inside. (16.21.11) -The projective spaces P2,,-'(R) are orientable (n 2 1).
We have seen in (16.14.10) that, for each m >= 1, the sphere S, is a two+ P,(R) is the canonical projection, then sheeted covering of P,(R). If K :s,,, for each z E P,(R) the two points of n-'(z) are antipodal on S, . If m = 2n - 1 is odd, we shall show that there exists on PznT1(R)a (2n - 1)-form a' such
21 ORIENTABLE MANIFOLDS A N D ORIENTATIONS
157
that ' a ( d )= a, in the notation of (16.21.10). Each point z E P,n-l(R) has a connected open neighborhood U on which are defined two Cm-sections, u, : U + a-'(U) and u, : U + n-'(U), which are diffeomorphisms of U onto two disjoint open subsets U1,U, of a-'(U), whose union is a-'(U); also we have u,(z) = s(ul(z)) for all z E U. It follows immediatelyfrom (1 6.21.10.2) that 'ui(a I U,) = 'uZ(a I U,); if 06 denotes this (2n - 1)-form on U, then it is clear that for each open subset V of PZn-,(R)over which the covering n-'(v) is trivial, the restrictions of a; and a; to U n V are the same. Hence the existence of the (2n - 1)-form d,which is clearly #O at each point. (1 6.21.12)
The projective spaces P,,(R) are not orientable (n 2 1).
With the same notation as in (16.21.11), suppose that there does exist a continuous differential 2n-form p on P,,(R) which is nonzero at every point. Then the same is true of 'a(p) on S2,,and therefore ' ~ ( p= ) f * a, wherefis a continuous real-valued function on S,, which is never zero. However, by definition we must have 's('n(p)) = 'a@), because a = a s; and since by (16.21.10.2) we have 's(o)(x) = -a(x), it follows thatf(-x) = -f(x) for all x E S,, . Since S,, is connected, this contradicts the fact thatf(x) # 0 for all x E s,, . 0
Let X be a pure complex-analytic manifold. Then the dzTerentia1 manifold Xo underlying X is orientable.
(16.21.13)
Let 2I be an atlas of X, and consider two charts (U, cp, n) and (U', cp', n) belonging to 2I, such that U n U' # 0.Let $' = cp'I(un U'),8 = $-I; $ = cpl(U n U'), 8 is a holomorphic mapping of an open set in C" onto an open set in C". For is a C-linear bijective mapping of each z E cp(U n U'), it follows that C" onto C". The proposition will therefore result from the following lemma: (16.21.13.1) r f u : C" + C" is a C-linear mapping and q u o : R2"+ Rz"is the same mapping u considered as an R-linear mapping, then
(1 6.21.I3.2)
det(uo) = 1 det(u) 1 '.
To see this, take a basis (bj)l of u is upper triangular (A. 6.10):
of C"with respect to which the matrix
158
XVI
DIFFERENTIAL MANIFOLDS
If r j = sj + i t j , where sj and ti are real, then the matrix of uo relative to the basis of Rz" formed by the bj and the ibj (1 S j 5 n) is of the form
R1 P12 ..*
P1n
a triangular array of blocks of order 2, in which
The formula (1 6.21 .I3.2) follows directly by calculating the determinant of this matrix (A. 7.4). We remark that there is a canonical orientation on Xo, with the property that for each chart (U, cp, n) of the complex manifold X the corresponding chart (U, cp, 2n) of X, preserves the orientation, where R2"is endowed with the canonical orientation (16.21.4) and C" is identified with R2"via the mapping ([',t2,. . . , H (W[',Y[', . . . , a(?', Yr). It is easily verified that the forms which belong to the canonical orientation of Xo are those which, for each chart (U, cp, n) of X , have a restriction to U which can be written as
r)
f * dcp'
A
d@' A dcpz A d?
A
-
*.
A
dcp" A d p ,
where f(x) > 0 for all x E U. (16.21.14)
The manifold underlying a Lie group G is orientable.
Suppose that dim G = n, and let z : be a nonzero n-covector at the identity element e of G. Then XHY(X)Z: is a C" differential n-form on G (16.20.13) which clearly is everywhere # 0. We remark that a homogeneous space of a Lie group is not necessarily orientable; for example, we have just seen that P,,(R) is not orientable ((16.11.8) and (16.21.12)). (16.21.15) We have seen in (16.21.12) that the nonorientable manifold P,,(R) admits an orientable two-sheeted covering. This is a general fact: (16.21 .I 6) Every pure manifold X of dimension n admits a canonical orientable
two-sheeted covering.
21 ORIENTABLE MANIFOLDS AND ORIENTATIONS
159
In the bundle A T(X)* consider the open set Z which is the complement of the zero section. The multiplicative group R* acts differentiably and freely n
n
on Z, because AT(X)* is a line bundle, and by taking a fibered chart of n
A T(X)* it is immediately seen that Z is a principal bundle over X with struc-
ture group R*.Now apply (16.14.8), taking H to be the subgroup RT of R consisting of the positive real numbers; since R*/R*, is the group of two elements, it follows that X' = Z/RT is a two-sheeted covering of X. To show that X is orientable, we shall construct an atlas of X' satisfying condition (b) of (16.21.1). To do this, we start with an atlas CU of X such that, for each chart (U, cp, n) belonging to CU, the open set U is connected and the inverse image of U in X' is the disjoint union of two open sets U', U" such that the canonical projections p' : U' + U and p" : U"+ U are diffeomorphisms. If n : Z +X, n' : Z + X are the canonical projections, then by hypothesis there is a canonical morphism of fibrations (16.15.4) $ : cp(U) x R* +z-'(U), and z'-'(U') and n'-l(U") are each equal to one of the images under $ of q(U) x R*, and q(U) x (-RT). Let s be the reflection of Rn with respect to the hyperplane t' = 0. If n'-'(U') = $(cp(U) x R:), we take as chart of U' the triplet (U', cp op', n); otherwise we take (U', s 0 cp op', n ) ; and similarly for U". We have now to show that the condition (b) of (16.21.1) is satisfied by the atlas of X so defined. For this we may limit ourselves to considering two charts corresponding to charts (U, cp, n), (U, cp', n) of X having the same domain of definition. Let $' : cp'(U) x R* + n-'(U) be the canonical morphism corresponding to the second chart; if F : cp'(U) 4cp(U) is the transition diffeomorphism, then the composite morphism $' $ - I is given (16.20.9.4) by 0
(A t
) ( F ~( 4 , J(x)-'t),
where J(x) is the Jacobian of F at the point x. Suppose for example that n'-'(U') = +(cp(U) x R:). If J(x) > 0 in cp'(U), then we have also n'-'(U') = $(cp'(U) x RT), and the transition diffeomorphism for the charts (U', cp "P',n), W', cp' "PI,n>
is then F. If on the other hand J(x) < 0 in cp'(U), then we have z'-'(U') = $'(cp'(U) x (-RT)), and the transition diffeomorphism for the charts (U', cp op', n) and (U', s 0 cp' op', n) is then F 0 s. In both cases, the transition diffeomorphism has Jacobian >O. The argument is similar when n'-'(U) = @(P(U) x (-RT)), and the proof is complete. We remark that if X is orientable, then the covering X' is trivializable, because Z admits a section over X and is therefore trivializable.
XVI
160
DIFFERENTIAL MANIFOLDS
PROBLEMS
1. Let G be a connected Lie group, H a closed subgroup of G. Suppose that, at the point xo E G/H which is the image of e, the endomorphisms h x 0 wt * hXoof T+,(G/H), where t E H, have determinant 1. Show that G/H is orientable. (Using the hypothesis, show that there is a differential form on G/H of highest degree which is invariant under the action of G.) Hence give another proof of the orientability of spheres (16.11.5). Generalize to Stiefel manifolds. Show in the same way that the homogeneous spaces
Sob, R)I(SO(P,R) x Sob - P, R)) = G,p(R) are orientable. G.4.p(R)is in one-one correspondence with the set of orientedp-dimensional subspaces of R". Show that GL,,(R) is a two-sheeted covering of the Grassmannian Gm, p(R). 2.
Show that the Mobius strip and the Klein bottle (16.14.10) are not orientable (same method as for projective spaces). Generalize to the situation of a principal bundle over S , , obtained by making an arbitrary finite subgroup of S1 = U act by translations.
3. Let X be a pure differential manifold of dimension n. Define a canonical differential
+ 1)-form on the manifold A TO()* which does not vanish at any point. I
(n 4.
If M is any pure differential manifold, show that the tangent bundle T(M) is orientable (use the chart construction of T(M) (16.15.4)).
22. CHANGE O F VARIABLES IN MULTIPLE INTEGRALS. LEBESGUE MEASURES
(16.22.1) Let U, U' be two open subsets of R",and let u be a homeomorphism of U onto U' such that both u and u-' are of class C'. For each x E U , let J(x) be the Jacobian of u at x (8.10). Let I , and I,, be the measures induced on U and U' by Lebesgue measure I on R".Then the image under u (13.1.6) of the measure I JI * I , is equal to I,. .
This means that iff is any function in S ( R 7 with support contained in U', then (16.22.1 .I)
f(x) w 4=
I
f(u(x)>I J(x)I d4-4
("formula for change of variables in a multiple integral").
22 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS
161
The proof is in several steps.
(1) Let (U,) be an open covering of U, so that the U: = u(U,) form an open covering of U', and let u, : U, -+ U: be the restriction of u to U, . Sup-
pose that the theorem is true for each u,. Then it is true for u. For if &,= is the restriction of 1 to U, and if pu, is the image under u of I J I * A", then its restriction to U: is the image under u, of (I J I 1 U,) * AUI , hence is the restriction to U: of A", , by hypothesis; hence we conclude from (13.1.9) that pus= A,. . (2) Let u' be a homeomorphism of U' onto an open subset U" of R",such that u' and u'-' are of class C', and put u" = u' u, which is a homeomorphism of U onto U". If the theorem is true for u and u', then it is true for u". For if J'(x) is the Jacobian of u' at the point x , then for any functionfe S ( R " ) with support contained in U" we have 0
kx)
d m ) = Sf(uYx)) I JW I d4x) = S f ( u W ) ) ) I J'(u(x)) I
*
I J(x)I a x )
and the Jacobian of u" at x is J'(u(x))J(x) (8.10.1). (3) The theorem is true when u is the restriction to U of an affine mapping X H U + w(x). For then we have Du = w (8.1.3); whence J(x) = det w for all x E R",and the formula (16.22.1.1) follows from (14.3.9). (4) The theorem is true for n = 1. We have then J(x) = Du(x), and every point of U is the center of a bounded open interval in which Du is bounded and keeps the same sign. By (1) above we may therefore assume that U = ]a, b[ is a bounded interval in R,u being the restriction to U of a continuous function, differentiable and monotonic in [a, b]. Then we have U' =]u(a), u(b)[ if D(x) > 0 for x E U, and U' = ]u(b),#(a)[ otherwise, and the formula (16.22.1 .I)reduces to (8.7.4). ( 5 ) The theorem is true for n arbitrary and u of the form
where 8 is of class C' and J(x) = D,B(x) # 0 for all x E U. For each point x' = (t2, .. , tn)E Rn-' such that the section U(x') # the mapping tl H O(t, , t2,. . ., 5") is a homeomorphism of the open set U(x') c R onto an open set in R,and both it and its inverse are of class C'. Hence, by (4),
.
a,
162
XVI
DIFFERENTIAL MANIFOLDS
and therefore, by virtue of the definition of the product measure on (1 3.21.2),
R"
(6) The results established so far show immediately that in the general case the theorem will result from the following lemma: (16.22.1.3) Under the hypotheses of (16.22.1), for each x E U there exists un open neighborhood V of x such that the homeomorphism of V onto uw), obtained by restricting u, is of the form up 0 0 * 0 u l , where each uj is a homeomorphism of an open subset of R"onto an open subset of R", of one of the types considered in (3) and ( 5 ) above. 9
By replacing u by t u t'-', where t and t' are translations, we may assume that x = u(x) = 0. Replacing u by (Du(O))-' u, we may assume moreover that Du(0) = lRm. Hence we may write u(x) = (ul(x),. . . , un(x)), where, for 1 4 j 5 n, ujisa C'-mapping of Uinto R sucb that Di ~ ~ (= 0 6, ) (Kronecker delta). Put 0
0
0
(where x = (tl, ...,&)); it foliows from the implicit function theorem (10.2.5) that there exists an open neighborhood V of 0 in U such that, for each j, wj = vj I V is a homeomorphism of V onto an open neighborhood of 0. We may therefore write
22 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS
163
where u is a permutation of {1,2, . . . ,n}, or a mapping of the form
so
0
(Wj
0
w,:-l1) s, , 0
where the permutations u and z are chosen so that this mapping is of the type (16.22.1.2). Q.E.D.
For an important example of the appljcation of (16.22.1) (“change to polar coordinates” in R”),see (16.24.9). (16.22.2) Let X be apure differential manijdd of dimension n. Then there exists apositive measure p on X with the following property: for each chart ( U , q, n) of X , the image under cp of the induced measurep, is of the form f 0 (A,,,,), where A is Lebesgue measure on R”and f is a function of class C“ which is # 0
at every point of cp(U). Moreover, any two measures p, p’ on X with this property are equivalent and each has a density of class C“ with respect to the other.
There exists a sequence of charts (uk,qk,n) of X such that the u k are relatively compact and form a locally finite open covering of x. Let v k be the image under the homeomorphism cp;‘ of the measure induced by A on Vk(uk). Also let (fk) be a C“-partition of unity subordinate to the covering (U,) (16.4.1). Then there is a measure pk on x which coincides w i t h f , * vk on u, and with the zero measure on the complement of Supp(f,) (13.1.9), and this measure pk is clearly bounded. Moreover, since each compact subset of X meets only finitely many of the uk, the sum p = pk is defined and is a k
positive measure on X. We have to show that it has the property stated above. Let g be a function in .X(X) with support contained in U . By definition, we have j g dp =
!/
dpk =
J:dk)
dvk
(9 pF’)(fk pL1) dA,
=
‘?k(unuk)
the summation being over the finite set of indices k such that Uk intersects U. If 8, : cp(U n U k )-+ qk(Un U,) is the transition diffeomorphism and J(6,) its Jacobian, then by (16.22.1.1) we obtain J g dP = J,(uk7
where h
= k
(A
0
c p - 9 dA,
q-’)I J(8,)l is a function which does not depend on g but
only on the charts. Since J(8,) is nonzero at each point of cp(U n U,) and
164
fk
XVI
DIFFERENTIAL MANIFOLDS
9-l vanishes in a neighborhood of each point of q ( U ) - q(U n Uk),it follows that h is of class C" and f O at each point z E cp(U), because at least one of the functionsf, q - l is #Oat this point. The last assertion of (16.22.2) is evident, because the inverse of a nonvanishing C"-function is of class C". 0
0
Measures p satisfying the condition of (16.22.2) are called Lebesgue measures on X . The set of negligible functions (resp. measurable functions) is the same for all these measures. 'So also is the set of locally integrable functions, because the density of one Lebesgue measure relative to another is continuous and locally bounded. Whenever we speak of negligible, measurable, or locally integrable functions on X without specifying the measure, it is always a Lebesgue measure that is meant. We remark that submanifolds of X of dimension < n are negligible, because vector subspaces of dimension < n in R" are negligible for Lebesgue measure ((14.3.6) and (13.21.12)). If E is a fiber bundle over X, the notion of a measurable section of E over an open subset of X is well defined, independently of the choice of Lebesgue measure on X. So also is the notion of a locally integrable section if E is a vector bundle; this is clear when the bundle is trivial, and since the notion is local with respect to X, it is enough to verify that on an open set over which E is trivializable, the notion is independent of the trivialization chosen, and this follows immediately from the definitions (1 6.1 5.3).
PROBLEMS
1. Let u be a C'-mapping of an open subset U of R" into R",and let J(x) be its Jacobian atxEU. (a) Let K be a compact subset of U such that J(x) = 0 for all x E K. Show that there exists a real number c > 0 and, for each sufficiently small E > 0, a real number &(&) > 0 with the following property: for each x E K and each cube C with center x and side length 26 < a&), contained in U, we have A(u(C)) 5 cd(C). (By using the uniform continuity of J on K,show that if the image of R" under Du(x) has dimension p < n, and if (bJ1 6,8p is an orthonormal basis of this space, and (c& S k 5 n - p an orthonormal basis of its orthogonal supplement, then u(C) is contained in the parallelotope with center u(x), constructed from the p vectors ZSnMb,, where M is the least upper bound of JIDu(y)IIin K,and the n - p vectors e8n''*ck .) (b) Deduce from (a) that, for each A-measurable subset A of U, we have h*(u(A)) 5 JA*
I J(x) Idx(x).
(Reduce to the case where A is relatively compact. Let K be the set of points x E A at which J(x) = 0, and let V be a relatively compact open neighborhood of K such that
22 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS
h(V) < h(K)
165
+ E . Cover K by small closed cubes with pairwise disjoint interiors, and
use (a) to show that if W is the union of the concentric open cubes of twice the side
length, then h*(u(W)) can be made arbitrarily small. Next, show that there exists a partition of A n CW into a finite number of integrable sets G,, each of which is contained in an open set U,, such that the restriction u, of u to U, is a homeomorphism of U, onto an open set u(U,) such that both u, and u;' are of class C'. Finally, use (16.22.1) in each U, .) (c) Deduce from (b) that the image under u of each set N c U of measure zero is of measure zero. If also N is closed in U, then u(N) is a meager set, a denumerable union of nowhere dense compact sets of measure zero. (d) If E is the closed subset of points x E U such that J(x) = 0, show that uQ is a meager set, a denumerable union of nowhere dense compact sets of measure zero. Deduce that if M is any meager subset of U, then u(M) is meager in R".(Show that if B is any compact and nowhere dense subset of U, then u(B) is nowhere dense. For this purpose, consider a decreasing sequence (V.) of open neighborhoods of E, whose intersection is E, and consider as in (b) above a suitable partition of B n CV,,into integrable sets.) 2.
With the notation of (16.22.1) show that if u is a homeomorphism of U onto U', of class C' (but whose inverse is not necessarily of class C'), the formula (16.22.1.1) remains valid. (Use Problem 1.) Furthermore, the set E = {x E U :J(x) = 0) is nowhere dense, but not necessarily of measure zero. (To prove this last point, use Problem 4 of Section 13.8.)
3. (a) Let F be a closed subset of R".Show that there exists a real-valued function g of class C" on R",such that g(x) = 0 for all x E F and g(x) > 0 for all x $ F. (Use Problem 4 of Section 16.4.) (b) Let fl ,f2, . ..,f. be n real-valued functions of class C' on an open set A c R". Show that if the Jacobian of thefJ vanishes on A, then for each compact B c A there exists a Cm-function g on R" such that the set g-'(0) is nowhere dense in R"and such that g(f,(x), .. .,fn(x)) = 0 for all x E B. (Use (a) and Problem l(d).) 4.
(a) Let f be a holomorphic function in an annulus S : r < l z l < R in C, and let
f(z) =
+m
C
a. Z be its Laurent expansion (9.14.2). Show thatf(S) is open in C and that
11E-W
h*(f(S)) 5
+m
C n l ~ . 1 ~ ( R-~v'") "
-m
(the right-hand side being interpreted as +a, if the series does not converge). Iffis injective on S, the two sides are equal. (b) Let fbe a holomorphic function for IzI > 1, and suppose that its Laurent series is of the form
+ c b.z-". m
f ( z )= 2
"=O
Iff is injective, show that
(Use (a).) Under what conditions is Ibl 1
= 1?
166
XVI
DIFFERENTIAL MANIFOLDS
(c) With the same assumptions on f as in (b), suppose moreover that f(z) # 0 for 1 z 1 > 1. Show that 1 b0 I 5 2. (Remark that there exists a function g, holomorphic for (z(> 1, such thatf(.z2) = (g(z))', by using Section 10.2, Problem 8.) (d) With the same assumptions o n f a s in (b), show that
for IzI > 1. (a) Let fbe a function holomorphic in the disk D : ( z I < I , with Taylor series of the formf(z)=z+azz2+~~~+a.~+....Showthatif~isinjectivein then D , la2/5 2 . For which functions is 1 a2 1 = 2? (Bieberbach's theorem: consider the function g(z) = f(z-')-l for (z(> 1 and use Problem 4.) (b) With the same hypotheses, show that the open setf(D) contains the open disk with center 0 and radius &. (If c .$f@), consider the functionf(z)/(l - c-'f(z)).) Let B be a positive definite symmetric bilinear form on R", and let A > 0 be its discriminant relative to the canonical basis of R".Show that
1,.
exp(-B(x, x)) dh(x) = n"/zA-l'z.
(Consider an automorphism u of the vector space R" such that the matrix of the transformed form B(u(x), u(y)) is diagonal.) Let P. denote the open subset of RnZ= M.(R) consisting of the positive definite symmetric matrices. (a) Let
is a row matrix and Xl is a positive definite matrix of order where 'z = (xlzxla... xln) n - 1 . Show that (det X1)-'(det X)= x ,
- 'z . X;'
*
Z.
(Reduce to the case where X , is a diagonal matrix by means of an orthogonal transformation in R"-'.) (b) If Y EP, and s > )(n l), show that
+
(Reduce to the case Y = Z by means of an orthogonal transformation. Integrate first with respect to xll,by taking as new variable of integration u=(detXl)(xll-f~~Xi'~~) and then with respect to x1z , . . . ,x l nby using Problem 6, and finally with respect to XI; hence obtain a reduction formula for the integral.)
23 SARD'S THEOREM
167
23. SARD'S THEOREM
Let X,Y be two differential manifolds,f : X + Y a C"-mapping. Generalizing the definition of (16.5.11), a point x E X is said to be critical for f if f is not a submersion at x (16.7.1), in other words if
r k ( f 1 < dim,&I.
If E is the set of critical points o f f , then Y -f(E) is called the set of regular values off. For each y E Y -f(E), the fiberf-'(y) is therefore either empty or a closed submanifold of X (16.8.8). (16.23.1) (Sard's theorem) Let f : X Y be a C"-mapping, E the set of criticalpoints ofJ Then f(E) is negligible in Y , and Y -f(E) is dense in Y.
The latter assertion follows from the former and the fact that the support of any Lebesgue measure on Y is the whole of Y. To prove the first assertion of the theorem, observe that if (U, , qk,n,) is a sequence of charts of X such that the U, cover X and such that eachf(U,) is contained in a chart of Y, it is enough to show that f(E n U,) is negligible for each k (13.6.2). We may therefore assume that Y = Rp and that X is an open subset of R". The proof will be by induction on n ; the case n = 0 is trivial. Put f = (fi , . . . ,f,)where each4 is a real-valued function of class C" on X. Put E, = E, and for m 2 1, let Emc E denote the set of points x E X such that all the derivatives of order S m of all t h e 4 vanish at x. The theorem will be established if we prove the following two statements: (i) For each m, f(Em- Em+,)is negligible. (ii) For m ZnIp, f(Em)is negligible. (16.23.1.1) Proof of (i). Since the topology of R"has a denumerable basis, it is enough to show that, for each xo E Em- E m + 1there , exists an open neighborhood V of xoin X such that f(Em n V) is negligible. By hypothesis, there exists an index j and a derivative D'fi of order [ a ( = m + 1 which is f O at x, . By permuting the coordinates, we may assume t h a t j = 1 and that DYl = D1w, where w is a C"-function. Consider the mapping h : X +R" defined by h(x) = (
~ ( ~ 15 23 , ..
*
9
5n)T
where x = (5, , Tz, . . . ,5.) E X. Since Dlw(x,) # 0, it follows immediately from (16.5.6) that there exists an open neighborhood of x, in X such that h 1 V is a diffeomorphismof V onto an open subset W of R".Let g = ( g1 , . . . , gp) denote the restriction o f f h-' to W, which is therefore a Cm-mappingof W into RP. 0
168
XVI DIFFERENTIAL MANIFOLDS
We distinguish two cases: (a) m
= 0 and
(b) m 2 1.
(a) m = 0. By definition, we may assume that w =fi,sothat g,(x) = t1 for all x E W. Next, the set E of critical points of g is equal to h(E n V), hence f(E n V) = g(E') and it is enough to prove that g(E') is negligible. Identify R" with R x R"-', and for each x = (c, z) E W, put g(x) = gc(z)). Then the Jacobian matrix of g at the point x is of the form
(c,
hence in order that x E E' it is necessary and sufficient that z E E;, where E; is the set of critical points of gr. Consequently, for each C E R,
((51
x R P - 9 (7 g(E') = {el x g m .
Now the inductive hypothesis implies that g,.(E;)is negligible in RP-'; on the other hand, E' is closed in W, hence is a denumerable union of compact sets, and consequently g(E') is a denumerable union of compact sets, hence is Lebesgue-measurable in RP (1 3.9.3). It follows now from (13.21 .lo) that g(E') is negligible. (b) m 2 1. By definition, we may assume that w(x) = 0 for x E Em, hence h(E, n V) c (0)x R-"I. For each point (0,z) E W n ((0)x R"-'), put g(0, z) = go(z). Since all the first derivatives of g vanish at each point of h(E, n V), all these points are critical points of go.The inductive hypothesis therefore implies that go(h(E, n V)) is negligible in RP,and this set is precisely f(E, n V). (16.23.1.2)
Proof of (ii). Let IJxI(denote the norm sup i
I til on R" and on
RP. For each real number a > O and each k = (k,,..., k,)ER", let I(k,a) denote the cube in R" defined by the inequalities k j 5 ti5 ki+ a (1 5 i 5 n). Then it is evidently enough to show that f(E, n I(k, a)) is negligible for some a > 0 such that I(k, a) c X. Let M be the least upper bound of l(f('"+')(x)lIi n I(k, a); it follows from Taylor's formula (8.14.3) that if x E E, n I(k, a) and x + t E I(k, a), then
Now observe that, for each integer N > 1, the cube I(k, a) is the union of the N" cubes I(s, a/N), where 5 = (k1
+ ( a , / N ) , . . . > kn +(asnlN))
and the integers sirange independently from 0 to N - 1. The set f(E, n l(k, a)) is therefore contained in the union of the N" sets f(E, n I(s, a/N)); but for
23 SARD'S THEOREM
169
each s such that Emn I(s, a/N) is nonempty, if xo is a point of this set, it follows from (1 6.23.1.3) that for every other point x E Emn I(s, a/N) we have [If(x)- f(x,)(I 5 M(a/N)"+', and therefore
I(f(E, n I(s, a/N))) S MP(~/N)P(m+l), where 1is Lebesgue measure on Rp. Hence I(f(E,,, n I(k, a))) S Mpap(m+l)Nn-p(m+l). Since by hypothesis m 2 n/p, the right-hand side of this inequality tends to zero with l/N, and the proof is complete. Sard's theorem implies, in particular: Let X, Y be pure differential manifolds of dimensions n, p , respectively, such that n < p , and let f:X + Y be a C"-mapping. Then Y -f(X)is dense in Y . (1 6.23.2)
I n other words, for C"-mappings there do not exist phenomena of the type of the "Peano curve" (Section 4.2, Problem 5, or Section 9.12, Problem 5).
PROBLEMS 1. (a) Let m, n, p , r be integers >O. Let f be a C"-mapping of a n open set U C R" into R",and let g be a Cm-mapping of a n open set V 3 f(U) in R" into RP. Put h = g f. Show that for each x E U we have 0
c c. . I
D'h(x) =
q = o (11..
o,(il, . . . , iq)Dqg(f(x)) (Dilf(x), ... 0
* i,)
where in the inner sum, (il ,. .., i,,) runs through all sequences of q integers 21 such that il ... iq = r . Furthermore, in this formula, the constants o,(il , ... , i,) are rational numbers which depend only on the ij, q, m,n,p , and r, and nor on the functions f and g. (b) Let xo E U. Assume now only that, for some s < v, f is a mapping of class C'-" of U into R",and that g is a mapping of class C' of V 3 f(U) into Rp. Suppose also that Dkg(f(xo))= 0 for k $ s. For each x E U and each integer k E [0, r], let
+ +
c c.... k
hk(x)=
q=s+1
(ii.
uk(il, . . . , i,)Dqg(f(x))
iq)
0
(D"f(x),
.. . ,D',f(x))
E
-Yk(Rm;Rp),
the second sum being over the same sequences ( i , ,.. . ,i,,)as in (a), so that ij 5 k - s for all j , and hence the function h, is well-defined on U. Ry applying Taylor's formula, show that hk(x) can be written in the form r-k
hdx) =
C
j=O
a,
*
(x
- x0)(j)
+ Rdx),
170
XVI
DIFFERENTIAL MANIFOLDS
where the ajk are constant elements of 9,(Rm; ..Yk(Rm;Rp)) = 9k+,(Rm;Rp), and IIRk(x)111JIx- xoIr-’ tends to 0 as x +xo . Further, the aJkdepend only on the values of the derivatives o f f at xo and of g at f(xo). Deduce that j!=jk = hk+J(xO)
f o r j $ r - k . (Use(a).) (c) More generally, let A be a closed subset of U, B a closed subset of V containing f(A), and suppose that Wg(y) = 0 for all y E B and 0 5 k 5 s. Defining the hkas in (b) above, show that for xo E A, h&’) =
J=oJ.
hk+j(X) *
(X’
- X)(’)
+ Rk(X’,
X),
where IIRr(x’, x)ll/Ilx’- x IY-k tends to 0 as x, x’ tend to xo whilst remaining in A. (d) Under the hypotheses of (c), deduce that there exists a mapping H : U+RP of class C‘, such that H(x) = g(f(x)) for all x E A and DkH(x) = 0 for all x E A and k 5 s (Kneser-GIueser rheorem). (Use Whitney’s extension theorem (Section 16.4, Problem 6).) Let U be an open subset of R”, f a mapping of U into Rp, of class C‘. Show that if r 2 max(1, n - p l), the image f(E) of the set of critical points off is of measure zero in Rp.(Proceed as in the proof of (16.23.1), using the result of Problem 1 to deal with the case m 2 1 in (i).)
+
The notation is that of Section 13.21, Problem 2. The simple arc K =g([O, 71) is therefore a subset of R2 of measure >O. Let KObe the union of {go(0)},{g0(7)} and the three sets Kol = g0([l, 21), Koz = g0([3,4]), and KO3 = go([5, 61). Let FObe the function on KOwhich takes the value 0 at g0(0), 4 on KOI, 4 on KO2,2 on K03, and 1 at g0(7). Define sets K. inductively as follows: K. is the union of K.-1 and the sets K,, where I is any sequence of n terms (il , .. ., in),each term of which is equal to 0,2,4, or 6; for any such s, K, is the union of Dethe three sets Ka, I = Bn(Ua([l, 21)), Ka. 2 = ~dua([3,41)),and K.. 3 = 8n(01([5,61)). fine functions F. on K. inductively as follows: F,(K.-, = Fn-l, and F. is constant on each K,, (1 5j S 3): putting a = g.(u,(O)), p = gm(us(7)), then
F. = fFjn-l(ar) F. = W n - h ) F n = bFn-i(a)
+ &Fn-lCs> + +F.-I@) + PFn-lGB)
on L 1, on IL.2 , on Ka, 3 .
(a) Let G be the function on the union of the K. which is equal to F. on K. for each n. Show that G extends by continuity to a real-valued function F on K. (b) Show that there exists a constant c > 0 such that, for any two points x, y E K, we - ~ 1 1 for ~ ’ a ~suitable choice of the sequence (a”). (Consider have 1 F(x) - F(y)l =< c * JJx the least integer II such that the two points g;l(x), g;’(y) do not belong to the same interval o,([O, I),for some sequence s of n terms.) (c) Deduce from (b) and Whitney’s extension theorem (Section 16.4, Problem 6) that there exists a function f:RZ-+R of class C1 for which all the points of K are critical points, and yet such that f(K) is the interval [0,1].
24 INTEGRAL OF A DIFFERENTIAL n-FORM
171
=< inf(p, n), let M, be the subset of the space Rp" of p x n matrices consisting of the matrices of rank k . Show that Mk is a differential submanifold of RpD,of dimension
4. For k
k(p
+ n - k). (Observe that an p x n matrix of the form
(z
:),where
A is a k x k
invertible matrix, is of rank k if and only if D = CA-'B; for this purpose, multiply the matrix under consideration on the left by a suitably chosen invertible matrix so that the product is of the form
(; :/).
5. Let U be an open set in R" and let f : U RP be a C" mapping. Suppose that p 2 2n. Show that for each E > 0 there exists a p x n matrix A = (a,,) with I u,, I 6 E for all i, j , such that the mapping x ~ g ( x = ) f(x) A * x of U into RP is an immersion.(For each k < n, consider the C" mapping Fk : Mt x U -+ Rp" (notation of Problem 4) defined by F&Z, x) = Z - Df(x), and remark (16.23.1) that the image of Fk is of measure zero in Rp", by using Problem 4; the complement of the union of the images of the FI for 0 =< k 5 n - I is therefore dense in RP".) -+
+
24. INTEGRAL O F A DIFFERENTIAL *FORM PURE M A N I F O L D O F DIMENSION n
OVER A N ORIENTED
(1 6.24.1) Let X be an oriented pure manifold of dimension n 2 0. Then there exists a C" differential n-form uo on X, belonging to the orientation of X. We shall define on X a positive Lebesgue measure (16.22) pvo depending on uo . To do this it is sufficient to define a Lebesgue measure on U, for each chart (U, cp, n) of X with U connected, and then to show that if (U',cp', n) is another chart with U' connected, and if p, p' are the measures defined on U and U', then the restrictions of p and p' to U n U' are equal (13.1.9). By hypothesis, if x = (t', .. . , E cp(U), we may write
r)
uo(cp-'(x))
=f(C', . . . , 5")
dt'
A
dt2 A
A
dt",
and since U is connected, f is of the same sign throughout cp(U). More precisely, if R" is endowed with the canonical orientation (1 6.21.4), thenf(x) > 0 (resp. f ( x ) < 0) in cp(U) if cp preserves (resp. reverses) the orientation. By definition, the measure p on U is the image under 'p-' of the measure Ef. I, where 1 is the measure on q(U) induced by Lebesgue measure on R",and E is + 1 or - 1 according as cp preserves or reverses the orientation. Likewise, if x E cp'(U'), we have uo(cp'-'(x)) = f ' ( x ) dC;' A dtz A A d y , and we define the measure p' on U' as the image under q'-' of ~ ' *fA', ' where I' is the measure on cp'(U') induced by Lebesgue measure, and E' is + I or - 1 according as cp' preserves or reverses the orientation. Now let -
0 : cp(U n U')
+ cp'(U n U')
0
172
XVI
DIFFERENTIAL MANIFOLDS
be the transition diffeomorphism, and let J(0) be its Jacobian, which is > O in cp(U n U') if E' = E , and c0 if E' = - E . For each x E cp(U n U') we have
Ax) =f'(e(x))J(e)(x)* To prove that the measures induced on U n U' by p and p' are equal, we may assume that U = U'. Then, by (16.22.1), the image under 0 of the measure cf * 1= ( c ( f ' 0)) * (J(0) .1)is equal to cf' * At if e' = E , and to - ~ f l A' if E' = - E . Hence in both cases it is equal to ~lf'* I, which proves our assertion. The fact that the measure puois a Lebesgue measure follows from the fact that uo(x) # 0 for all x E X; moreover it is clear that this measure is positive. If u1 is another C" differential n-form on X, belonging to the orientation of X, then u1 = hu,, where h is a C"-function which is >O at all points of X, and it follows immediately that p,, = h * puo (13.14.5). 0
1
(16.24.2) With the same hypotheses and notation as above, consider now an arbitrary differential n-form u on X. We have u = go,, where g is a realvalued function on X. The n-form u is said to be integrable (or integrable over X) if g is p,,-integrable, and the number g dp,, is called the integral of u (or
s
/
Ix
the integral of u over X) and is denoted by u, or u, or Jx u(x). We must show that this definition is independent of the C" n-form u, chosen in the orientation of X. Now, if u1 is another n-form of class C" in the orientation of X, then u1 = hu, , where h is a C"-function on X, everywhere >O. Hence u = gh-'ul; but also p,, = h * p u o , and g is puo-integrableif and only if gh-' is h * p,,-integrable (13.14.3). Moreover we have P
P
(1 3.14.3), which proves the assertion.
The integrable differential n-forms on X clearly form a real vector space, and UH u is a linear form on this space, taking positive values on forms u 2 0 (in the sense defined in (16.21.2)). If we fix a C" n-form u in the orientation of X, it is clear that the linear form f- fu on 2 ( X ) is a positive Lebesgue measure, and conversely all positive Lebesgue measures on X are of this type for a suitable choice of u. These measures are also called volumes on the oriented manifold X, and the corresponding forms u are called volume-forms.
s s
(16.24.3) A differential n-form u = gu, on X is said to be measurable (resp. locally integrable, resp. negligible) if the function g is measurable (resp. locally
24 INTEGRAL OF A DIFFERENTIAL n-FORM
173
integrable, resp. negligible) with respect to the measure p v 0 . It is immediately verified that these notions are independent of the choice of the n-form u, in the orientation of X (13.15.6). If u is locally integrable and of class C", then fu is integrable for each measurable functionf which is bounded and of compact support. Let u be a locally integrable differential n-form on X, let (uk) be a locally finite covering of by relatively compact open sets, and let (uk) be a continuous partition of unity subordinate to (uk). Then u is integrable if and only if
x
the series
zj1gukl dpuois convergent; in which case the series juku k
k s
converges and has u as its sum. If each U, is the domain of definition of a chart ( U k ,q k ,n), where (Pk is orientation-preserving, and if
(16.24.4) Let X, be an orientable pure manifold of dimension n ; let X be the oriented manifold obtained by endowing X, with an orientation, and let -X be the oppositely oriented manifold. If u is an integrable differential n-form on X, then u is also integrable over - X , and
(1 6.24.5) Let f:X' + X be a diffeomorphism of a connected oriented manifold X' onto an oriented manifold X. Then, for each integrable n-form u on X, the form 'f(u) is integrable over X', and we have n
n
(16.24.5.1)
where the sign is + or - according asf'preserves or reverses the orientation. For by the method of calculation indicated above, we reduce immediately to the situation where X and X' are open subsets of R", where the result follows immediately from the definitions and the formula (16.22.1.1) for change of variables.
174
XVI DIFFERENTIAL MANIFOLDS
(16.24.6) Let Y be an oriented manifold of dimension m, X a manifold of dimension n >= m, and f : Y + X a mapping of class C ' (r 2 1). We have seen (16.20.9) how to define the inverse image 'f(u) of an m-form (i on X. If this inverse image is integrable, we may consider its integral 'f((i).When Y is an oriented submanifold of X, andf is the canonical injection, we shall often write
sy
.Jy (i in
place of .Jy ~((i).
Example (16.24.7) Integration on a sphere. Let n > 0. Consider the closed parallelotope P in R" defined by the inequalities
-nienjn (the @ being the coordinates of a point of P). The interior P of P is defined by -tnrej6$n
(isjsn-i),
the same inequalities with all the signs 5 replaced by <. It is immediately checked (by induction on n) that the mapping $ : P + S, which maps the point (O', . .., 0") of P to the point of S, with coordinates
t'
= sin 8',
c2 = cos e1 sin e2, (16.24.7.1)
5" = cos el cos e2 -. cos en-' to = cos e1 c o e2 ~ . - cos e-,'
sin en, cos en,
is surjective; its restriction to f' is a difleomorphism of this open set onto an open subset U, of S,, whose complement is contained in the hyperplane = 0 and is therefore negligible (16.22.2). Let U denote the open subset of U, which is the complement of the intersection of Uo with the hyperplane to = 0, so that the complement of U in S, is also negligible; U is the image under $ of the open set Q c P consisting of the points such that 0" f 3n, and the 8' form a system of coordinates in U. If V is the complement of the hyperplane to= 0 in Rd+',then the n-form (i on U defined in (16.21.10) is induced by the n-form on V
<"
(16.24.7.2)
because in V we have
24 INTEGRAL OF A DIFFERENTIAL n-FORM
175
and r = 1 on S, . The " triangular" form of the equations (16.24.7.1) shows that the form (16.24.7.2) is equal to
and consequently, on U,
(16.24.7.3)
0
= cos"-l 8'
*
COS'-~
8'
cos P-' do'
* *
A
* *
A
68".
It follows that for a function f on S, to be such that the form f * u is integrable, it is necessary and sufficient that the function
f($(el, ..., e n ) ) ~ ~ s n - 1 e 1 ~ ~ ~ n - 2 e z - - ~ ~ S e n - 1 should be Lebesgue-integrable over P, and then
l.f* (r
-
L2 n1.2
cos"-' 8' do'
n l2 ~-x12-.*
n/2
J - n l c o s On-' do"-'
J:j($(€l', . . .,On)) do".
The form u (also written d")) is called the solid angle form on the sphere S, oriented toward the outside. (16.24.8) Let X , Y by two orientedpure differential manifolds, of dimensions n and m , respectively, f : X +. Y a surjective submersion, u an integrable dgferential n-form 2 0 on X , and [ a locally integrable differential m-form on Y such that [ ( y ) > 0 (16.21.2) almost everywhere on Y. Then for almost ally E Y the (n - m)-$orm o/[(y) (16.21.7) is defined and integrable over f - ' ( y ) (endowed with the orientation induced by f from the orientations of X and Y ) ;the form y~ [ ( y ) /I- ,(y) u / [ ( y ) is integrable over Y, and n
( u k
n
n
There exists a denumerable open covering ( u k ) of X and charts , ( P k , n), ( f ( u k ) , $ k , m ) of X and Y, respectively, such that (Pk(Uk)
= $k(f(uk)
I"-",
where I = ] - I , I[, and f l u k = Fk ( P k , where Fk is the restriction to q , ( u k ) of the canonical projection of R" onto R". Using the integrability 0
0
176
XVI
DIFFERENTIAL MANIFOLDS
criterion of (16.24.3), we reduce immediately to the situation in which Y is an open set in R", X = Y x I"-" and f = pr, . Then we have u(x) = u(x) d t l
A
dt2 A
A
dy,
[ ( y ) = w(y) dt'
A
* *.
A
dt",
where u is integrable with respect to Lebesgue measure R on X, and w is locally integrable and # O almost everywhere with respect to Lebesgue measure A' on Y. The form u/[(y), defined for almost all y = (t', ..., t"), may be written as
. . c)
Now the function (t', . , H u(t', . .. , 5")/w(t1,.. . , 5") is measurable with respect to the measure w * R on R",and this measure may be thought of as the product measure (w * A') 63 A", where 1" is Lebesgue measure on R"-" (13.21.1 6). The proposition is therefore a consequence of the LebesgueFubini theorem (13.21.7) and the definition of the orientation on the fibers f +(Y). (1 6.24.9) Application : calculation of integrals in polar coordinates. Let n 2 2. We shall apply (16.24.8) by taking X = R" - {0},Y = R*,= 10, co [, f(x)= llxll = ((5')' * * * ((?)2)"2, so that f -'(u), for u > 0, is the sphere u Sn-' homothetic to S,,-,. Also take [(t)= t-' d t , and take u to be an n-form g * u o , where uo is the canonical n-form dt' A * - * A d y , and g is Lebesgue-integrable over X. The (n - 1)-form uo/[(u) on u * Sn-, may be calculated as follows: Let 0' be an (n - 1)-form on a neighborhood of u * S,,-l in X, such that uo = 'f([) A a'. Let h, : x w u x be the homothety of ratio u on X. Then we have
+
+ +
9
-
'h,(oo) = t~u(Y(o) A 'W') in a neighborhood of S,,-'; but it is immediately seen that 'h,(uo) = 24" * uo
9
'h,(Y(CN
=
m.
In view of the uniqueness of uo/[(u),it follows that 'hu(uo/[(u))= U" . S,-l, whence by (16.24.5.1)
1
s(uo/~(u)) = u" I s n - g ( u * z)a("-')(z).
u.s,-,
Hence the formula (1 6.24.8.1) gives
on
24 INTEGRAL OF A DIFFERENTIAL n-FORM
177
Using the formula (16.24.7.4), we obtain finally the formula for calculating an integral over R" - {O} in polar coordinates:
Of course, we should arrive at the same formula by calculating the Jacobian of the diffeomorphism of P x R; onto an open set in R" - (0) with complement of measure zero. In the notation of (16.24.7) (with n replaced by n - 1) this diffeomorphism is u * b,t. Applying (1 6.24.9.1) to the particular case where g is the characteristic function of the unit ball B,, : llxll 5 1 in R", we obtain
=Is
1 V,=-C&, n
where Q,, a("-"iscalled the solidangle in R" or the superjicialmeasure n- 1 of Sn-l. From the preceding calculations and the formula (14.3.11.3), its value is (1 6.24.9.3)
nn=
n71"J2
T($n
+ 1)'
or equivalently,
(1 6.24.9.4)
2nKn - I
1 3*5
0
.
.
(2n - 3)'
We remark that, with the definition of do)given in (16.21.10), the formula (16.24.9.1) remains valid for n = 1 , for it then takes the form
178
XVI
DIFFERENTIAL MANIFOLDS
(16.24.10) The interesting thing about the preceding method is that it applies without modification when f is any positive and positively homogeneous function of degree 1 on R" - {0} (i.e., such that f(u * x) = uf(x) for all u > 0) of class C" and whose differential df is nonzero at allpoints of R" - (0).If El is the submanifold given by the equation f ( x ) = 1 in R",and of is the form uo/y([(l)) on E l , then by the same reasoning as before we shall obtain the following generalization of (1 6.24.9.1) :
From a practical point of view, if the set U in El of points where aflat' # 0 for some index i has a complement of measure zero, we may determine of by the method of (16-21.9.1): We consider, in a neighborhood of U in R" - (0) the following (n - 1)-form: (16.24.10.2)
of = (- l)i-'
d5'
0
A
A
dt'
A
A
dc
amti
and we take for of the form induced by
0;
on U.
(Stokes' formula, elementary version) Let V be an open subset of R"-', U an open subset of R", F a function of class C" on U such that D,F = dF/a[' # 0 in U and such that the mapping
(16.24.11)
T")H(F(tl,* . . , T"), t2,. . ., 5") is a diffeomorphism of U onto I x V, where I is an open interval in R. For each u E I, let E, be the closed submanifold of U defined by the equation F(C1, . . ., = U, whose image under $ is { u } x V , and let o, be the ( n - l)-form on E, equal to $ : (5'
9
* *
3
r)
uo/'F(e,*),where e,* is the unit covector in T,(R)*. Let [a,b] be a closed interval contained in I. Then, for every C'-jiunction f on U,we have (1 6.24.11.I)
1.., 1.
D l f ( t l , . .. , 5") dt' d t 2 * - . d r
=
f(z)DIF(z)ob(z)
-
6.
f(z)Dl
F(z)ae(z),
where Ua,bis the set of x E U such that a 5 F(x) b, and the orientations on E,, and Eb are induced by F from the canonical orientations of R"-' and R (16.21.9.2).
24 INTEGRAL OF A DIFFERENTIAL n-FORM
Put ui,b= ]a, b[ x V, and let g Fubini theorem, we have
s
D,g(u, t2,...,Y) du
U'a.b
= Jv d5'
A
A
*
dr"
A
J:
= f o
I)-' on I x V. By the Lebesgue-
dt2 A . .*
A
dc
D,g(u, t', . . . , r") du
t', .. .,5") - g(a, t2,... ,5")) d5'
= s?(b,
179
A
* * *
A
dy.
5', .. .,Y), the projection of z on R"-' is
Observe that if z = $ - I @ ,
The calculation of a b indicated in (16.21.9.1) shows that this (n - 1)-form is induced on E, by the (n - 1)-form x ~ d 5 'A * . . A d<"/D,F(x)on U. There is an analogous calculation for a,, and hence we see that the value of the right-hand side of (16.24.11.1) is
s s
D, g(u,
e', . . . , Y) du
A
dt2 A
A
dY.
U'a,b
However, we have D, g($(x)) = D,f(x) (D,F(x))-' (8.2.1) for each x E U, and since 'I)(uO)(x)= D,F(x) . uo(x), we obtain finally
D, g(u, t', . . . , r") du A dt2 A . *
A
dy"
U'a,b
Dlf(tl, . . . , t ' ) d t ' ~ . . . ~ d r " , - sun..
which proves (16.24.11.I).
Remark (16.24.12)
In the preceding calculation, replacef by the function
xc@f(t. a and b by t and t
t', . . . ,r") dt,
+ h ; divide by h and let h + O ; j ( c l , .. . , 5") d t l
* * .
then we obtain
dr" =
I.
~(z)o,(z).
In particular, takingf= 1, this gives an interpretation of 0, as the derivaJEt tive of the volume of U,, ,.
180
XVI DIFFERENTIAL MANIFOLDS
(16.24.13) The notion of the integral of a differential n-form on an oriented manifold X is easily extended to the case of a differential n-form with values in a vector space F (16.20.15). For example, let I be an open interval in R, yo a C'-mapping of I into an open subset A of C,and f:A + C a continuous function. If y is the restriction of yo to a compact subinterval J of I, then the integral denoted by f(z) dz in (9.6) is none other than
J,
+ i dt2)),
fYo(f' (&'
in which we have taken the inverse image under yo of the complex-valued differential 1-formf. (dt' + i dtz) defined on A c RZ.
PROBLEMS
to> (l$l(fJ)z)l'z, where (to,...,$) are the coordinates in R"+l . If a = (a",al,...,a")E H,show that there Let H be the subset of R"+' defined by the inequality
exists a constant cn > 0, independent of a, such that
1"
exp( -
zo
aJtJ)dt" d f l
* * *
(
d p = cn (a")' -
2
1=1
- (n + 1)/2 (a?))
.
(Use a suitable Lorentz transformation.) Let H be a hyperplane in Rn.There exists a Euclidean ciisplacement transforming Rn-l (identified with the subspace of R"generated by the first IZ - 1 vectors of the canonical on R"-' is a basis) into H. The image under this mapping of Lebesgue measure measure on H which does not depend on the displacement chosen, and we denote it again by Show that ifu is a unit vector orthogonal to H, and A an integrable subset of H,then the orthogonal projection p(A) of A on R"-' has measure X.-l(~(A))
I
= (enlu) lhn-l(A).
Let P be a compact convex polyhedron in R"with nonempty interior. If F k ( 1 5 k r ) are the faces of P (Section 16.5, Problem 6), the ureu (or the (n - 1)-dimensionalureu) of P (or of the frontier of P) is defined to be the number dm-l(P) = x h . - l ( F k ) . For k
each vector u E S.-l, we denote byX.-,(P, u) the measure of the orthogonal projection of P on the hyperplane (xlu) = 0. Show that
(Cuuchy's formula for convex polyhedra). (UseProblem 2.) Deduce from this formula that if P' is another compact convex polyhedron containing P, then dm-,(P)= dn-*(P).
24
4.
INTEGRAL O F A DIFFERENTIAL n-FORM
181
Let C be a compact convex body in R" (Section 16.5, Problem 6). There exists a sequence (P,) of compact convex polyhedra tending to C with respect to the Hausdorff distance h defined in Section 3.16, Problem 3 (Section 16.5, Problem 8(a)). Show that the sequence (-W~' -~(P,)),~ tends to a limit which does not depend on the particular sequence P, converging to C. This limit is called the area (or the (n - 1)-dimensional urea) of C (or of the frontier of C), and is denoted by S'"-~(C). For u E S.-l,we denote again by h.-,(C, u) the measure of the orthogonal projection of C on the hyperplane (XI u) = 0. Show that u ~ h , - ~ ( u) C is , continuous on S.-l (use Problem 3) and that
(Cuuchy'sformula for convex bodies). If C' is another compact convex body containing C, then .dn-l(C) c .!dn-l(C').If (C,) is a sequence of compact convex bodies tending to C with respect to Hausdorff distance, then lim -W'n-l(C,) = S#"-~(C). Consider m-w
the case C = S.- 1 .
5. Let @" be the set of all compact convex bodies in R",endowed with the Hausdorff distance (Section 3.16, Problem 3). Define by induction on n a sequence of n 1 realvalued functions Win(0 i 5 n) on &, as follows: W,,(C) = hl(C), W11(C) = 2. For n > 1, Wo,(C) = A,@), and for 1 5 i 5 n,
+
where pu is the orthogonal projection on the hyperplane (x I u) = 0. Show that each of the functions WI, is increasing and continuous on @", and that W,.(aC) = a"-'Wl.(C) for all a > 0. If A , B are convex bodies belonging to L, , such that A u B is convex and A n B has a nonempty interior, then W&
u B)
+ W d A n €3)
= Wln(A)
+ W,.(B).
(Observe that if u E A and b E B, there exists a point of A n B in the segment with a,b as endpoints. Consequently, if H is a supporting hyperplane of A n B, then H is a supporting hyperplane of either A or B. Deduce that P,(A n B) =pU(A) np,(B).)
In particular, nW,,.(C) 6.
= .dn-l(C)
(Problem 4).
If C is a compact convex body in R" containing the origin, recall that the function of support of C is the function H(z) = sup(x1z) (Section 16.5, Problem 7). For each
+
XEC
u E Sm-], let b(C, u) = H(u) H(- u) (the "width" of C in the direction u, cf. Section 14.3, Problem 9(a)). Show that, in the notation of Problem 5,
182
XVI
DIFFERENTIAL MANIFOLDS
(Proof by induction on n. For each u E S.- ,let E(u) denote the hyperplane (XI U) = 0 in R",and let u("-*) be the differential (n - 2)-form on n E(u) which is the image of a("-') by a rotation of R" transforming Sn-' into Sn-l n E(u). Show that the integral n- 1
a - l nE(u)
b(P"(C),v). U:-')(v)
can be written in the form
where P is the submanifold of S"-lx S.-l consisting of pairs (u,v) such that (u Iv) = 0, and w is a (2n - ])-form on P obtained by the procedure of (16.21.7); use (16.24.8).)
7. (a) Show that if P is a compact convex polyhedron of dimension n in R",then the area S ~ " - ~ ( Pis) equal to the Minkowski area of P (Section 14.3, Problem lqc)). Deduce that, for each p > 0, in the notation of Section 3.6,
(b) Show that, for each compact wnvex body C in R",we have
(Steiner-Minkowski formula). (Prove the formula first in the case that C = P is a compact convex polyhedron of dimension n, by using (a) and induction on n. Then pass to the limit in L. .) (c) Deduce the formula
sn
and r 2 0. (Observe that V,+,(C)= V,(V,(C)) and use the Steinerfor 0 5 i Minkowski formula.) (d) Deduce from (b) and (c) that formula (1) is valid for any compact convex body C . In particular, .dn-l(C)is equal to the Minkowski area of C (Section 14.3, Problem 1O(C)). 8.
Let LA be the set of all nonempty compact convex sets in R",endowed with the Hausdorff distance (Section 3.1 6, P.roblem 3). 9; is a compact space in which L,,is dense. Show that the functions W , . defined on 9. extend by continuity to S;(induction on n). If A is a compact convex set of dimension t n in R",then
25 EMBEDDING AND APPROXIMATION THEOREMS
183
9. Let A c RP, B C Rq be compact convex sets. Show that
(Apply the Steiner-Minkowski formula to A x B, and use the Lebesgue-Fubini theorem.) Hence calculate the values of Wl.(C) when C is a cube in R". 10. Let C C Rn be a compact convex body of dimension n. Show that the set CV,(CC) is convex (express it as an intersection of translates of C). For each r > 0, we have
V,(CV,(CC)) c C. By using the continuity of the functions W,.(C) with respect to C, deduce that the function rH/\.(CV,(CC)) has a derivative on the right at the origin, equal to - .dn-l(C). 11. Let X, Y be two oriented pure differential manifolds, of respective dimensions n and m ;let f:X .+ Y be a submersion, x a point of X, y =f ( x ) . We shall use the notation of (16.21.7). Let 5 be a C"m-form on Y belonging to the orientation of Y.For each k
k 6 m, 5 defines a canonical isomorphism z Y w@ r ( y ) ( ~ y ) of A T(Y) onto A T(Y)*, such that @r(y)(z,) = zy _I ( ( y ) . Let u be a C" (n - m k)-form on X with compact k
A
rn-k
+
support. To each k-vector zy E T,(Y) there corresponds an (n - m)-form f - ' ( y ) such that, for x E f - ' ( y ) , we have
&, on
in the notation of (16.21.7). This form is independent of the choice of 5 in the orientation of Y. We give each fiberf - I ( y ) the orientation induced by f from the orientations of X and Y (16.21.9.1). Show that there exists a unique C" k-form y on Y such that, for all y
EY
and all z,
k
E
A T,(Y),
This form is denoted by ub and is called the integral of u along the fibers ofJ (Reduce to the case (16.7.4).) If 3!,' is a C" k'-form on Y,then cd A j?= (a A *f(j?))b.
25. EMBEDDING A N D A P P R O X I M A T I O N THEOREMS. T U B U L A R NEIGHBORHOODS
(16.25.1) Let X be a dijierential manifold, U a relatively compact open subset of X. Then there exists an integer N and an embedding (16.8.4) of U in RN. There exist a finite number of charts (uk, (Pk,nk) of X (1 6 k 5 m) such that the U, cover the compact set 0. Next, there exists a family (Vk)lSksrn of open subsets of X which cover 0 and are such that ?k c U, for each k
184
XVI
DIFFERENTIAL MANIFOLDS
x,
(12.6.2). Finally, there exists a family of C"-functions (fk)lsksm on with values in [O, 11, such that ~upp(f,)c uk andfk(x) = 1 for all x E 0,. We shall show that the mapping : XH((fk(X))i~k~m? (fk(X)(Pk(X))ljk4rn)
of 0 into RN = R" x
n m
R"kgives the desired embedding by restriction to
k= 1
U. It is clear that g is of class C" (16.6.4). Next, g is injective. For if x, x' are two distinct points of X such that fk(x) =fk(x') for I r k r m , then since x E v k for some k, it follows that &(X) = 1 and thereforefk(x') = 1, so that x' E uk; but then, since x # x', we have fk(x)qk(x) = (Pk(X)# (Pk(X')=f,(x')qk(x') so that g(x) # g(x'). Since 0 is compact, it follows that g is a homeomorphism of 0 onto g ( 0 ) (3.17.12), hence of U onto g(U). Hence it remains to show that g is an immersion (16.8.4) at each point x of U. Let k be such that x E vk , and let p be the projection of RN onto the factor Rnk,so that p o g = (Pk on v k . Since T,(q,) = Tecx,(p)0 T,(g) is of rank nk, it follows that T,(g) is of rank nk . This shows that g is an immersion at x (1 6.7.1) and finishes the proof. One can in fact show that there exists an embedding of the whole of X into an RN,and that if X is pure of dimension n , one can take N = 2n + 1 (Problems 2 and 13(c)). The above embedding theorem will enable us to extend to manifolds the Weierstrass approximation theorem (7.4.1), polynomials being replaced by C"-functions. We shall begin by establishing two extremely useful auxiliary results on " tubular neighborhoods " of a submanifold of RN. (16.25.2) Let X be a pure submanifold of RN,of dimension n. Let j : X -+ RN be the canonical injection and U a relatively compact open subset of X . For each x E X let M, be the n-dimensional subspace of RN which is the image of T,(X) under the mapping r , o T,(j) (16.5.2). Let d be the distance function on RN derived from the scalar product (x I y) = t j q j , and let N, be the orthogonal i
supplement of M, in RN (6.3.1). Suppose that we have de$ned, on an open neighborhood V of 0 in X , N - n mappings uj (1 5 j 5 N - n ) of V into RN, of class C", guch that for each x E V the uj(x) form a basis of N,. Then there exists an open neighborhood T of U in RN and a dixeomorphism Y H (n(Y), B(Y))
of T onto U x RN-"such that, for each y whose distance from y is equal to d(y, X).
E T, ~
( yis) the unique point of X
25 EMBEDDING AND APPROXIMATION THEOREMS
185
Consider the mapping g : V x RN-"+ RN defined by g ( x ; t1 7 . .
9
tN-,,)
=x
+
N-n j = 1
tj
uj(x).
Clearlyg is of class C". Moreover, for each point a E V, the tangent mapping T(,, ,&)is bijective, hence (16.5.6) there exists an open neighborhood W, c V of a in X and an open ball B, with center at the origin in RN-"such that the restriction of g to W, x B, is a diffeomorphism of this open set onto an open neighborhood T, of a in RN. Let y ~ ( n , ( y ) O,(y)) , be the inverse diffeomorphism. We shall show that there exists a ball S, c T, with center a such that, for each y E S, , n,(y) is the only point x E X such that d(y, x ) = d(y, X). We shall use the following lemma: (1 6.25.2.1) For each a E X there exists a ball S: in RN with center a such that, for each y E S: , there exists at least one point x E X for which d(x, y ) = d(X,y). For such a point x, the vector y - x is orthogonal to M, .
Since X is locally closed in RN,there exists a closed ball S L with center a and radius r, in RN such that X n Sg is closed in Se, and therefore compact. Let S: be the closed ball with center a and radius Sr, . For each y E S: ,we have d ( y , a) S +r, , and for each z E X such that z 4 S: , we have d(y, z) 2 %r,, so that d ( y , X) = d ( y , X n Se). Hence X n contains a point x such that d ( y , x) = d(y, X) (3.17.10). Moreover, the function z-h(z) = (d(y,2)' = n
j= I
(q' - [ j ) 2 is of class C" on X and admits a minimum at the point x ; hence
d,h = 0 (16.5.10). However, relation d,h = 0 may be written as N
1 (qj j= 1
cj)
dcj = 0 ;
SJs
also, for each vector t E M,, the numbers (t, d") ( I N) are the components o f t with respect to the canonical basis of RN.Hence the vector y - x is orthogonal to M,. Having established the lemma, we shall now argue by contradiction. Suppose that there exists a sequence (y,) of points of T,, tending to a, and a sequence (x,) of points of X such that x , # n,(y,) and d ( y , , x,) = d ( y , , X). Since d ( y , , x,) 5 d ( y , , n,(y,)), it follows that d(y,, x,) +0 and hence that x, + a ; so we may suppose that x , E W,. By virtue of the lemma (16.25.2.1), we can write y , - x ,
N--n
= j = 1
tj,uj(x,);and since x , # x,(yY), the point
186
XVI
DIFFERENTIAL MANIFOLDS
ctit,
..
N--n
does not belong to the ball B,. If we put r,' =
therefore bounded. Since
N-n
i= 1
the sequence (r;') is
(r;1tjv)2= 1, we may, by passing to a sub-
j= 1
sequence of the sequence (yv),assume that each of the sequences (r;'tjV) has a limit ti (1 S j S N - n); we have then t j Z = 1; but on the other hand,
1
rv-'(yv - x,) --f 0 as v + 03, and hence
i
i
t j ui(a) = 0. This is impossible, since
the tJ are not all zero and the uj(a) are linearly independent. If b is another point of V such that S, n Sb # @, then it follows from above that n, and q,agree on S, n s b . Hence there is a unique function n, defined on the union S of the open sets S, (a E V), which extends each of the functions n, . For each a E V, let W: c W, be an open neighborhood of a in X , and B: c B, an open ball with center 0 such that g(W: x B:) c S,. Cover 0 by a finite number of open neighborhoods W:, (1 5 i s r), and let B" be an open ball with center 0 in RN-" (hence diffeomorphic to RN-") contained in the intersection of the B:, . Then g(U x B") is an open subset of RN contained in S and containing U, and for each (x, t) E U x B" we have n(g(x, t)) = x
and
t = g(x, t) - x .
This proves that g l ( U x B") is a bijection of U x B" onto g(U x B"), and hence a diffeomorphism of U x B" onto an open neighborhood of U in RN (16.8.8(iv)). (16.25.3)
There does not always exist a system (ui) of mappings of V into
RN having the properties of the statement of (16.25.2). For example, if X is a
nonorientable compact manifold embedded in RN (16.25.1), the existence of an open neighborhood of X in RN diffeomorphic to X x RN-" would contradict (16.21.9.1), since every open subset of RN is an orientable manifold. However, there is the following weaker result:
(16.25.4) Let X be apure submanifold of RN of dimension n. Then there exists an open neighborhood T of X in RN and a C" submersion n of T onto X with thefollowing property:for each y E T, n(y) is the onlypoint of X whose distance from y is equal to d(y, X), andfor each x E X, the3ber A - ' ( x ) is the intersection of T and the linear manifoldx + N, (the space N, being defined as in (16.25.2)). It is enough to prove that, for each a E X, there exists an open neighborhood T, with center a in RN such that X n T, is closed in T,, and a surjective submersion n, : T, + X n T, such that, for each y E T,, n,(y) is the unique point of X whose distance from y is equal to d(y, X), and such that, for each
25
EMBEDDING AND APPROXIMATION THEOREMS
187
+
x E X n T,,, the fiber a;'(x) is the intersection of T,, with x N,; the union T of the T,, will then have the required properties. By virtue of (16.25.2), it is enough to show that there exists a relatively compact open neighborhood V of a in X on which N - n functions uj can be defined so as to satisfy the conditions of (16.25.2). By means of a displacement we may assume that a = 0 and that Ma is the space R" spanned by the first n vectors of the canonical basis ( e i ) 1 5 i sof N RN.Consequently (16.8.3.2) there exists a relatively compact open neighborhood U of 0 in R" such that a neighborhood V of 0 in X is formed by the points of U x RN-"which satisfy the equations ( * ' j = jj(t', . .., 5") (1 j 5 N - n), where the fi are C"-functions defined on U which vanish, together with their first derivatives, at the origin. For each point x = (5'
9
**.,
t",fi(t',. . * , t"),
. * * , f N - n ( t l , * *51) *,
of V, the space M, is defined by the N - n linear equations ["+j
-
n
i= 1
Difi(t1,... , 5") - [' = 0
(1
sj 5 N - n)
and consequently the functions uj(x) =
-
n
i= 1
DiJ;.(tl,. . . , Y)ei
(1
5 j S N - n)
satisfy the required conditions. We can now state the approximation theorem: (16.25.5) Let X, Y be two dzTerentia1 mangolds, K a compact subset of X, andf : K + Y a continuous map. Let d be any distance which defines the topology of Y, and let E > 0. Then there exists an open neighborhood U of K in X and a C"-mapping g : U + Y such that
4 f 0 ,d-9)5 E for all x E K.
Let U, be a relatively compact open neighborhood of K in X, and Vo a relatively compact open neighborhood of f ( K ) in Y (3.18.2). We may assume that Uo is embedded in R" and V, in R" (16.25.1). Let T be an open neighborhood of V, in R"having the properties of (16.25.4), and let 6 < +E be a positive real number such that every point of R" whose distance from f ( K ) is j S lies in T (3.17.11). By the Weierstrass approximation theorem, there exist n polynomials hi (1 51 n) in m variables such that if h = (h, , , . . ,h,,),we have
188
XVI
DIFFERENTIAL MANIFOLDS
d(f(x),h(x)) =< 6 for all x E K (7.4.1). Since h(K) c T, there exists an open neighborhood U c Uo of K in X such that h(U) c T. For each x E U, put g(x) = n(h(x))E Vo . Sincef(x) E Vo , we have by definition (16.25.4) and therefore dcf(x), g(x)) 6 E for all x E K.
PROBLEMS
1. Let M be a pure submanifold of R", of dimension n, and let A be a compact subset of
M. Let Rl be the set of linear mappings u :R" +RZ"+'such that U I M is of rank n at every point of A, and let R2 be the set of linear mappings u :Rm-+R2"+' such that u I A is injective. Show that, if m 2 2n 1, R l and R2are dense open sets in the vector space 9 ( R m ;R2"+l).(Show first that the complements Ql ,Q2 of a,, R2 are closed, by using the compactness of A; then show that DZ are meager subsets of 9(R"; Rz' l), by covering A with a finite number of charts, and using Sard's theorem.)
+
+
2.
Let M be a pure manifold of dimension n, and let (u&1 be a locally finite denumerable open covering of M such that each uk is relatively compact and is the domain of a chart (uk,pk, n) of M, where p k = (pi, ...,pi) is such that pk(uk) is the cube in R" defined by 18' 1 I < 4 for 1 s j 5 n. For each k, let v k be an open set such that v k C u k and the v k cover M. Let gk be a Cm-mappingof M into [0,1], with support contained in uk, and equal to 1 on v k .
-
m
(a) Show that the function uo =
k=l
kgk is of class Cmand is a proper mapping (Sec-
tion 12.7, Problem 2) of M into R+ . (k 2 1, 1 6 i $ n ) ar(b) Let (fdh)hZl denote the sequence of functions gk and ranged in some order. For each x E M, let u(x) denote the point of E = R") whose coordinates (all but a finite number of which are zero) are the uh(x)for h >= 0. Show that u is injective. (C) k t Ak (resp. B,) denote the union of the uh (resp. t h ) for h 5 k. Then U(&) 3 U(&) is contained in a vector subspace Ek of E, of finite dimension which we may assume to be 2 2 n 1. Show that the restriction of u to Ak is an embedding of Ak in & . (d) Let E'+ denote the topological product RT, and let F+ = E Y + l . An element v E F+ is therefore of the form v = (vl, ..., V Z . + ~where ), vj = (tj,,),,~~, the sequence (.fJ;h)htOof numbers 2 0 being arbitrary. We may identify u with the linear mapping of E into RZ"+'which maps the point ( q & O of E (in which all but a finite number of
+
F+ consisting of the v E F, such that the restriction of u to u(Ak)is an immersion at each point of u(Bk),and such that the restriction of v to u(Bk) is injective. Use Problem 1 to show that Rkis a dense open subset of F+ . Deduce that the intersection of the 0, is dense in F+ . In particular, there exists w = ( w l , . . .,w ~ ~in +this ~ intersection ) such that the.coordinate of index 0 in w1 is fO. Show that the mappingf= w u is aproper embedding of M in R2"+'(Whitney's embedding theorem). 0
25 EMBEDDING A N D APPROXIMATION THEOREMS
189
3. Let M be a submanifold of dimension n in R". Show that for each point a E M there exists a number E > 0 with the following property: for each b E B n M, where B is the open ball in R" with center a and radius E , the orthogonal projection of B n M on the linear manifold Lb tangent to M at b (16.8.6) is a bijection of B n M onto a convex open subset of Lb . (Reduceto the case where a = 0 and T,(M) is the subspace Rnof R" generated by the first n vectors of the canonical basis (el)'6 1 s m , so that in a neighborhood of the origin M is defined by n - m equations ...,& (1 5 n - m).k t u l (1 5 i 5 n) be the vectors in Tb(Tb(M))which project orthogonally onto the el (1 5 i 5 n). Orthonormalize the sequence (ul, .. .,u., en+,, . ..,em), thus obtaining an orthonormal basis (vJl of R" whose elements are C"-functions of b, and the first n vectors of which form a basis of Tb(Tb(M)). Using the impkit function theorem, show that if E is sufficiently small, B n M is identical with the set
@+'=fJ(t',
sj
of points
m '=1
71vL,where the point y = (T'),
S16m
runs through an open set Hb in
R", and the q"+*= Fl(y,b) for 1 5 i 5 m - n are C"-functions of (y, b) in a neighborhood of the origin in R" x M. Finally show that if E is sufficiently small, the funcn
tion G(y)= C (7'- at)' 1=1
+I=C (Fj(y,b) - F,(a, b))' m--n
1
is convex in a neighborhood of
0 in R",where a = (a')E R" is sufficiently small.) 4.
5.
Let M be a pure manifold of dimension n. Show that there exists a locally finite denumerable open covering (Ar) of M such that every nonempty intersection of a finite number of the sets Ar is diffeomorphic to R".(Use Problems 2 and 3.) Let X be a pure submanifold of R", and let T : X
-+
B be a surjective submersion. Let
Y be the graph of T,which is a submanifold of X x B diffeomorphic to X, and consider Y as a submanifold of R" x B. Show that there exists an open neighborhood T of Y in Rm x B and a submersion p of T onto Y with the following property: for each 6 E B and y E T n pr;'(b), p ( y ) is the only point of Y n pr;'(b) whose distance from y in R" x {b} is equal to d ( y , Y n pr;'(b)) (dbeing the Euclidean distance on R" x {b}). (Use (16.7.4).)
6.
Let M be a pure differential manifold of dimension n and f a continuous mapping of M into R". Let F be a closed subset of M such that the restriction o f f t o F is of class C' (where r is an integer >0, or +a)in the following sense: at each point x E F there , such thatfo 9-l is of class C' at q ~ ( x )in , the sense of Section 16.4, is a chart (V, q ~ n) Problem 6. Let S be a neighborhood of the graph off in M x R". Show that there exists a mapping g : M +R" of class C, which coincides withfon F and is such that ( x , g(x)) E S for all x E M. (We may assume that M c R'"+' and extendfto R'"+' by the Tietze-Urysohn theorem. Let S' be a neighborhood of the graph offin R2"+'x R" such that S' n ( M x R") c S, and for each x E R'"+' let r ( x ) be the distance from (x,f(x)) to CS'. Show first, using Weierstrass' theorem and a partition of unity, that there exists a mapping h : R'"+' 4 R m of class C" such that Ilf(x) - h(x)ll< ar(x). Using Whitney's extension theorem (Section 16.4, Problem 6) show that there exists a mapping u : Rz"+'+R" of class C' which is equal to f- h on F and is such that Ilu(x)ll< t r ( x ) for all x.)
7. Let (X, R, T)be a fibration, F a closed subset of B, s a continuous section of X over B. Suppose that s is of class C' (r an integer >0, or f a ) in F,in the following sense: for
190
XVI
DIFFERENTIAL MANIFOLDS
each b E F there exist charts (U, p, n) of B and (V, $, m) of X such that s(U) c V and such that $ s 0 q~-' is of class C' at p(b) in the sense of Section 16.4, Problem 6. Show that for each neighborhood S of the graph of s in B x X, there exists a section s1 of X over B, of class C' in B, such that (s(b), sl(b)) E S for all b E F.(Embed X in R" (for some m) and apply the results of Problems 5 and 6.) 0
8. A Co-fibration h = (X, B, T)is defined by replacing, in the definition (16.12.1), dif-
ferential manifolds by topological spaces, and diffeomorphisms by homeomorphisms. X i s also said to be a Co-fiber bundle. If (X, B, T)is a fibration in the sense of (16.12.1), it defines a Co-fibration (called the underIying Co-fibration) by regarding X and B as topological spaces. Generalize to Co-fiber bundles the definitions and results of Section 16.12. Define likewise the notion of a Co-principd bundIe and a Co-vectorbundle, and generalize the definitions and results of Sections 16.14-16.19. Show that if a vector bundle E (in the sense of (16.1 5)) is such that the underlying Co-vector bundle is Co-trivializable, then E is trivializable. (Use Problem 7.) Show likewise that if two vector bundles (in the sense of (16.15)) E, F over B are such that the underlying Co-bundles are isomorphic, then E, F are isomorphic. (Consider the bundle Hom(E, F) and use (16.16.4)J 9.
Let (X, B, n-) be a Co-fibration. It is said to have the section extension property if every continuous section of X over a closed set A which is the restriction of a continuous section of X over an open neighborhood of A is also the restriction of a continuous global section of X (which need not agree with the preceding one on the open neighborhood of A). Suppose that B is separable, metrizable, and locally compact. Let (U.) be an open ) the section covering of B such that for each a the fibration induced on T - ~ ( U =has extension property. Then (X,B, T) has the section extension property. (Let A be a closed subset of B, and let Vo be an open neighborhood of A which is equal to the set of points a t which a continuous mapping go : B [O, I ] is strictly positive, the function go being equal to 1 at all points of A. Let (Vn)nzl be a locally finite open covering of B, and let (g.).21 be a continuous partition of unity such that V. = { x E B :9.h) > 0) for all n 2 1. Then the functions ho = g o and R, = ( 1 - go)gm (n 2 1 ) form a continuous partition of unity. Let s be a continuous section of X over A which extends to a continuous section of X over VO. Proceed by induction on n by introducing the functions fn = ho hI * h., and reduce the problem to the following one: Let u,v be two continuous functions on B with values in [0, I], and let U, V be, respectively, the open sets on which u(x) > 0, u(x) > 0; suppose that T - ' ( V ) has the section extension property. At the points x E U n A (resp. V n A) we have u(x) = I (resp. u(x) = I ) and there is a continuous section s over A such that s((U n A) extends to a continuous section s' over U. Show that sI ((U U V) n A) extends to a continuous section s" over U u V. For this purpose, consider the function w on U u V which is equal to 1 at points x such that u(x) u(x), and equal to u(x)/v(x) otherwise; observe that the set V n w-'(I) is closed in V, and extend the section which is equal to s' in V n w - I ( I ) and to s in V n A, to a continuous section over V. --f
+ + + - 7
10.
Let M be a pure differential manifold, and (U.) an open covering of M.Show that there exists an integer N, depending only on the dimension of M,and a denumerable locally finite refinement (V.) of the covering (Urn), consisting of relatively compact open sets V., such that no point of M belongs to more than N of the sets V.. (Embed M in some
25 EMBEDDING AND APPROXIMATION THEOREMS
191
Rm (Problem 2) and consider the U, as the intersections of M with open sets UL in Rm;this reduces us to considering only the case M = R". Let K,be the closed cube in Rmdefined by lz'l 6 n for 1 6 j 5 m ; decompose each set K. - K,-l into equal closed cubes, sufficiently small that each one is contained in some U: (3.16.6); then enlarge each of the cubes slightly to a concentric open cube.)
11. Let (gn).ro be a continuous partition of unity in a separable, metrizable, locally compact space B. Let V,, be the set of b E B at which g.(b) > 0. For each finite subset J of N let W(J) be the set of all b E B such that gi(b) > gi(b) for all i E J and j $ J. If J and J' are two distinct subsets of N having the same number of elements, then W(J) n W(J') = 0.For each 6 E B, let J(b) be the set of integers n such that gn(b)> 0, so that J(b) is finite. Then W(J(b)) C V. for all n E J(b). For each integer m > 0, if W, is the union of all the sets W(J(b)) such that J(b) has m elements, the W, form an open covering of B. Using these results and Problem 10, show that if B is a pure differential manifold and X a fiber bundle over B, then there exists afinite open covering (WJlsiSm of B such that X is trivialiable over each W r. Generalize Problem 5 of Section 16.19 by replacing the relatively compact open subset U of B by B itself. 12.
(a) Let M be a differential manifold of dimension n, let f : M +RP be a Cm-mapping, and let N be a compact subset of M such that f is of rank n at each point of N. Also let v be a chart of M with domain W, let V be a relatively compact open set such that 0 c W, let U be a relatively compact open set such that 0 c V, and let u : R"-+ [0,11 be a C"-mapping such that u(z) = 1 for z E @) and u(z) = 0 for z $ @). Suppose that p 2 2n. Show that for each E > 0, there exists a p x n matrix A such that: (1) IIA .I(( 5 E for all z E v(V); (2) the function ZH~(V-'(Z)) + A .z is of rank n in y(V); (3) the function ZH~(P)-~(Z)) u(z)(A .z) is of rank n in v(N nv). (Use Problem 5 of Section 16.23.) Deduce that the function g : M +RP defined by g(x) = f(x) for x $9,g(x) = f(x) u(v(x))(A . cp(x)) for x E W, is of class C", is of rank n at each - f(x)/lg E for all x E M. point of N u 0, and is such that (b) Let M be a differential manifold of dimension n, let f : M + Rpbe a C"-mapping. and let N be a compact subset of M such that f is of rank n at each point of N. Let 6 be a continuous real-valued function on M, everywhere >O. If p 2 2n, show that there exists an immersion g : M + RP which agrees with f on N and is such that Ilg(x) - f(x) I[ 2 6(x) for all x E M. (There exists an open set T 3 N in which f is of rank n. Consider a denumerable locally finite open covering (W,) of M such that the W, are domains of definition of charts and are contained in either T or M - N. Construct g by induction, using (a).)
+
+
13. (a) With the same notation as in Problem 12(a), suppose that f is an immersion, and that the restriction off to an open set S is injective. Put u(x) = u(~+-)), and suppose that p 2 2n 1 . Show that, for each E > 0, there exists a vector a E Rp such that: (1) /lal/5 E ; (2) the function xt+f(x) o(x)a is an immersion of M in RP;(3) therela tion g(x) = g(y) implies v(x) = o(y) and f(x) = f(y). (To show that the third condition can be satisfied, consider the open set D in M x M consisting of pairs (x, y ) such that u(x) # u(y), and the image of the mapping (x, y ) - (f(x) ~ - f(y))/(o(x) - a(y)) of D into Rp.) (b) Let M be a pure differential manifold of dimension n, let f : M +RP be an immersion, let S be an open subset of M such that f 1 S is injective, let N be a closed subset
+
+
192
XVI
DIFFERENTIAL MANIFOLDS
of M contained in S, and let 6 be a continuous real-valued function on M, everywhere >O. If p 2 2n 1, show that there exists an embedding g : M + Rp which agrees with f on N and is such that Ilg(x) - f(x)ll5 8(x) for all x E M. (Consider a denumerable locally finite open covering (W,) of M, where the W, are domains of definition of charts, the restrictions f I Wk are injective, and each Wk is contained either in S or in M - N. Construct g by induction on k, using (a).) (c) Deduce from Problems 12@) and 13(b) a new proof of Whitney's embedding theorem. (Problem 2; first define a proper mapping of class C" of M into R2"+'.)
+
+
Let M be a manifold of dimension n. Show that there exist 2n 1 real-valued functionsf, of class C" on M (I zj 5 2n 1) such that every C"-function on M is of the form F(fl, .. ,fZn+J, where F is a C"-function on R2"+'. (Use Whitney's embedding theorem.) Show that the B(M)-module B,(M) of C" differential p-forms on M is generated bythep-forms df,, A ... A df;, (1 6 jl <j z < * - - < j , 5 2n l).(Same method.) (b) Let M , N be two differential manifolds. Show that, for each p, the B(M x N)-module B,(M x N) of C" p-forms on M x N is the direct sum of the ( p 1) B(M x N)-modules 8,,p-, ,where B,, p - , is theB(M x N)-modulegeneratedby thep-forms of the type 'prl(a) A*pr,@), where a E B,(M) and E Bp-,(N) (0 5 r s p ) .
14. (a)
+
.
+
+
15. Show that there exists an immersion of the Klein bottle (16.14.10) into R3, the image where of which is the set of points (el,
t',t3),
5' = (a + cos )u sin t - sin )u sin 2r) cos u, 8' = (a + cos )u sin t - sin )u sin 2r) sin u,
t3= sin )u sin t + cos t u sin 2t
(0 t 5 27~,0
u 5 27~).
16. Define a real-analytic mapping f : R"+'-+R2"+'as follows: If x = (to, ..., z = f(x) = (5". . . . , C2") is given by the formulas
e),then
The restriction fo o f f to S. factorizes into P
S . + Pn(R) -+ R2"+',
where
is the canonical mapping. Show that h : 7T(x)Hg(7T(x))//Ig(~(x))llis an ems,. . Define similarly an embedding of P,(C) in %-'and of P,(H) in S,.-3 (replace Pe-" by &?-.). (James embedding.) 7~
bedding of Pm(R)into
17. (a) Let M, N be pure differential manifolds of dimensions n, p. respectively; let f:M + N be a C"-mapping, and Z a submanifold of N of dimension p - q . Let A be a closed subset of M such thatfis transversal over Z at all points of A nf-'(Z) (Section 16.8, Problem 9). Let # be a chart of N, with domain T, such that #(T) = H x I, where H is an open set in Rp-4, I an open set in R4, and f l T n Z) = H (16.8.3). Also let 'p be a chart of M with domain W c f - ' ( T ) , let U and V be relatively compact open sets such that 0 c V a n d v c W, and let u be a C"-mapping of R" into [0, 11 such
26
DIFFERENTIABLE HOMOTOPIES AND ISOTOPIES
193
rp(v).
that u(z) = 1 for all z E (p(0)and u(z) = 0 for all z 4 Show that for each E > 0 there exists a vector b E Rqsuch that: (I) Ilb(J5 E ; (2) the function XH
#(f(x))
+ u(p(x))b
takes its values in #(T) for x E V ; (3) the mapping g = gb : M +N defined by g(x) = # - ' ( f l f ( x ) ) u(p(x))b) for x E W, g(x) = f ( x ) for x E M - V is transversal over Z at the points of g-'(Z) n 0 ; (4) the mapping g is transversal over Z at the points of g - ' ( Z ) n A. (To satisfy (3), use Sard's theorem (16.23.1) for the mapping
+
Z H Prz(?4f(rp-'(ZN)
of rp(V) into R '. To satisfy (4j,it is sufficient that it should be satisfied at the points of g - ' ( z ) n K, where K = A n (0 -U). For this, consider the mapping (4 b) H (gb(x),
D(prz
#
gb O rp-')(rp(x)))
of K x R4 into N x Rq"; observe that it is continuous and transforms K x (0) into a subset of the open set ((N - A) x Rq")u (A x M,) of N x Rq",in the notation of Section 16.23, Problem 4.) (b) Let M, N be differential manifolds of dimensions m,n respectively, letf: M -+N be a C"-mapping, let Z be a submanifold of N of dimension p - q, and let A be a closed subset of M such that f is transversal over Z at all points of A n f - ' ( Z ) . Finally let d be a distance defining the topology of N, and let 6 be a continuous function on M,everywhere >O. Show that there exists a C"-mapping g : M + N which is transversal over 2, coincides with f o n A, and is such that d (f(x ),g(x)) 5 6 ( x ) for all x E M (Thorn's transversality theorem). (There exists an open neighborhood S of A such thatfis transversal over 2 at the points off-'(Z) n S. Consider a denumerable locally finite open covering (T,),,, of N, where To = N - Z and the Tk(k 2 1) are domains of definition of charts 4, of N, such that $h.(Tk) = Hk x I t , where Hk is open in RPwq and I I , is open in R4,and t,bk(Tk n 2) = Hk . Then take a locally finite open covering (W,) of M, for which the Wk are domains of definition of charts of M, and which refines the covering formed by the intersections of the open setsf-'(T,) with M - A and S. Construct g by induction on k, using (a).)
26. DIFFERENTIABLE H O M O T O P I E S A N D ISOTOPIES
(16.26.1) The notion of homotopy, defined in (9.6) for paths, extends t o arbitrary continuous mappings. Given two continuous mappings f , g of a topological space X into a topological space Y, a homoropy off into g is a continuous mapping cp of X x [a, fl] (where c( < /3 in R) into Y such that p(x, a) = f ( x ) and q ( x , B) = g(x) for all x E X. The mapping g is said t o be homotopic to f if there exists a homotopy offinto g. Just as in (9.6) it is immediately shown that if g is homotopic tof, thenfis homotopic t o g ; and that if h is homotopic to g, and g is homotopic toS, then h is homotopic tof; in other words, the relation "fis homotopic to g " is a n equivalence relation on the set of continuous mappings of X into Y.
194
XVI
DIFFERENTIAL MANIFOLDS
(16.26.2) Now suppose that X and Y are diferential manifolds and that f and g are mappings of class Cp (wherep is an integer 2 1, or 00) of X into Y. Then a CP-homotopy (or a homotopy of class Cp) off into g is a mapping cp of class Cp of X x J into Y, where J is a nonempty open interval in R,such that for two points a < p in J we have cp(x, a) =f(x) and cp(x, fi) = g(x) for all x E X.The mapping g is said to be CP-homotopicto f if there exists a CPhomotopy off into g.
+
(16.26.3) Let X , Y be differential manifolds. Then the relation " f is CPhomotopic tog " is an equivalencerelation on the set of Cp-mappingsof X into Y .
It is immediate that the relation is reflexive and symmetric. To show that it is transitive, consider three mappingsf, g , h of class Cp from X to Y, a CPhomotopy cp o f f into g, and a CP-homotopyt,b of g into h. By a linear change of variable we may assume that cp and $ are both defined on the same space X x J, where J is an open interval in R, and that there exist a < p in J such d x , 8) = g ( 4 , $(x, a) = gW, t , b k = h W , forallx E X. that cpb, a) =fM, There exists a C"-mapping R : R + [a,83 such that A(t) = a for t < 3, R(t) = for t > 3 (16.4.2). The functions of class Cp,
(x, t )H c p k
and
(x, t )H$(x, 4 t - 1))
defined on X x R agree on X x 13, $[. We may therefore define a CP-homotopy 0 o f f into h by putting O(x, 2) = q(x, R(t)) for t 5 1 and 0(x, t ) = $(x, A(t - 1)) for t 2 1. (16.26.4) Let X , Y be two differential manifolds, K a compact subset of X , and f : K -+ Y a continuous mapping. If d is a distance de$ning the topology of Y , then there exists E > 0 such that every continuous mapping g : K -+ Y satisfying d(f ( x ) ,g(x)) 5 E for all x E K is homotopic to$ Moreover, if the restrictions o f f and g to K are of class Cp, then f and g are CP-homotopic.
Let V be a relatively compact open neighborhood o f f ( K )in Y. By (16.25.1) we may assume that V is embedded in some R". Let T be an open neighborhood of V in R"with the properties of (16.25.4). If d' is the Euclidean distance on R",there exists a number tt > 0 such that all the points z E R" for which d'(z, f(K)) q belong to T (3.17.11). Next, there exists E > 0 such that d(y,f(x)) 4 E for y E Y and x E K implies that d'(y,f(x)) S q (this is easily proved by contradiction, using the compactness of K) and hence y E T. Now let g :K 4 Y be a continuous mapping such that d(f(x),g(x))_I E for all X E K. For X E Kand t E [0,1] we have
d ' ( f ( x ) ,tf(x) -t (1
- t)g(x)) S d'Cf(x),
S tt
26 DIFFERENTIABLE HOMOTOPIES AND ISOTOPIES
195
+
and therefore the point tf(x) (1 - t)g(x)belongs to T . If we put
cp(x, t ) = .(tf(x>
+ (1 - tlg(x))E y ,
then it is clear that cp is a homotopy of g into f. For by virtue of (3.17.11) applied to cp-'(T), there exists a > 0 such that tf(x) (1 - t)g(x) belongs to T for -a < t < 1 tl and all x E K. Moreover, iff1 K and g I K are of class Cp, then the restriction of cp to K x ] -a, 1 a [ is of class Cp, from which the second assertion follows.
+
+
+
Remark (16.26.4.1) The above proof shows that if f ( x o ) = g(xo) for some xo E K, then the homotopy cp constructed above is such that cp(x, , t ) = f ( x o ) for all t E [O, I]. (16.26.5) Let X , Y be diferential manifolds, K a compact subset of X , and f a continuous mapping of K into Y . Then there exists a relatively compact open neighborhood U of K in X and a C"-mapping g of U into Y such that g I K is homotopic to j:
Let d be a distance defining the topology of Y , and let E > 0 be a real number for which (16.26.4) holds. By virtue of (16.25.5) there exists a relatively compact open neighborhood U of K in X and a C"-mapping g of U into Y such that dcf(x), g(x))5 E for all x E K. The result now follows from (1 6.26.4). (1 6.26.6) Let X , Y be diferential manifolds, K a compact subset of X , andf, g homotopic continuous mappings of K into Y . I f f I K and g I K are of class Cp,
then they are CP-homotopic.
By hypothesis, there exists a compact interval I = [a,81 in R and a continuous mapping cp : K x I + Y such that cp(x,a) =f ( x ) and cp(x,p) = g ( x ) for all x E K. Choose E > 0 for which (16.26.4) holds for X, Y , f , and K ; for X, Y , g , and K ; and for X x R, Y , cp, and K x I. Then there exists an open neighborhood U (resp. J) of K in X (resp. of I in R) and a C"-mapping $ : U x J + Y such that d(cp(x, t), $(x, t)) 5 E for all ( x , t ) E K x I. If we put f i ( x ) = $(x, a) and g,(x) = $(x, p) for x E U, then $ is a C"-homotopy of f i into 9 , . However, we have d(fi(x),f(x)) E and d(g(x),g,(x)) E ; hence b y (16.26.4) it follows that f 1 K and fi I K are CP-homotopic, and that g , I K and g1 K are CP-homotopic. Hence, by (16.26.3), f I K and g I k are Cphomotopic.
196
XVI DIFFERENTIAL MANIFOLDS
Iff, g are two diffeomorphism of a differential manifold X onto a differential manifold Y, a C"-homotopy cp o f f into g , defined on X x J, where J is an open interval in R,is said to be a C"-isotopy (or an isotopy of class C") if for each t E J the mapping x Hcp(x, t ) is a diffeomorphism of X onto Y. The same argument as in (16.26.3) shows that the relation "there exists a C"-isotopy off into g " is an equivalence relation on the set of diffeomorphisms of X onto Y. (1 6.26.7)
(16.26.8) Let X be a differential manifold, U a connected open subset of X , and a, b two points of U.Then there exists a C"-isotopy cp of the identity map l X into a direomorphism h of X onto X , such that h(a) = b and such that cp(x, t ) = x for all t and all x # U.
(I) We consider first the following particular case: X ball llxll <
,/;
(relative to the Euclidean norm llxll =
= R",U
1
I1l2
is the open
), a = 0 and
llbll < 1. Let (ei)lsisn be the canonical basis of R", and suppose first that b = pel, with 0 < p < 1. We shall use the following lemma: (16.26.8.1) Let E be a real Banach space, f:E --+ E a bounded and p times
continuously diflerentiable mapping. Then for each zE E there exists a unique solution t H F(t, z) of the differential equation dxldt =f ( x ) , dejined on the whole of R and such that F(0, z) = z. Furthermore, for all s, t E R we have F(s, F(t, z )) = F(s + t, z), and for each t E R, the mapping ZH F(t, z) is a homeomorphism of E onto E which together with its inverse is p times continuously dyeren tiable.
The first assertion is proved as in (10.6.1), by remarking that because of the mean-value theorem every solution u of the differential equation in a relatively compact open interval J of R is bounded in J, which allows us to apply (10.5.5) becausef(u(t))is bounded in J. Every integral of the differential equation which takes the same value as t H F(t, z) at a point of R must be identical with this function, by (10.5.2). Now, for each S E R , the function t I+ F(t + s, z)is a solution of the equation which takes the value F(s, z)when t = 0; hence F(t, F(s, z ) ) = F(t + s, z). In particular, F(t, F( - t , z)) = zfor all t E R and zE E; also by (10.7.4) we know that (t, Z ) H F(t, z) is p times continuously differentiable, hence the proof of the lemma is complete. To apply this lemma, consider a C"-mapping g : R + [0, 11 such that g ( t ) = 0 for I t I 2 1, g ( t ) 0 for I t I < I and g(0) = 1 (16.4.1.4). We apply the lemma to the system of differential equations
=-
26 DIFFERENTIABLE HOMOTOPIES AND ISOTOPIES
197
(16.26.8.2)
Hence we obtain a mapping ( x , t ) F(x, ~ t ) of class C", from R" x R to R", such that F(x, 0) = x , and such that X H F(x, t) is a diffeomorphism of R" for all t E R. If I c j l 2 1 for some index j , then F(x, t) = x for all t , since the right-hand side of the first equation (16.26.8.2) is then zero. It remains to show that F(0, t) = b for a suitable choice of t ; but if we put F(0, t) = (u(t), 0,...,0), then u is strictly increasing on R,and the inverse function is 5 g([)- 'dr, defined for - 1 < t < 1, and which tends to + 00 as t --f 1 by reason of the fact that all the derivatives of g vanish at the point 1. It remains to consider the case where b is arbitrary (subject to I(b(l 1). There exists a rotation r of R" transforming b into a point of the form be,, and in place of F we consider the function ( x , t ) ~ r - ' ( F ( r ( x t)). ),
-1;
-=
(11) Now consider the general case. For any two points p, q of U, let R(p, q) denote the relation obtained by replacing a, b by p , q in the statement of (16.26.8). It follows immediately from (16.26.3) that R is an equivalence relation on U. Moreover, each equivalence class of R is open in U: for each p E U, there exists a chart (V, $, n) of U such that ~ ( p=) 0 and such that $(V) contains the closed ball with center 0 and radius in R";let W c V be the inverse image under $ of the open ball with center 0 and radius 1. If cp, is an isotopy defined on V x J and such that cpo(x,t) = x for x E V - W, we extend it to an isotopy cp defined on X x J by putting ~ ( xt), = x for all x E X - V. Then part (1) of the proof shows that the relation R(p, q) is true for all q E W, which proves our assertion. The equivalence classes of R are therefore both open and closed in U ; since U is connected, it follows that there is only one equivalence class. Q.E.D.
fi
(1 6.26.9) Let X be a connected diferential manifold, xo,x1 , .. ., xp distinct points of X, and U a neighborhood o f x o . Then there exists a difleomorphisrn h of X onto X such that h(xo) = xo and such rhat h(xi)E U for 1 5 i 5 p . Consider a chart (V, $, n) of X at xo such that V c U, and such that $(V) contains the open ball with center $(xo) and radius 1. Let V k(1 5 k 5 p ) be the inverse image under $ of the open ball with center $(xo) and radius k/p, so that 0,c V,,, . By induction on k it is enough to show that there
198
XVI DIFFERENTIAL MANIFOLDS
exists a diffeomorphism hk+l of X onto X which fixes the points of v maps xk+l into V k + l . The only case to be considered is that in which
6 vk+l
xk+l
k
and
*
Let W be the connected component of in X - v k , which is open and closed in X - v k (3.19.5). We have W n v k + 1 # 0,otherwise the closure of W in X would be equal to its closure in X - v k , and therefore W would be both open and closed in X, which is absurd. The result now follows from (16.26.8) applied to the connected open set W in X. Let X be a connected diferentiul manifold of dimension 2 1, and let a, b be two points of X. Then there exists an injective C"-mapping f : R --f X such that f(R)contains both u and b. (In geometrical terms, u and b can be joined in X by a simple C"-arc.) (16.26.10)
By means of a diffeomorphism of X onto itself, we may assume that b lies in an open neighborhood U of a such that there exists a chart (U, p, n) of X for which p(U) = R" ((16.26.9) and (16.3.4)). We have then only to take a parametric representation t w g ( t ) of the line joining p(u) to p(b), and then take f = p - l o g .
PROBLEMS
1. Let X, Y be two pure differential manifolds. Show that there exists a neighborhood V
of the diagonal in Y x Y with the following property: Iff, g : X -+ Y are mappings of class Cp (p an integer >0, or co) such that (f(x), g(x)) E V for all x E X, then f, g are Cp-homotopic. (Embed Y in R" (Section 16.25, Problem 2) and use (16.25.4).) Deduce that, for each continuous f:X -+ Y,there exists a C"-mapping g :X -+ Y homotopic to f, and that if two Cp-mappings f, g : X -+Y are homotopic, then they are Cp-homotopic.
+
2. Two topological spaces X,Y are said to have the same homotopy type if there exist continuous mappings f:X -+ Y and g : Y +X such that g 0 f is homotopic to Ix and f o g homotopic to l u . In that case f'(resp. g ) is said to be a hornotopy equivalence. If X and Y have the same homotopy type, then for two continuous mappings u :Z -+ X (resp. u : X -+ Z), where 2 is a topological space, to be homotopic it is necessary and sufficient that f u and f v (resp. u g and v g ) should be homotopic. If Y is a differential manifold and X is a closed submanifold of Y,show that there exists an open neighborhood U of X in Y such that X and U have the same homotopy type. (Embed Y in some R" and use (16.25.4).) 0
0
0
0
26
DIFFERENTIABLE HOMOTOPIES AND ISOTOPIES
199
3. (a) Let X be a metrizable topological space, A a closed subset of X, U a neighborhood of A , f a continuous mapping of U x [0, 11 into a topological space Y. Suppose that there exists a continuous mapping g : X +Y extending the mapping X H ~ ( X , 0).Show that there exists a continuous mapping h : X x [0, 11 + Y such that h(x, t ) = g(x) for x E X - U and all t E [0, 11, and such that h(x, I ) = f ( x , t ) for x E A. (Use the TietzeUrysohn theorem.) (b) Deduce from (a) that if Y is a differential manifold and i f f is a continuous mapping of A x [0, I ] into Y, such that there exists a continuous extension to X of the mapping x H f ( x , 0) of A into Y, then fcan be extended to a continuous mapping of X x [0, I ] into Y. (Embed Y in some R'" and use the Tietze-Urysohn theorem and (16.25.4).) In particular, the mapping X H ~ ( X , 1 ) extends to a continuous mapping of X into Y. Consider the case A = 0. (c) Suppose that X is a differential manifold, A a closed submanifold of X, and Y a topological space. Let f : A x [0,1] -+Y be a continuous mapping such that the mapping X H ~ ( X , 0)admits a continuous extension to X. Then fcan be extended to a continuous mapping of X x [O, 11 into Y. (Embed X in some R", then use (a) and (16.25.4).) Consider the case A = 0. 4.
Let h = (X, B, T)be a Co-fibration (Section 16.25, Problem 9), where B is of the form A x [a,61, with [a,b] c R. Suppose that there exists c E ]a, 6[ such that the fibrations induced by h on A x [a,c] and A x [c, 61 are trivializa ble. Show that h is trivializable. (Observe that an A-isomorphism of a trivial bundle A x F over A can be extended to an (A x 1)-isomorphism of (A x I) x F, for any interval I c R.) Consider the analogous theorems for principal bundles and vector bundles.
5.
Let h = (X, B, T)be a Co-fibration, where B is of the form A x [0, I]. Show that there exists an open covering (U.) of A such that h is trivializable over each open set U,, x [0, 1 I. (Use Problem 4.) Consider the analogous theorems for principal bundles and vector bundles.
6.
Let B be a separable, metrizable, locally compact space, A a closed subset of B, and h = (X, B x [0,1], T)a Co-fibration. Suppose either that the fibers of X are differential manifolds or that B is a differential manifold and A a closed submanifold of B. Show that every continuous section of X over the closed set (A x [0, I]) u (B x {0}) extends to acontinuous sectionofx over B x [0, I]. (UseProblems 3 and 5 above, and Section 16.25, Problem 9.) Consider the case A = (21.
7.
(a) Let B be a separable, metrizable, locally compact space and let E l , Ez be two Co-vector bundles over B x [0, I]. Show that if the bundles induced by El and EZover B x {0} are isomorphic, then E l and Ez are isomorphic. (Apply Problem 6, with A = 0, to the fiber bundle Isom(E, , EJ, which is an open subset of the vector bundle Hom(El, Ez) over B x [0, I], and the fibers of which consist of the vector space isomorphisms between the corresponding fibers of El and E, .) (b) Deduce from (a) that if E is a C"-vector bundle over B x [0, I ] and if EDis the vector bundle induced by E over B x {0}, and p : (b, t ) ~ ( 60), the projection onto B x {O}, then E is isomorphic to p*(Eo). In particular, if El is the vector bundle induced by E over B x {I}, and if B x (0)and B x { I } are identified canonically with B, then EDand El are isomorphic vector bundles over B.
200
XVI
DIFFERENTIAL MANIFOLDS
(c) Let f, g be homotopic continuous mappings of B into a separable, metrizable, locally compact space B'. If E is a Co-vector bundle over B', show that f'*(E') and g*(E') are B-isomorphic. In particular, if B is contractible (16.27.7), then every Covector bundle over B is trivializable. 8.
Let E be a real Co-vector bundle of rank k over a separable, metrizable, locally compact space B. Let rn : E --f B be the projection. Let u, u be two Gaussian mappings of E into R" (Section 16.19, Problem 8), and suppose that there exists a Gaussian mapping w of E x [0, 11 (regarded as a vector bundle over B x [O,l]) into R" such that w(x, 0) = u(x) and w(x, 1) = v(x). Iff; g are the continuous mappings of B into G,, k corresponding to u and v, respectively (loc. cit.), show that f a n d g are homotopic.
9. Let q+ and q- be the two injective linear mappings of R" into R2" defined by q+(ei)= ez,,q-(eJ=e21-l (1 s i s n ) .
(a) Show that each ofq+, q- is homotopic to the injective linear mappingq :R" --f R2" defined by q(ei)= e, (1 i 5 n). (b) Consider the Gaussian mappings q+ prz and q- prz of Unrkinto R2";to them correspond canonically (Section 16.19, Problem 8) continuous mappings f+, f - of G.,k into G Z n , k . Deduce from (a) and from Problem 8 that f + , f - are each homotopic to the canonical injection j : G., k --f Gzn, (Section 16.19, Problem 2(a)). (c) Let E be a real Co-vector bundle of rank k, with base B and projection n, and let u, v be two Gaussian mappings of E into R" (where n 2 k). Then q+ 0 u and q- u are Gaussian mappings of E into Rz". Show that the mapping 0
0
0
( x , t ) H w(x, t ) = (1
- tlq+(u(x))+ t q - ( d x ) )
is a Gaussian mapping of E x [0, 1 ] into R2". (d) Let f, g be two continuous mappings of B into G., k . Show that if the Co-vector bundles f*(U., k ) and g*(U,, k ) are B-isomorphic, then the mappings j o f and j g of B into Gzn, are homotopic. (Consider a vector bundle E of rank k over B and the Gaussian mappings u, v :E --f R" corresponding tof, g (Section 16.19, Problem 8); use successively (c), Problem 8, then (b).) 0
E > 0 such that, for each a E R" with /la115 E (Ilall being the Euclidean norm) and each endomorphism S of R" with 111- SII 5 E (the norm here being the norm on End(R") induced by the Euclidean norm (5.7.1)), there exists a C"-mapping F : R" x R - t R", satisfying the following conditions: (1) for each t E [0, 1 I, the mapping X H F(x, t ) is a diffeomorphism of R" onto R" such that F(x, t ) = x whenever llxll> 2; (2) F(x, 0) = x for all x E R";(3)F (x, 1) = a S . x whenever llxll< 1. (Take F(x, f) = x th(llxllz)(a S * x - x), where h is a C"mapping of R into [O,11 which is equal to 1 for 181 5 1 and zero for 181 2 4. To show that condition ( I ) is satisfied for all small E , use Section 16.12, Problem 1.)
10. Show that there exists
+
+
11. With the hypotheses of (16.26.81, let (uj), sjjn be a basis of T.(X) and (vJ,
+
s j 6 n a basis of T8(X). If X is orientable, suppose that these two bases are direct. Show that the n. isotopy p may be constructed so that h(a) = b and also T,,(h) uj = v, for 1 (Follow the second part of the proof of (16.26.8), but work in the frame bundle R(X) (20.1.1), and use Problem 10.)
26
DIFFERENTIABLE HOMOTOPIES AND ISOTOPIES
201
12. Letf, ,. . . ,f. be real-valued C"-functions defined on a neighborhood of 0 in R",such that the Jacobian matrix of Fo= (fj), j 4 n at 0 is the identity matrix. Show that there exists a real number E > 0 and a C"-mapping F : R" x R R"satisfying the following conditions: (1) for each t E [0, I], the mapping XH F(x, t ) is a diffeomorphism of R" onto R"such that F(x, t ) = x for llx//2 2.5; (2) F(x, 0) = x for all x E R";(3) F(x, 1) = (f,(x), ...,&(XI) for ((XI( <= E . (Take F(x, t ) = x t h ( ~ Ilxl12)(Fo(x) -~ - x), where h is chosen as in Problem 10; then use again Problem 1 of Section 16.12.) --f
+
13. Let X be a connected differential manifold, U an open neighborhood of a point a E X, and f a Cm-mapping of U into X such that T.(f) is a bijection of T.(X) onto T,(&). If X is orientable, assume also that TJf) preserves the orientation. Show that there exists a C"-mapping F : X x R + X satisfying the following conditions: (1) X H F(x, t ) is a diffeomorphism of X onto itself, for all t E [O, I]; (2) F(x, 0) = x for all n E X; (3) F(x, 1) =f(x) for all x in some neighborhood V c U of a. (Using Problem 11, reduce to the case wheref(a) = a and T,(f) is the identity mapping. Then use Problem 12.)
+
14. Let f, g be two diffeomorphisms of the open ball B ={x E R" : Ijx I[ < 1 a ) (where llxll is the Euclidean norm) onto open subsets of a connected differential manifold X. If X is orientable, assume also that f and g are orientation-preserving. Show that there exists a diffeomorphism F of X onto itself such that F(f(x)) = g(x) for IIxII < 1. (Using Problem 13, show that there exists E > 0 and a diffeomorphism H of X onto itself such that H(f(x)) = g(x) for //XI/ < E . Then remark that there exists a C" realvalued function h on B and diffeomorphisms GI, G2 of X onto itself such that G,(f(x)) =f(h(x)x) and G,(g(x)) = g(h(x)x) for all x E B, and such that I h(x)l 2 E for llxll< 1 , and G , ( y ) = y (resp. G , ( y ) = y ) for y not in the image off(resp. g).) 15.
Let X, ,X, be two connected differential manifolds of the same dimension n and let
u1 (resp. u,) be a diffeomorphism of the open ball B : /(XI/ < 2 in R" onto an open subset of X1(resp. X,). If B is the closed ball I(xI(2 & in R",let U1 (resp. U,) be the open set in XI (resp. X2) which is the complement of uL(B) (resp. uz(B)). Let U12=
ul(B - B), Uzl = u,(B - B'). Let X be the topological space. obtained by patching together U1 and U, along UL2and Uzl by meansofthe homeomorphismhzl :Ul2 +UZl defined by
Identify U1 and Uz (endowed with their structures of differential manifolds) canonically with their images in X. Show that the structures of differential manifolds induced by U1 and U, on U1 n Uz are the same, and hence are induced by a structure of a connected differential manifold on X. The space X, endowed with this structure of differential manifold, is called the connected sum of X, and X, relative to u1 and u2 , Using Problem 14, show that up to diffeomorphism X is independent of the choice of u , and u 2 , provided that each of the manifolds X 1 ,X2is either nonorientable, or orientable but admitting an orientation-reversing diffeomorphism onto itself. By abuse of notation we write X = X1 # Xz. If X3 is another connected differential manifold of dimension n with the same properties, show that (with the same abuse of notation) (X1 #X,) # X3 = XL# (X, # X,). Show also that X, # S. = X 1 .
202
XVI DIFFERENTIAL MANIFOLDS
27. T H E FUNDAMENTAL GROUP O F A CONNECTED MANIFOLD
The definitions in (9.6) of the notions of path, loop, opposite paths, and juxtaposition of paths apply without any alteration when the open subset of C is replaced by an arbitrary topological space X. An unending path in X is by definition a continuous mapping of an open interval J c R into X. If X is a differential manifold, an unending path in X is said to be of class Cr (r 2 1 or r = co) if it is a Cr-mappingof an open interval J c R into X. A path in X is said to be of class C' if it is the restriction to a compact interval of an unending path of class C. Let X be a connected differential manifold. Given two points a, b in X, we denote by Q,, b the set of paths defined on I = [0, 11 with values in X, with origin a and end-point b. This set Q,, is not empty (16.26.10). We define an equivalence relation R,,b on Q,,b by considering two paths y1 ,y 2 E Q,,b as equivalent if there exists a homotopy cp : I x [tl, b] + X of y1 into y 2 such that cp(0, = a and q(1, = b for all 5 E [a,j3] (for brevity, we say that cp leaves a and bfixed).
r)
r)
We have the following lemma: (16.27.1) Let p be any continuous mapping of I into I such that p(0) = 0 and p(1) = 1. Then for each path y E Q,, b , the paths y and y 0 p are equivalent for Ra,b*
This follows from considering the homotopy cp : I x I + X defined by V(t, 0= Y((1 - 5)t + tP(t)). Let E,, b denote the set of equivalence classes for the relation R,,, b . Given three points a, b, c E X, we shall define a mapping E,, x Eb, + E,, c , which we shall denote by (u, v) I-+ u o (or simply uo). Consider two paths y1 E Q,,,b , y 2 E Qb, c . From these we construct a path y E Qu, by putting (1 6.27.1.I)
y2(2t - 1)
for 0 5 t S + , for 3 5 t 5 1.
We write y = yly2 (this is a juxtaposition of paths equivalent to y1 and y 2 , chosen in a particular way). Let y ; ,7; be paths equivalent, respectively, to yl, y 2 . Then y ; y ; is equivalent to yly2. For we may suppose that the homotopies cpl of y1 into y ; and cp2 of y 2 into y i are defined on the same set
27 THE FUNDAMENTAL GROUP OF A CONNECTED MANIFOLD
I x [oi,
PI, and then we construct a homotopy cp of ylyz cp(t, 5 ) = y
2
f
into y;y; by defining
for 0 5 t s t , for 3 5 t 5 1 ;
8
7
203
q 4 2 t - 1, 5 )
the function cp so defined is continuous, because cpl( I , 5) = cpz(O, 5 ) = b for all 5 E [a, B]. Consequently the class of ylyz in E , , b depends only on the classes u of y1 and v of y z , and this class we denote by u . v. For each a E X we shall denote by e, the class in E,, ,of the constant path t Ha. Also, given a, b E X and a path y E a,, b , the class of the opposite path yo in Eb,,depends only on the class of y in E,, b . For if y' is equivalent to y under a homotopy cp, then y" is equivalent to yo under the homotopy
(4
5)HcpU
- t,
0.
If u is the class of y, we denote by u-' the class of yo. Clearly (u-')-'
= u.
(16.27.2) (i) Let a, 6, c, dbe fourpoints O f x . IfU E E,, b , 0 E E b , c , W E Ec,d, then (uu)w = u(uw) (and therefore we write this product as uuw, without brackets). (ii) I f a , b E X , t h e n e , u = u a n d u e b = u , f o r a l l u E E a , b . (iii) If a, b E X , then uu-l = e, and u - ' u = eb,for all u E E , , b .
(i) Let 71 u, YZ E u, Y3 Then we have
w ;put 7 = (ylYZ)y3 y' 9
l i
Y 1(40
y ( t ) = y2(4t
- 1)
y3(2t - 1)
and
YI(2t)
y'(t)
=
y2(4t - 2) y3(4t - 3)
= yl(y2 73).
for 0 5 t 5 $ , for t 5 t 5 f, for 3 5 t 5 1, for 0 5 t 54, for 5 t 2 $, for 9 5 t 1 .
+
Now consider the function p : I -+I defined by
p(t)=
[ : : i
+(t+l)
Clearly p is continuous; p(0)
for 0 5 t 5 t , for 4 5 t 1 3 , for & S t S l .
= 0, p(1) = 1,
and we have
y ( t ) = Y'(P(t))
for all t E I. Hence the result (16.27.1).
204
XVI DIFFERENTIAL MANIFOLDS
(ii) We shall prove that ueb = u ; the proof that e,u = u is similar. If y E u, a path in the class ueb is defined by for
3 6 t 6 1.
Define a homotopy cp : I x I + X of y' into y as follows:
for
3(1 + 5 ) 5 t 5 1.
(iii) We shall prove that uu-l = e,,; the proof of the other relation is similar. Let y E u, then y' = yyo is defined by for 0 5 t 6 + , Define a homotopy cp : I x I + X of the constant path t H a into y' as follows: y(25 - 2t5)
for 05 t 6 f, for 3 6 t 6 1.
From (16.27.2) we deduce : (16.27.3) (i) For each a E X, the mpping (u, v) Huv deJines agroup structure on Ea, a * (ii) If a, b E X andf E Ea,b , then the mapping U H fuf -'is an isomorphism Of the group Eb, b Onto the group E,, , . Remarks
(16.27.3.1) (i) With the notation of (16.27.3(ii)), if y E u and u ELthen the loops y and y' = (ay)uo are homotopic (as loops): for by definition, y' is given by the equations 440 for 0 6 t S $ , y(4t - 1) for 4 5 t 5 f, 4 2 - 2t) for 3 5 t 5 1, and we define a loop homotopy cp of y' onto y by
4 5 + 4(1 a({ OH
- 5)t)
+ 2(1 - <)(l - t))
for O j t j t and O j y j l , for $ S t ( = + and OS<SI, for 3 6 t 6 1 and 0 6 5 6 1.
(ii) It follows from (16.27.2) that for each uo E Eo,b , the mapping b ) is bijective. uov (resp. W H wuo) of Eb, into E,,, (resp. of Ec, ,into Ec,
27 THE FUNDAMENTAL GROUP OF A CONNECTED MANIFOLD
205
(1 6.27.4) For each pair (a, b) of points of X,the set E,,,b is at most denumerable.
We shall first prove the following lemma, which generalizes (7.4.4) : (16.27.4.1) Let E be a metrizable compact space, F a separable metrizable space, d (resp. d ) a distance defining the topology of E (resp. F). For any two mappingsf, g : E + F, pur P(f, 9 ) = SUP teE
d'(f(0,g(t)).
Then p is a distance on the set VF(E) of continuous mappings of E into F, with respect to which %'p,(E) is a separable metric space. The fact that p is a distance follows immediately from the fact that p ( f , g )is$nite wheneverf and g are continuous (3.17.10). For each pair myn of integers >O, let G,, be the set of functions f E %g'(,E) such that the relation d(x, x') 5 l/m implies that d'(f(x), f (x')) 5 l/n. Since each f E V,(E) is uniformly continuous (3.16.15), it follows that for each fixed n > 0, V,(E) is the union of the G,, for m > 0. Let {al, ... ,apcm,}be a finite subset of E such that the open balls with centers a, and radius l / m cover E. Also let (br),L be a denumerable dense sequence in F. For each mapping : { 1,2, . . .,p(m)} N , let H, be the set off E G,, such that d'( f (ak),bp(k))S l/n for 1 5 k 5 p(m). From the definition of the b, , G,, is the union of the H, for all cp E Np(,). Let C, be the set of cp E Np(,) such that H, # 0, and for each cp E C,, choose an element g , E H,; finally, let L,, denote the denumerable set of g , for cp E C,, Let f E G,, and let cp E C,, be such thatfE H, . Then it follows immediately from the definitions that d'( f (x), g,(x)) 5 4/n for all x E E; in other words, p(f, g,) S 4/n. Hence the union of the L,,,,, is dense in %F(E): for each integer n > 0 and each f E %?F(E) there exists m such that f E G,, , and we have just seen that the distance fromfto Lmnis 5 4/n.
.
We now apply this lemma to the set
Sla,b,
endowed with the distance
p(yl, y z ) = sup d(yl(t), yz(t)), where dis a distance defining the topology of X. tsl
It follows that there is a sequence (y,) which is dense in b . On the other hand, it follows from (16.26.4.1) that, for each y E a,, b , there exists E > 0 such that the relation p(y, y') 5 E implies that the paths y, y' are equivalent; but since there exists a y, such that p(y, 7), 5 8, it follows that E,, b is the set of equivalence classes of the y,; hence it is at most denumerable. (16.27.5) The group E,,, , is called the fundamentalgroup of the manifold X at the point a and is denoted by nl(X, a). Up to isomorphism it is independent of the point a, and we shall use the notation nl(X)to denote any one of the groups n,(X, a), and refer to nl(X)as the fundamentalgroup of X.
206
XVI DIFFERENTIAL MANIFOLDS
A connected differential manifold X is said to be simply connected if a,(X, a) is the group of one element for some a E X (and hence for all a E X ) ; or, equivalently, if every loop with origin a is homotopic to the constant loop t H a under a loop homotopy leaving the point a fixed. (16.27.6) Let Y be another connected differential manifold, f :X -,Y a continuous mapping. For each pair of points a, b in X and each path y E Q,, b ,f y is a path belonging to Q,,,,, f ( b , . If y, y' E Q,,, b are equivalent paths, under a homotopy cp, then it is clear that f y andfo y' are equivalent under the homotopyfo cp. Hence, as y runs through a class u E E,,b , the paths f y all belong to the same class f*(u) E Eft,,,, f ( b ) . Moreover, it is easily verified that if u E E,, and u E Eb, , then f*(uu) =f*(u)f*(u). In particular, the restriction off* to xl(X, a) is a homomorphism of this group into the group al(Y,f(a)). Finally, if Z is a third manifold and g : Y --+ Z a continuous mapping, we have (g of)* = g* of*. 0
0
0
Examples
(1 6.27.7) A manifold X is said to be contractible if there exists a point a E X
and a homotopy cp of the identity mapping 1, onto the constant mapping X H U which leaves a fixed: in other words, cp is a continuous mapping of X x I into X such that q(a, 5 ) = a for all 5 E I and cp(x, 0) = x, cp(x, 1) = a for all x E X. For each loop y E Q,, ,,the mapping ( t , <)H cp(y(t),<) is then a homotopy of y onto the constant loop t H a which leaves a fixed. Consequently a contractible manifold is simply-connected. For example, R"is contractible, because the mapping cp(x, 5) = 5x satisfies the above conditions for a = 0. (1 6.27.8) For n 2 2, the sphere S, is simply-connected. Let y : I -,S, be a loop with origin a. If y(1) # S, , then y is homotopic in S, to the constant loop. For if b $ y(I), then S, - (b} is homeomorphic to R" (16.8.10), and therefore y is homotopic to the constant loop already in S, - {b).Hence we have to show that there exists a loop y' with origin a which is loop-homotopic to y and is such that y'(1) # S, . Let d be the distance on S, induced by the Euclidean distance on R"", and let E E 30, 1 [ be a number satisfying the condition of (16.26.4) for the mapping y. Then there exists a mapping y1 : I-, S, which is the restriction of a C"-mapping defined on a neighborhood of I, such that d(r(t),y l ( t ) ) 5 # E for t E I(16.25.5). Let b = yl(0), c = y1(l), so that b and c belong to the open neighborhood V of a in S, consisting of the points x such that d(x, a) < $ 8 , which is homeomorphic to an open ball in R" (16.2.3). Let 6 €10,+[ be such that the relation I t - t'I < 6 (for t, t' E I) implies that d(y(t'),y ( t ) ) < $6 (3.16.5).Then there exist continuous
27 THE FUNDAMENTAL GROUP OF A CONNECTED MANIFOLD
mappings yo : [0,6] -+ V and y 2 : [I - 6, 13 4 V such that yo(0) = a, yo(@ y2(l - 6) = c, yz(l) = a. Now define y' on I as follows: for 0 5 t
for
207 = b,
5 6,
l-dstsl-
Since I t - t' I < 6 implies that d(y,(t),y,(t')) < +&, it follows immediately that d(y(t),y'(t)) 5 E for all t E I. Hence, by (16.26.4), y and y' are equivalent in Q,,,, . On the other hand, by virtue of the hypothesis n 2 2 and Sard's theorem, yl(I) is nowhere dense in S,, (16.23.2); by construction, this implies that y'(1) # S,, , and the proof is complete. We shall prove later (Chapter XXIV) that for n 2 1 the sphere S, is not contractible (cf. Section 16.30, Problem 9(b)). On the other hand, the circle U = S, is not simply-connected. More precisely, n,(S,)is isomorphic to Z. In fact, we have already defined in Chapter IX and the Appendix to Chapter IX the indexj(0; y) of a loop y in S, and shown that it depends only on the class u of y in n,(S,,a), where a is the origin of y. Moreover, we have shown that if this index is denoted by i(u), then i is a homomorphism of n,(S,,a ) onto Z (9.8.4). Finally, this homomorphism is injectiue, for ifj(0; y) = 0 the proof of (Ap.2.8) shows that y is homotopic to a constant loop. (16.27.9)
(16.27.10) Let X, Y be two connected differential manifolds, a E X and b E Y. Then the group q ( X x Y, (a, 6)) is isomorphic to the direct product nl(X, a ) x n,(Y, b). In particular, the product of two simply-connectedmanifolds is simply-connected.
We define a canonical bijection f : Q,,, x f & , b -+ Q,,,,,),,,,,) as follows: if a E Q,,,, and p E f i b , b , then f ( a , p) is the loop t F+ (a(t),p(t)). The inverse bijection f takes a loop y E Q(,,,b ) , (,, b ) to the pair (pr, y, pr2 y). Clearly we have f ( a l a 2 , &f12) = f ( a , , Pl)f(a2, p2). Finally, if a, a' are two loops belonging to a,,,,,, which are equivalent under a homotopy cp : I x I + X, and if p, p' are two loops belonging to b , which are equivalent under a homotopy I,+ : I x I Y, then the loops f ( a , p) and f (a',p') are equivalent under the homotopy (cp, I,+) : I x I + X x Y. Conversely, iff (a, p) andf(a', p') are equivalent under a homotopy 0 : I x I -+X x Y, then a and a' are equivalent under pr, 0, and f? and p' are equivalent under pr, 0 0. Hence we have a bijection f o f E,,, ,x E b , b onto E(,,,b ) , (,, b ) which assigns to each pair of classes (u, u) the class off(a, p) for a E u and p E u ; and f is a group isomorphism.
-'
0
-+
0
0
208
XVI
DIFFERENTIAL MANIFOLDS
PROBLEMS
1. A topological space X is said to be urcwise-connectedif, for each pair of points a, b in X, there exists a path in X with origin a and endpoint b. A subset A of X is said to be arcwise-connected if the subspace A is arcwise-connected. (a) An arcwise-connected space is connected. In R2 the graph A of the function y = sin(l/x), defined on 10, w[, is arcwise-connected, but its closure A is connected but not arcwise-connected. (b) Show that propositions (3.19.3), (3.19.4), and (3.19.7) remain valid when “conarcwise-connected” throughout. nected ” is replaced by ‘‘ (c) The union C ( x ) of all the arcwise-connected subsets of X which contain the point x E X is an arcwise-connected set, called the arcwise-connected component of x in X; it is a closed subset of X. (d) A topological space X is said to be locally arcwise-connectedif each point of X has a fundamental system of arcwise-connected neighborhoods. Show that this condition is equivalent to the following: the arcwise-connected components of each open subset U of X are open in X. The arcwise-connected components of U are then equal to the connected components of U (in the sense of (3.19)).
+
2.
a,
x
If is any topological space, the sets b can be defined as in (16.27) whenever a and b belong to the same arcwise-connected component of X. If X is separable and metrizable, then so is b with respect to the topology defined in (16.27.4.1). Two paths yl , y2 E b are equivalent with respect to the relation R ,b if and only if they belong to the same arcwise-connected component of b . The group rrl(X, a) is defined as in (16.27.5).
a,
a,,,
a,,.
(a) Generalize the result of (16.27.10), and show that the canonical bijection
!&,a
b
!&.
b). (0, b )
is a homeomorphism. defined y ~ in (16.27.1.1) is a continuous (b) Show that the mapping (yl,y ~ ) + - + y ~ mapping of a,,,I, x f i b , into a ,e . (c) For each y E an, a and each [ E [O, I], let #(y, 0 denote the loop y’ E Clap defined by (I
If F. is the constant loop equal to a,show that i ) is a homotopy of the mapping YH YE. ,. into the identity mapping of of , into (d) Let f, g be homotopic mappings of X into Y , and let a E X. Show that there exists an isomorphism u of rl(Y,f(u))onto r l ( Y ,g(a)) such that g+ = u of*. Deduce that if X, Y are arcwise-connected and of the same homotopy type (Section 16.26, Problem 2), then rl(x)and rrl(Y)are isomorphic.
a,,.
a,,,
28 COVERING SPACES A N D THE FUNDAMENTAL GROUP
209
3. Deduce from (16.27.10) that the Hopf fibration of Sa over S2(16.14.10) is not trivializable. 4.
Let X be a simply-connected,arcwise-connected space. If a, b are any two points of X, show that the set E., b consists of a single element. (If y, y' E a ,b , show that y' is equivalent (with respect to Ra, b ) to (yyo)y'.)
5.
Let X be a locally compact metrizable space, and let A, B be closed subsets of X such that X = A u B. Suppose that A, B are each arcwise-connected and simply-connected; suppose also that A n B is arcwise-connected and that, for each x E A n B, there exists a fundamental system of open neighborhoods V of x for each of which V n B is arcwiseconnected and simply-connected, and V n A n B arcwise-connected. Show that X is arcwise-connected and simply-connected. (Let y : I +X be a loop with origin a E A, and suppose that ~ ( 1meets ) B. Then the inverse image under y of ~ ( 1 n ) (X - A) is the union of a (finite or infinite) sequence of pairwise disjoint open intervals I,,, in I. For each integer n, cover fiI) by a finite number of open subsets Ua. of X, of diameter l/n, such that Ue, nB is arcwise-connectedand simply-connected, and Ua,,nA nB arcwise-connected. Then consider the finite set of intervals I k such that y(Ik) is contained in Ua," but not in U., "+ ; finally, use Problem 4.) Hence give another proof of the fact that S, is simply connected for n 2 2.
28. COVERING SPACES A N D T H E FUNDAMENTAL GROUP
(16.28.1) Let ( X , B,p) be a covering of a direrential manifold B (16.12.4), f a continuous mapping of a connected topological space Z into By and g l , g 2 : Z + X two continuous liftings (16.12.1) o f f . I f there exists a E Z such that S l ( 4 = g z ( 4 , then 91 = Q2 *
The set of points z E Z such that g,(z) = g2(z) is nonempty and closed in Z (12.3.5); hence it is enough to show that it is also open in Z. Now, if zo E Z is such that gl(zo) = gz(zo) = xo , there exists an open neighborhood U of xo such that p I U is a homeomorphism of U onto the open set p ( U ) c B. Since g l , g 2 are continuous, there exists a neighborhood V of zo in Z such that glcv) c U and g2(V)c U. However, for all z E V , we have p(g,(z)) =f ( z ) = p(g,(z)); since p I U is injective, it follows that g,(z) = g2(z)for all z E V. This completes the proof. (1 6.28.2) Let ( X , B, p ) be a covering of a diferential manifold B.
(i) Fpr eachpath p : I B with origin b E B, and for eachpoint a Ep-'(b), there exists a unique path y : I -+ X with origin a such that p = p y (called the lifting of with origin a).
a,
0
210
XVI DIFFERENTIAL MANIFOLDS
(ii) Let /?: I + By B' : I + B be two paths with origins by b', respectively, and let y : I -+ X be the lifting of /? with origin a. If cp : I x I -+ B is a homotopy of B into F , then there exists a unique homotopy JI of y into a path y' : I + X such that cp = p JI. (JI is called the lifting of cp such that JI(0,0) = a.) 0
(i) The uniqueness of y follows from (1 6.28.1). To prove the existence of y, let E denote the set of s E I for which there exists a continuous mapping ys : [O, s] + X such that ys(0)= a and p(y,(t))= B(t) for 0 5 t 5 s. This set E
does not consist only of 0, because there exists a neighborhood U o of a such that p I U o is a homeomorphism of U o onto the open set p(Uo) of B ; if so > 0 is such that B(t) € p ( U o )for 0 6 t 5 so, put yso(t) = qo(B(t)) for 0 5 t 5 so , where go :p(Uo)--f U o is the inverse of the homeomorphism pi U , ; then p(yso(t)) = B(t) for 0 It 6 so, so that so E E. By virtue of (1 6.28.1), if s < s' in E, the restriction of ys. to [0, s] is equal to ys . Consider now the least upper bound 1 of E in [0, 11 ;there exists a continuous mapping y' of [0, A[ into X such that y'(0) = a andp(y'(t)) = B(t) for 0 5 t < 1. Let V be a connected open neighborhood of B(1) in B over which the covering space X is trivializable; then p-'(V) is a (finite or infinite) union of pairwise disjoint connected open sets U,, such that p I U,, is a homeomorphism onto V for each n. Choose to < 1 and sufficiently close to d so that /?([to,4) is contained in V , and let n be such that y'(to) E U,, Since y'([to , A[) is connected and contained in p-'(V), it must be contained in U,,;consequently, if q,, : V --f U,, is the inverse of the homeomorphism p I U,,,then y'(t) = q,,(B(t)) for to 5 t < A. This shows first of all that y' can be extended by continuity to the point 1,by defining ~'(1)= q,,(8(1)). Second it shows that 1 = 1, otherwise there would exist s1 such that 1 < s1 < 1 and such that B([1, s l ] )c V; we can then extend y' to ys, by putting ysi(t) = q,,(fi(t)) for A 4 t 5 slyand we shall have s1 E E, contrary to the definition of 1. This completes the proof of (i). (ii) For each t E I, there exists a unique lifting of the path C H cp(t, () with origin y(t). Let 5 H JIt(c) denote this lifting. Then we have to prove that the mapping JI : (t, ()HJI,(~) is continuous on I x I. Let d be a distance defining the topology of B. Since cp(1 x I) is compact, the same argument as in (9.6.3) shows that there exists a number p > 0 such that, for each b E cp(1 x I), the ball B(b; p) is connected and the covering space X is trivializable over B(b; p). Next, there exists E > 0 such that the relations I t - t' I S E , 15 - <'I S E imply that I d t , C) - d t ' , 571 < + p . Let ( t i ) o s i s r be an increasing sequence in I such that to = 0, t, = 1, and I t i + l - tiI E for 0 5 i 5 r - I. We shall prove by induction on i that JI is continuous on I x [0, t i + ' ] . For this, it is enough to prove that JI is continuous on Qij = [ t j , t j + l ] x [ t i , t i + ' ] for each j = O , l , . . . , r - l . Put b . . = c p (t j , t i ).Then !J p-'(B(bij;p ) ) is the union of a sequence of pairwise disjoint connected open sets (U,,)such that p I U,, is a homeomorphism of U,,onto B(bij; p ) for each n.
.
28 COVERING SPACES AND THE FUNDAMENTAL GROUP
211
Since by hypothesis t I-+ $(t, ti) is continuous, the image under this mapping of [ti, t j + J , being connected and contained in p-'(B(bij;p)) by virtue of the choice of E , is contained in some U,; then, for ti t 5 since $r(ti)E U, and $t([ti,t i + J )is contained in p-'(B(b,; p)) (again by the choice of E), it follows that Gr([ti,t i + J )c U,, so that finally $(Qii) c U,; but then, if q, : B(bij;p ) + U, is the inverse of the homeomorphism p I U, , we have $(t, 5 ) = q,(q(t, 5)) in Qii, and the proof is complete. (16.28.3) Let (X, B, p) be a covering of a differential manifold B, let b E B, and let xb = p-'(b) be the (discrete) fiber of X over 6. We shall define a right action of the fundamental group nl(B, b) on the fiber X, : (a, u) H a u, as follows: Let y E u be a loop in B with origin b. Then it follows from (16.28.2) that there exists a unique path yo in X with origin a, such that p y, = y. Further, if y' is another loop in B with origin b, belonging to the class u, then it is homotopic to y by a homotopy q : I x I + B which leaves b fixed. Now this homotopy has a unique lifting, by virtue of (16.28.2), to a homotopy $ : I x I + X of y, into yi, leaving a fixed, and such that p 0 $ = q. In particular, for each 5 E I, we have p($(l, 5)) = cp(l,t) = b ; in other words, 5 H $(l, t)is a continuous mapping of I into X b ; but since xb is a discrete space, this mapping is constant (3.19.7), so that y,(l) = yi(1). The point y,(l) E X b therefore depends only on the class u E n,(B, b), and it is this point that is denoted by a u. We have thus defined an operation of q ( B , b) on xb; for we have 9
0
a*eb=a
for all a E xb,
directly from the definitions. (16.28.4) Let X be a connected differential manifold, (Y, X , p ) a covering of X , a apoint of X , and Y , = p - ' ( a ) theJiber over a.
(i) Y is connected if and only if q ( X , a) acts transitively on Y , . If this condition is satisfied, then for each point c E Y , the homomorphism p* : nl(Y, c) + nl(X, a) is injective and its image is the stabilizer S , of c (for the action of n , ( X , a) on X,) (12.10). (ii) I f Y is connected and i f p , is surjective (hence bijective), then p is a diffeomorphism, so that the covering Y is trivializable. I n particular, every connected covering of a simply-connected manifold is trivializable. (iii) In order that Y should be connectedandsimply-connected, it is necessary and suficient that the group q ( X , a) should act transitively and freely on Y, (which is therefore homeomorphic to n,(X, a)).
212
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DIFFERENTIAL MANIFOLDS
(i) If Y is connected, then for any two points c,, c2 E Y, there exists a path y in Y with origin c1 and endpoint c 2 . The path p y is a loop with origin a in X, and y is its unique lifting with origin c, (16.28.2); hence c2 = cl* u, where u is the class of p y , and so n,(X, a ) acts transitively on the fiber Y, . Conversely, if the action is transitive, then any two points c,, c2 E Y, can be joined by a path in Y, by definition (16.28.3). Next, if b is any point of Y, there exists a path in X joining p(b) to a, and the lifting of this path with origin b (16.28.2) has its end point in Y,. Hence every point of Y can be joined to a given point c E Y, by a path and therefore Y is connected (3.19.3). Now let y be a loop in Y with origin c, whose image p y is homotopic to the constant loop with origin a. Then, by (16.28.3), the loop y , being the lifting of p 0 y, is homotopic to the constant loop with origin c. This shows that the homomorphism p* : nl(Y, c) -,n,(X, a) is injective. Its image consists of the classes u of loops with origin a in X whose lifting with origin c is a loop with origin c, i.e., the u E nl(X, a) such that c u = c. (ii) The hypothesis implies that c * u = c for all u E nl(X, a), and since nl(X, a) acts transitively on Y,, we have Y, = {c}. Since X is connected, all the fibers are isomorphic to Y,, hence consist of a single point; in other words, p is bijective, and is therefore a diffeomorphism (because it is a local diffeomorphism). The hypothesis that p* is surjective is clearly satisfied when X is simply connected, because then nl(X, a) is the group of one element. (iii) If n,(Y, c) = { e } , the stabilizer S , consists only of the identity element, by (i), and since q ( X , a) acts transitively on Y,, the stabilizer of each point of Y, consists only of the identity element, so that q ( X , a) acts freely. The converse is an immediate consequence of (i). 0
0
0
Example (16.28.5) From (16.14.10) we know that S, is a two-sheeted covering space of the projective space P,(R). Since S, is connected and simply-connected for n 2 2 (16.27.5) it follows that nl(Pn(R)) = 2 / 2 2 for n 2 2. (For n = 1, however, Pl(R) is diffeomorphic to S, (16.11.I2) and its fundamental group is therefore isomorphic to Z (16.27.9).) (16.28.6) Let X be a connected differential manifold, (Y, X, p ) a covering of X. Then f o r each connected component Z of Y, ( Z , X, p I Z ) is a covering of X. I n particular, each covering of a connected and simply-connected manifold is trivializable.
Let x E~(Z) and let U be a connected open neighborhood of x in X such that Y is trivializable over U ; in other words, p - ' ( U ) is the union of a
28 COVERING SPACES AND THE FUNDAMENTAL GROUP
213
(finite or infinite) sequence of pairwise disjoint open sets U, such that the restriction p , : U, + U of p is a homeomorphism for each n. At least one of the U, intersects Z, and since U is connected and Z is a connected component of Y, it follows that the U, which intersect Z are contained in Z. This shows already that x e p ( Z ) ;hencep(Z) is both open and closed in X and is therefore equal to X. Moreover, Z n p - ' ( U ) is the union of a subsequence of the U,, which proves that Z is a covering space of X. The last assertion of (1 6.28.6) follows from the first and from (1 6.28.4). We remark that if Y is an n-sheeted covering of X, then Y has at most n connected components, by virtue of (1 6.28.6).
Example (1 6.28.7) Let' X be a nonorientable connected differential manifold. Then nl(X) # {e}. For by (1 6.21.I6) there exists an orientable two-sheeted covering Y of X. If X were simply-connected then Y would have two connected components Y,, Y,, and each of the projections Y, X, Y, + X would be a diffeomorphism; but this is absurd, because Y, is orientable and X is not. --f
(1 6.28.8) (Monodromy principle) Let (Y, X, p ) be a covering of a di#erential manifold X, and let f : X' + X be a C"-mapping of a connected, simplyconnected, dflerential manifold X into X. Let a' E X' and let b e p - l ( f (a')). Then there exists a unique C"-mapping g : X' --f Y such that g(a') = b and p g =f (the mapping g is said to be a lifting off). 0
Consider the inverse image Y under f of the covering Y of X, which is a covering of X (16.12.8). Let p' : Y + X and f ' : Y' + Y be the canonical projections, and let b' be the point of Y' such that f '(b') = b, p'(b') = a'. The connected component Y& of Y containing b' is a connected covering of X' (16.28.6); hence p'I Yh is a diffeomorphism of Y& onto X' (16.28.4). If q' : X + Y&is the inverse diffeomorphism, then the mapping g =f ' o q' has the required properties, because
p og
= (p of')
oq'
= ( f o p ' ) oq' =f
and g ( d ) =f'(b') = b. The uniqueness of g follows from (16.28.1).
Example (16.28.9) Let U be a simply-connected open set in C" and let f be a holomorphic function on U which does not vanish on U. Then the monodromy
214
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DIFFERENTIAL MANIFOLDS
principle can be applied by consideringf as a mapping of U into X = C - {0}, and by taking Y to be the Riemann surface of the logarithmic function (16.12.4.2). It follows (1 6.8.11) that there exists a holomorphic function g on U such that f = ee (and g is unique up to a constant integer multiple of 2719.
PROBLEMS
1. Generalize the definitions and results of (16.28.1) to (16.28.4) by replacing coverings of differential manifolds by Co-coverings (Section 16.25, Problem 8) of arcwise-connected topological spaces. 2. Let X be an arcwise-connected space, (Y, X,p) a Co-covering of X and G the group of X-automorphisms of this covering (16.12.1).
(a) Show that G (considered as a discrete group) acts freely on Y. (Use (16.28.1).) (b) If Y is arcwise-connected and G acts transitively on one fiber of Y,then G acts transitively on all fibers of Y. (Join two points of Y by a path.) (c) A Co-covering (Y, X,p) is said to be a Galois covering of X if Y is arcwise-connected and G acts transitively on each fiber (which implies that G acts simply transitively on each fiber, by virtue of (a)). Then (Y, X , p ) is a principal bundle with the discrete group G as structure group. Let a E X and b ~ p - ' ( u ) For . each u E ml(X, a), there exists a unique element : u-g, is a surjective homogo E G such that go(b)= b . u.Show that the mapping morphism of m , ( X , a ) onto the group Go opposite.to G, and that its kernel is the (normal) subgroup of m l ( X , a) which is the image of ml(Y, b) under the homomorphism p* : in other words, we have an exact sequence Qb
1 +nl(Y, b) 2 ml(X, a)+Go + I . 3. Let G be a discrete group acting continuously and freely on a locally compact metrizable space Y; let C be the set of points (y, z) E Y x Y such that z = s * y for some (unique) s E G. Put s = ~ ( yz ), for (y,z) E C. Show that the following conditions are equivalent :
(a) G acts properly on Y (Section 12.10, Problem 1); (b) for each y E Y there exists a neighborhood V of y in Y such that s . V n V for all s # e in G ; (c) C is a closed subset of Y x Y and p : C + G is continuous.
=@
4.
Let Y be an arcwise-connected space, and G a discrete group acting continuously and freely on Y,and satisfying condition (b) of Problem 3. If p :Y + Y / Gis the zanonical mapping, show that (Y, Y/G,p) is a Galois Co-covering whose group of automorphisms is canonically isomorphic to G.
5.
Let (Y, X, p ) be a Galois Co-coveringof X (Problem 2) and let G be the opposite of its group of automorphisms. For each discrete space F on which G acts on the left, the
28 COVERING SPACES A N D THE FUNDAMENTAL GROUP
215
fiber bundle Y x F with fiber-type F associated with Y (16.14.7) is a Co-covering of X. For Y x F to be connected it is necessary and sufficient that G should act transitively on F, so that if r is the stabilizer of a point of F, then F can be identified with G / r ; and then Y x F can be identified with Y I P , and Y is a Galois Co-covering of Y x F, whose automorphism group is isomorphic to the opposite of I?. 6. Let X be an arcwise-connected, locally arcwise-connected topological space.
(a) Show that every arcwise-connected component of a CO-covering (Y,X, p) of X is a Co-covering of X. (Consider an arcwise-connected open neighborhood of a point of X, over which Y is trivializable.) (b) Let a E X, and let U be an arcwise-connected open neighborhood of a in X, such that the canonical image of .rrl(U, a) in .rrI(X,a)consists only of the identity element. Show that every Co-covering of X is trivializable overU.(If (Y, X,p) is aCo-covering, consider a connected component of p - ' ( U ) and show that .rrl(U,a) acts trivially on the fiber over a of this covering of U.) 7.
Let X be an arcwise-connected space, (Y, X,p) a connected Co-covering of X. Let X be an arcwise-connected, locally arcwise-connected space and f:X --f X a continuous mapping. Let a' E X , a =f'(a'), b E p-'(a). Show that f admits a continuous lifting g :X +Y such that g(a') = b if and only if the image of the homomorphism f* : xI(X',a') -+nl(X, a) is contained in the image of the homomorphism P* : nlw,b) +7r'(X, a).
(With the notation of the proof of (16.28.8), apply (16.28.qii)) to Y O.) 8.
(a) Let X be an arcwise-connected, locally arcwise-connected space, and let (Y, X, p ) , (Y',X, p') be two Co-coverings of X. If Y is arcwise-connected, show that for every X-morphism g : Y + Y , (Y', Y,g ) is a covering. (If U is an arcwise-connected open subset of X such that Y and Y' are trivializable over U, let V be a connected component of p-'(U), and consider g-'(V).) (b) Suppose in addition that Y' is arcwiseconnected. Let a E X, b E p-'(a), b' E p'-'(a). Show that there exists an X-morphism g :Y'+ Y such that g(b') = b if and only if the image under p i of n l ( Y ,b') in nl(X, a) is contained in the image under p* of nl(Y',b) in n,(X, a);the X-morphismg is then unique. (Use Problem 7.) (c) Deduce that, for each u E .rrl(X, a) and b E p-'(a), there exists an X-automorphism of Y mapping b to b . u if and only if u belongs to the normalizer in rrl(X, a) of the stabilizer of b (for the action of nI(X, a) on the fiber over a). (d) Deduce from (c) that an arcwise-connected covering (Y, X, p ) of X is Galois (Problem 2) is and only if, for some point b E p-'(a), the image under p* of nI(Y, b) is a normal subgroup of r,(X, a).
9. l n R3, let T be the union of the following sets: the segment with endpoints (0, - I , 1 ) and (0,2, I), the segment with endpoints (0,2, 1) and (1.2, O), the segment with endpoints ( I , 2 , O ) and ( I , 0,0).and the set of points ( x , sin(n/x), 0)for 0 < x 5 n. Let X
be the projection of T on RZ (identified with the plane z = 0), and let Y be the union of the sets T 4-ne3 with n E Z. Show that X i s arcwise-connected, that Y is a connected Co-coveringof X whose arcwise-connectedcomponents are the sets T ne, ,whichare not Co-coveringsof X; and that Y is not trivializable, although Xis simply-connected.
+
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10. Let B be a separable, metrizable, locally compact space, B' a topological space, and (X, B , n) a Co-fibration whose fibers are differential manifolds. Let f, g be two continuous mappinB of B into B' and let v : B x I B' be a homotopy offinto g. Suppose that there exists a continuous lifting h : B -+X off. Show that there then exists a continuous lifting I,$ : B x I -+ X of the homotopy 'p. Moreover if, for some bo E B, v(bo,0 is independent of 6 E I, then # can be chosen so that I,$(bo,0 E X is independent of 5 (homotopy lifting theorem: apply Section 16.26, Problem 6 and consider the inverse image 'p*(X)). -+
29. THE UNIVERSAL COVERING OF A DIFFERENTIAL MANIFOLD
(16.29.1) Let X be a connected diflerential manifold. Then there exists a connected, simply-connected covering (Z, X , p ) of X. If (Y, X, a ) is a connected covering of X, b a point of Y, a = a(b), and c a point of p-'(a), then there exists a unique X-morphismf :Z + Y (16.12.1) such that f (c) = b, and(Z, Y, f) is a covering of Y. In particular, i f Z is a connected, simply-connected covering of X, then there exists an X-isomorphism of Z onto Z .
We shall prove the second assertion first (the third will then follow immediately, by reason of (16.28.4)). The existence and uniqueness of the Xmorphism f follow from the monodromy principle (1 6.28.8) applied to p : Z + X and the covering Y. To show that (Z, Y,f) is a covering, let us first show that f is surjective. Let y E Y ; because Y is connected, there exists a path fl : I -+ Y from b t o y . The path a fl : I -+ X from a t o a ( y )then lifts t o a path y : I + Z from c to a point z ~ p - ' ( n ( y )(16.28.2). ) It is clear thatfo y is a path with origin b which lifts a 0 p, hence is equal to /I, and thereforef ( z ) = y . Now consider a connected open neighborhood U of a ( y ) such that n-'(U) is the union of a sequence of pairwise disjoint open sets V,, the restriction n, of n to each V, being a diffeomorphism of V, onto U, and such that p - ' ( U ) is the union of a sequence of pairwise disjoint open sets W , , the restriction p , of p t o each W, being a diffeomorphism of W, onto U. Suppose that y E V,,; if a point z' E W, is such that f ( z ) E V,, , thenf(W,) c V,, because W, is connected; but the definition offthen shows that the restriction o f f t o W,can be written as a;' p, ,and is therefore a diffeomorphism of W, onto V,, . This proves that ( Z , Y, f) is a covering of Y. We shall now establish the existence of the covering (Z, X , p ) with the properties stated, by the method of construction of (16.1 3.3). Consider a covering (U,) of X formed of open sets homeomorphic to open balls in R",and therefore simply-connected. For each index ti, fix a point a, E U , ; for each x E U , , we shall denote by c,(x) the element of E,,=* which is the class of all paths from a, to x which are contained in U, (by hypothesis, all these 0
0
29 UNIVERSAL COVERING OF A DIFFERENTIAL MANIFOLD
217
paths are equivalent in Qua, ,). Let a. be one of the indices, and write a = a,, to simplify the notation. For each x E U,, the mapping U C , u * c,(x) is a bijection of E,,, onto E,,, x ; we shall denote this bijection byf,(x). Given two indices a, b and x E U , n U , , we put f&) = (f,(x)) - 0.m) u H UCu(X)(C,(X)) which is a bijection of Ea,a~ onto E,, , , B . The mapping x i+fS,(x) is constant on each connected component V of U, n U,; for if x, y are two points of V, there exists a path c E Qx,, contained in V, and if s is the class of this path in Ex,,, we have c,(y) = c,(x)s and c,(y) = c,(x)s, whence the assertion follows. Finally, the above definitions show that for any three indices a, p, y, we have 9
(16.29.1.1) .f,,(x) =fYa(x) ofaa(x)
for all x E U, n u
p n
u,.
Since the sets E,,,aa are denumerable, they may be considered as discrete differential manifolds. Apply the method of (16.1 3.3) to these manifolds, by defining mappings +,a
: (Ua n U,) x
Eo.ua
+
(Ua n Up) x Eo,an
by the formula $,,(x, u ) = (x,ffl,(x)(u)). Since x ~ f @ , ( xis ) locally constant, the mapping (x, u)~-+f,,(x)(u)is of class C", and for each x E U, n U, we have seen thatf,,(x) is a bijection of E,,,,,~ onto E,,,,,B ; finally, it follows from (1 6.29.1 .I)that the $ , satisfy the patching condition (16.13.1.1). We have therefore defined a covering (Z, X, p ) of X, and it remains to show that Z is connected and simply-connected. We shall apply (1 6.28.4) by showing that on the fiber p - ' ( a ) = E,,,,, = 7c1(X,a), the fundamental group q ( X , a) acts by right multiplication (for the group structure of 7c1(X,a)); this will complete the proof. Consider a path 1: [0, I] -+ X with origin 1(0) = a . For each E [0, 13, let AE be the path [0, 51 --t X obtained by restricting A to [0,
<
(45)9h(45))-1(U<))
E
u, x E 4 , a a -
Notice first that if this is true for one index a such that I(<) E U a ,then it is true for all other indices p such that A ( 5 ) E U,. This follows from the definition of the transition diffeomorphisms given earlier. Let A c [0, 11 be the set of points 5 for which the property in question holds; clearly 0 E A, and it will suffice to show that A is both open and closed in [0, I] (3.19.1). Let then 5 E A; then there is a neighborhood V of 5 in [0, I] such that 1(V) c U, for
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DIFFERENTIAL MANIFOLDS
some index a ; hence p(V) is contained in an open set W , c Z such that p IW, : W, + U , is a diffeomorphism. Let be a point of A n V. We have to show that
U(t"
- '(Ur.1 =S,(Xr))- '(%J
for all t' E V. Now if we suppose for example that t' > q, then the relation A([v, t'])c U , implies that c,(1(q))-'cU(1(t'))is the class in of the path z H 1 ( t ) restricted to [q, 5'3 ; but, by definition, u;'ur. is also the class of this path, and the assertion is proved. Suppose now that 1is a loop with origin a. If u is its class in E,,, = n,(X,a), then the point p(1) is the image in p-'(Ua0) of (a,f,(a)-'(u));but c,,(a) = e, by definition, hence p(1) is the image of (a, u), and since u = e, * u, this completes the proof. The connected and simply-connected covering (Z, X, p), defined up to X-isomorphism, is called the universal covering of X. (16.29.2) We remark that (Z, X,p) is a principal bundle with structure group nl(X, a) acting on the Zeft: if q, : U , x E,, -+ Z is the mapping defined in the above construction, then for v E n,(X, a) and (x, u) E U, x Ea,as,we have u * %Ax, 4 = cp,(x, v .u).
,.
It is clear that if X is a real-analytic manifold (resp. a complex manifold), then the construction of (1 6.29.1) defines the universal covering space Z as a real-analytic manifold (resp. a complex manifold), p being an analytic (resp. holomorphic) mapping.
(16.29.3)
Example (16.29.4) The Riemann surface Y of the logarithmic function (16.8.11) is a universal covering of C* = C - {O}. For since C* is isomorphic to U x R*, (16.8.10), its fundamental group is isomorphic to 2 (16.27.10), and it acts simply transitively on the fiber of the point z = 1, as we have seen in (1 6.12.4).
PROBLEMS
1. (a) Let X be an arcwise-connected, locally arcwise-connected topological space. Show that there exists a simply-connected arcwise-connected CO-covering space Y of X if and only if there exists a covering of X by arcwise-connected open sets U, such that the canonical homomorphism rrl(U,, am)--f nl(X, a,) i s trivial, for some a, E U, . (To show that the condition is necessary, considera simply-connected Co-covering(Z, X, p ) of X,
29
UNIVERSAL COVERING OF A DIFFERENTIAL MANIFOLD
219
choose the U. such that Z is trivializable over U,, and consider the connected components of p-I(Ua). To show that the condition is sufficient, follow the existence proof in (16.29.1).) (b) If X satisfies the conditions of (a), show that every Co-covering space of a Cocovering space of X is a Co-covering space of X (reduce to connected covering spaces). (c) Suppose that X satisfies the conditions of (a), and let Y be the simply-connected covering space of X. Then Y is a Galois Co-covering space (Section 16.28, Problem 2) whose automorphism group G is isomorphic to the opposite of 7rl(X). Every connected Co-covering space of X is of the form Y x F (Section 16.28, Problem 2). (d) Show that there exists a canonical one-one correspondence between the subgroups of 7rl(X) and the isomorphism classes of connected Co-coverings of X. A subgroup r of rrl(X) corresponds to a connected Galois Co-covering of X if and only if r is normal in 7rl(X), and the automorphism group of the covering is then isomorphic to the opposite of 7rl(X)/r (" Galois theory of coverings "). 2.
Let X be the union of the two circles 1z- 112=1, l z - 2 1 z = 4 in C. Show that X admits a universal Co-covering space, and describe it explicitly. Let Y be the compact subspace of C which is the union of the circles I z - (I/n) 1' = l/n2for all n 2 1. Show that Y does not admit a universal Co-coveringspace (although Y is arcwise-connected and locally arcwise-connected). Describe the nontrivializable connected Co-coverings of Y.
3. Let K be a Galois extension of degree n of the field C(X) of rational functions in one indeterminate with coefficients in C. There exists an element 0 E K which generates K and is a root of an irreducible polynomial of degree n in Y:
F(X, Y) = Y"
+ al(X)Y"-' + ... + a.(X),
where the aj are polynomials in C[X]. Let A(X) be the discriminant of F (regarded as a polynomial in Y). (a) Let zl,... ,zN be the zeros of A in C, and let U be the complement in C of the set {zl, ... ,zN}.Let Z be the set of points ( x , y ) E Cz such that F(x, y ) = 0. Show that R = pri'(U) n Z is a connected n-sheeted covering of U. (Use Section 16.12, Problem 2(f). To show that R is connected, consider a connected component of R and the elementary symmetric functions of the second projections of the points where this component meets pr;'(x) for x E U; then use the fact that F is irreducible.) (b) Let r be the Galois group of K over C(X). For each u E I? we have
where the b,,, are rational functions in X. For each x E U other than the poles of the bO,,(u E r, 0 sj 5 n - 1) and each y E C such that (x, y ) E R, put
c b,.
n- 1
u d x , u) =
j=o
j(XlrJ.
Show that the mappings u, extend by continuity to the whole of R (9.15.2) and that u, is an automorphism of the covering R of U. The mapping U ' H U , is an isoniorphisrn of 1' onto the group of automorphisms of R, and R is a Galois covering.
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(c) Generalize the results of (a) by replacing K by an arbitrary finite algebraic extension of C(X). (Embed L in a Galois extension K of C(X), use (b) above and Galois theory.) 4.
With the notation of Problem 3 (but not assuming K to be Galois over C(X)), suppose that aF/aX and aF/aY do not simultaneously vanish at any point of Z. Then there exists a unique structure of a complex manifold on Z which induces the complex-analytic structure of U. If F(X, Y) = Y2- X2 X", show that F is irreducible over C(X) but that there exists no structure of differential manifold on Z compatible with the topology of Z as a subspace of C2.
+
5. Let X be a metrizable, arcwise-connected, locally arcwise-connected space, Y a metrizable space, p : Y + X a continuous mapping. Suppose that for each x E X the subspace p-'(x) of Y is discrete.
(a) Suppose that, for each a E X, each path p : I +X with origin a (where 1 = [O,l]) and each point b E p-'(a), there exists a unique path y : I + Y with origin b such that j? = p o y (cf. (16.28.2)). Show that for each a E X and each b ~ p - ' ( u ) there , exists 6 > 0 with the following property: for each loop p : 1+X with origin a, contained in the ball B(a; a), the unique path y : I + Y with origin b which lifts j? is a loop. (Argue by contradiction, and show that if the result is false there exist two sequences (Am),(p.) in [0,1] such that h. < pn< Am+] and A,,+ 1, vn+ 1, and a loop p : 1+ X with origin a having the following properties: The restriction j?" of B to [h., p.] is a loop with origin a and of diameter < I/n, and its lifting ym: [A,, p.]+ Y with origin b is not a loop. Then define j? on [pLn, &+,] so that up to equivalence the restriction of B to [p., is the opposite of the loop pn,and lift this loop to a path in Y with origin y&) and endpoint b.) (b) Deduce from (a) that, under the same hypotheses, every continuous mapping IJJ : 1 x 1 -+ X such that v(0,O) = a lifts uniquely to a continuous mapping $ : I x 1 + Y such that $(O, 0) = b. (First lift the path t Hp(0, t ) to a path y : r H y(r) such that y(0) = b ; then lift each path SHP)(S, t ) to a path with origin y ( t ) ; we have to prove that the function $ : 1 x 1 + Y so defined is continuous at each point ( S O , to). By virtue of (a), there exists a > 0 such that, for each loop in I x I with origin (sl, to) (where s, E [0, 11) and diameter ga,the image under of this loop lifts to a loop with origin $(sl, t o ) in Y. Prove that $ is continuous at each point (s, t o ) such that fka<sz&(k+l)ar
by induction on k, arguing by contradiction.) (c) Suppose that X, Y, p satisfy the hypotheses of (a) and also that X admits a simplyconnected arcwise-connected Co-covering space (Problem 1). Show that (Y, X,p) is a covering of X. (Using the result of (b), show that for each of the open sets U. defined in Problem I(a), and for each point of p-'(Ua), there exists a unique continuous section of p - ' ( U , ) passing through this point.)
6. Let X be a simply-connected, arcwise-connected topological space, and let Y be an arcwise-connected, locally arcwise-connected space which admits a simply-connected, arcwise-connected Co-covering P (Problem I ). Suppose that there exists a surjectiue continuous mapping f: X +Y such that for each y E Y the fiber f - ' ( y ) is connected. Show that Y is simply-connected. (Lift f t o a continuous mapping of X into 9.)
30 COVERING SPACES OF A LIE GROUP
30. COVERING SPACES OF A
221
LIE G R O U P
Let G be a connected Lie group and let m : (x, y ) ~ x be y the C"-mapping of G x G into G which defines its group structure. Let (Z, G , p ) be a universal covering (16.29.1) of the manifold underlying G, and let Z be a point of p-'(e). We shall show that there exists on Z a unique Lie group structure such that 8 is the identity element and p : Z + G a group homomorphism. The space Z x Z is connected (3.20.16) and simply-connected (1 6.27.10). Consider the composite mapping z x zPXP-G x G ~ G ; by virtue of the monodromy principle (1 6.28.8) this C"-mapping lifts uniquely to a C"-mapping f i : Z x Z + Z such that f i ( 8 , E) = Z. We have to show that f i defines on Z a group structure, which by construction will be the unique group structure for which p is a homomorphism. First of all, the multiplication f i on Z is associative, for the two mappings (x, y , z ) ~ & ( f i ( xy, ) , z ) and (x, y , Z)H f i ( x , f i ( y , z ) ) of Z x Z x Z into Z are both liftings of the same mapping (x, y , z ) ~ p ( x ) p ( y ) p (and z ) are equal at the point (Z, Z,8); hence they coincide because Z x Z x Z is simplyconnected (16.27.10). Secondly, Z is a neutral element for f i , because the mappings X H f i ( x , Z), x ~ f i ( Zx), and the identity mapping of Z are three liftings of the same mapping x ~ p ( xwhich ) take the same value at the point 8, and therefore coincide for the same reason as before. Finally, to prove the existence of inverses, consider the C"-mapping x w ( p ( x ) ) - ' of Z into G, which lifts to a C"-mapping 7: Z + Z such that ;(a) = 8. The three mappings XH 2 ( x , i(x)),XH 2 ( q x ) , x ) , and XH Z are liftings of the constant mapping X H C which take the same value at the point 2, and therefore once again they coincide. (16.30.1)
The manifold Z, endowed with the Lie group structure just defined, is called the universal covering group of G and is denoted by e. From now on we shall write xy in place of f i ( x , y) ( x , y E e). (16.30.2) Let x, y E 6,and let t hy ( t ) be a path in 6,defined on the interval I = [0, 11, with origin E and endpoint y . Then t H x y ( t ) is a path in e from x to xy and is the unique lifting with origin x of the path t Hp(x)p(y(t))in G. In particular, suppose that x and y belong to the kernel p-'(e) of the homomorphism p . Then the remark above shows that xy is equal to x . uy, where uy is the class in q(G, e) of all the loops with origin e which lift to a path from 2 to y (16.28.2). If z is another point of p-'(e) we have xyz = x * uyz = x (uyu,) by (16.28.2.1); hence uyz = u y u z . This proves that y++uy is an isomorphism of the discrete group .rr,(G, e) onto the discrete subgroupp-'(e) of 6.Furthermore :
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The subgroup p-'(e) of
e is contained in the center of e.
For p-'(e) is a normal subgroup of 6, and the assertion therefore is a particular case of the following general lemma: (16.30.2.2) In a connected topological group H , every discrete normal subgroup D is contained in the center of H .
For each d E D, the mapping x ~ x d x - ' of H into D is continuous and therefore constant (3.19.7); hence its value is ede-l = d. Consequently x d = dx for all x E H. From (16.30.2.1) it follows that the fundamental group q(G) of a Lie group G is commutative. It follows also that G is isomorphic to 6 / p - ' ( e ) (16.10.8). Conversely, if L is a connected, simply-connected Lie group, and N is a discrete normal subgroup of L (hence contained in the center of L), then L is a universal covering group of L/N (1 6.14.2). (16.30.3)
Let G be a simply-connected Lie group, G a connected Lie group,
el the universal covering group of G , andp ; 6'+ G' the canonicalprojection.
Then for each Lie group homomorphismf ; G + G , there exists a unique Lie group homomorphism g : G + 0 such that f = p g . 0
The existence and uniqueness of a Cm-mapping g : G + el such that f = p g and g(e) = I' (where e, I' are the identity elements of G and e', respectively) follows from the monodromy principle (16.28.8). To show that g is a homomorphism, consider the two mappings (x,y ) Hg(x)g(y) and (x,y)t-+g(xy)of G x G into 61. Since P ( & M Y ) ) = f ( x l f ( v )=f(w) = P(g(xy)), ) take the same value at both these mappings are liftings of (x,y ) ~ f ( x yand the point (e, e); hence they are equal (16.28.1). 0
Consider now an arbitrary connected covering space (Z,, G,p,) of a connected Lie group G, and let el be a point of the fiber p ; '(e). We shall show that there exists a unique structure of Lie group on Z, for which el is the identity element andp, is a homomorphism. Let q be the unique G-morphism of e onto Z, such that q(Z) = el (16.29.1), which makes (6,Z,, q ) a covering of Z,. We shall show that the multiplication m, on Z, is defined as follows: if x1 = q(x) and y1 = q(y), then m,(x,, y,) = q(xy). To show that mi is welldefined we must show that if &) = &'), then q(xy) = q(x'y) and q(yx) = q(yx'). For this, consider the two mappings z ~ q ( x zand ) z ~ q ( x ' z )they ; are (1 6.30.4)
30 COVERING SPACES OF A LIE GROUP
223
liftings to Z, of the same mapping of G into G, because pl(q(xz))= p(xz) = p(x)p(z) and p,(q(x'z)) = p(x'z) = p(x')p(z), and p(x) = p(x') because q(x) = q(x'); also, for z = 2 we have q(x2) = q(x'Z), and therefore q(xy) = q(x'y) for all y E d by virtue of (16.28.8). The proof that q(yx) = q(yx') is similar. It follows that rn, is well-defined and therefore defines a group structure on Z, such that q is a homomorphism. Furthermore, because q is a local diffeomorphism, it follows immediately that rn, is of class C", and that the inverse mapping x , HX;' is also of class C". It is clear that p , : Z, + G is a homomorphism for the group structure just defined on Z,, and this group structure is the unique Lie group structure with this property, for which el is the identity element. For if x,, y1 E Z , and if a, jl are paths in Z,, defined on [0, I], with origin el and endpoints, respectively, x, and y,, then the hypothesis that p , is a homomorphism implies that the path t ~ a ( t ) j l ( tis) a lifting to Z, of the path t wp,(a(t))p,(jl(t))in G, and this lifting has origin e l , and hence is uniquely determined (1 6.28.1). Since q is surjective,the kernel p ; ' ( e )is equal to q(p-'(e)). Hence, bearing in mind (16.14.2), we see that we obtain (up to isomorphism) all the coverings (G,, G, p , ) , where G, is a connected Lie group and p 1 a homomorphism, by taking the quotient group b/D, where D is a subgroup of p-'(e), and p;'(e) is isomorphic to p-'(e)/D (and hence isomorphic to a quotient of nl(G)). (16.30.5) Let G be a connected Lie group and H a Lie subgroup of G such that G/H is simply-connected. Then H is connected and q ( G ) is isomorphic to a quotient of nl(H).
Let H, be the identity component of H, so that H/H, is discrete (12.11.2). It follows from (16.14.9) that G/H, is then a covering space of G/H; since it is connected (3.19.7) and G/H is simply-connected, it follows that G/H, is canonically diffeomorphic to G/H (1 6.28.4), hence H = H, and H is connected. Consider now the universal covering (e,G, p ) of G , and the covering fi = p-'(H) induced by G (16.12.8). Since G = G/p-'(e) and H = fi/p-'(e) it follows, by (16.14.9) applied to the groups d 2 2 p-'(e), that G/H is diffeomorphic to consequently is simply-connected, and since e is connected, it follows from the first part of the proof that A is connected; but then it follows from (1 6.30.4) that n,(G, e) = p-'(e) is isomorphic to a quotient of n,(H).
e/R;
Examples (1 6.30.6) By (1 6.11.5) and (1 6.27.8), the spaces SO(n + l)/SO(n) are simplyconnected for n 2 2; SU(n + l)/SU(n) for n 2 1 ; and U(n 1 , H)/U(n,H)
+
224
XVI
DIFFERENTIAL MANIFOLDS
for n 2 0. Since SU(1) = U(0, H) = {e}, it follows from (16.30.5) that the Lie groups SU(n) and U(n, H) are simply-connected for all n 2 0. Since each matrix X E U(n, C) can be written uniquely in the form
where YeSU(n), the manifold U(n,C) is diffeomorphic to the product S, x SU(n), hence al(U(n, C)) = 2 for n 2 1 ((1 6.27.9) and (1 6.27.10)). As to the groups SO(n), observe that the mapping p which associates to each quaternion z of norm 1 the rotation u, : XHZXZ-' in R3 (identified with the space of pure quaternions) is a surjective homomorphism of the Lie group U(l, H) (the multiplicative group of quaternions of norm 1) onto S0(3), with kernel {- 1, l}. Hence ((16.9.9 (iv)) and (16.14.2)) U(l, H) is a two-sheeted covering of the group SO(3). Since U(l, H)is simply-connected, we have 7c1(so(3)) = 2/22, From the remarks above and from (16.30.5), it follows that a,(SO(n)) for n > 3 is either 2 / 2 2 or trivial. (We shall see later (Chapter XXI) that it is in fact equal to 2/22; cf. Problem 10.) Recall finally that SO(2) is isomorphic to the multiplicative group U = U(1, C) of complex members of absolute value 1, and therefore a1(SO(2)) = Z (16.27.9).
+
(16.30.7) Let G, G be two Lie groups such that G is connected and simplyconnected. Then every local homomorphism (resp. C" local homomorphism) hfrom G to G which is defined on a connected open neighborhood U of e has a unique extension to a homomorphism of topological groups (resp. of Lie groups) of G into G .
Let V be a symmetric connected open neighborhood of e such that V2 c U. Then every x E G can be written as a product x1 x2 . . * x,, with xiE V (1 5 i 5 n) (12.8). If fi is a homomorphism of G into G' which extends h, then we have h(x) = h(x,)h(x,) * h(x,,), so that li is unique. To prove the existence of h, we observe first that if we can establish the existence of a homomorphism of abstract groups which extends h, then this homomorphism will automatically be continuous (12.8.4) (resp. of class C" (16.9.7)). We shall first prove the following lemma: (1 6.30.7.1)
If xl,...,x,
EV
are such that xlxz
--
* *
. x,, = e, then
h(xl)h(x2) * h(x,,) = e'.
30 COVERING SPACES OF A LIE GROUP 225
Put yo = e, y j = y j - l x j for 1 5 j 1 sj 5 n, there exists a path
s n, so that y,, = e. Since y j E yj-lV for
from yj-l to y j . The path y : I = [0, I] + G, obtained by juxtaposing the y j , is therefore a loop with origin e. By hypothesis there exists a homotopy cp : I x I + G of y into the constant loop I + {e}, leaving the point e fixed. Let d be a left-invariant distance defining the topology of G (12.9.1) and let E > 0 be a number such that the relation d(e, z ) 5 E implies z E V ; next let 6 > 0 be sufficiently small that the relations I t - t' I 5 6 and 15 - I 5 6 imply that d(q(t, cp(t', C')) 5 E (3.16.5). We can always reduce to the case where l / n 6. For put ti = j / n , so that y ( t j ) = y j (05 j 5 n), and let t i , . .,t; be points of the interval ti] such that I ti 1 5 6, I tj - thl 5 6, and I t;+l - t;l 6 for 1 k m - 1. If we put z k = y(t;), then z k e y j V for 1 5 k 5 m, and Z k + l E Z k V . Hence, by induction on k, we conclude that
s s
e'
r),
.
s
h(yJ"zl)h(z;'zz)
* * *
h(z,=']zk) = h(yJ"zk)
by virtue of the hypothesis on h, and consequently h(y,:'z1)h(z;1z*)
* * .
h(z,l,z,)h(z,'yj+,)
= h ( y 7 1 y j + l )= h ( x j + l ) .
Hence the product h(x,)h(x,) * . . h(x,,) is not changed by replacing the -1 ..,x,,, sequence x,, ...,x,, by x,, ... ,x i , yy'z,, z ; ' z Z , ...,z, y j + l , which proves our assertion. This being so, put q(rj, t k ) = Y j k , so that yjo = y j and yjn = e for all j. It is enough to prove that h(YOklYlk)h(YlklYZk)
* '
h(y,='l, k Y n k ) = h ( y , : + 1 y 1, k +
1)
*
.
*
h(Yi-ll,
k + 1 Y n , k + 1)
for all k. This will follow by induction on j from the relation (*j )
h(yi,:+lyl,k+l)
= h(y,'y
since ynk= e. Now, to prove
' * '
1k ) (*j)
* *
h(yy-ll,k+lyj,k+l)
. h(yy-ll,k y j k ) h ( y s l y j ,k + 1)
by induction, it suffices to prove that
h ( y ~ : + l y j + l , k + l= ) (h(yik'yj,k+l))-lh(yiklyj+l,k)h(Yy~l,kyj+l,k+l);
but we h a v e y ~ : + l y j + , , , + l E V a n d ~ i , ' y j , k + lhence E V ; YS'yj+l,k+l EU, which shows that MY;' ~ jk + , ])Oil+ 1 yj+ 1, k + l ) = W i , ' y j + 1, k + l ) SimiIarly h(yi,'yj+l,k)h(yi;li,k~j+l,k+l) = N y i , 1 Y j + l , k + l ) * This completes the proof of (1 6.30.7.1).
226
XVI DIFFERENTIAL MANIFOLDS
Now each X E Gcan be expressed in at least one way as a product ... x,, , with x j E V for each j. If also x = x ; x ; . * . x; with xi E V for each k, then the relation x1 . . . x, xL-' * * = e implies, by ( I 6.30.7.1), that h(x,)h(x,) * h(xJ = h(x;)h(x;) * h(xh). We may therefore define h(x) to be h(x,)h(x,) . * - h(x,,), since this product depends only on x . If y = y1 * * . y, is another element of G, with y i E V for 1 5 i 5 p , then we have xy = x1 . . . x,y, * * * y,, so that xlxz
9
--
XI-'
and therefore h is a homomorphism. It remains to show that h(x) = h(x) for x E U. To do this, join x to e by a path a : I -+ U, then take a sequence of points e = z o , zl,... , z,,-,, z,, = x in a(1) such that z,Ll1zi E V for 1 i 5 n. Since zi E U for all i, we have h(z,)h(z;'z,) * . . h(zLll zi)= h(zJ by induction on i, and hence h(x) = h(x) as required.
PROBLEMS
1. Let X be a topological space on which is defined a law of composition ( x , y) H x * y , which is a continuous mapping of X x X into X. If yl :1 + X, y 2 : I +X (where I = [O, I]) are two paths, let yl * y2 denote the path t~ yl(r) * y2(t). (a) Show that if 7; is homotopic to y1 (resp. y ; homotopic to y2) under a homotopy which keeps fixed the endpoints of the paths, then y; * y2 (resp. yl * y;) is homotopic to yl * y2 by a homotopy which keeps fixed the endpoints of the paths. (b) For each u E X and each path y in X, let u * y (resp. y * a) denote the path r H u * y ( t ) (resp. tw y ( t ) * a). Let e, a,b be three elements of X such that e * e = e, yl E y2 E a ,c . Show that yl * y2 is homotopic to each of (yl * e)(u * y 2 ) and (e * y2)(yl * b). (Use (16.27.1)J (c) Suppose in addition that the two mappings X H X * e and x H e * x of X into itself are homotopic to l x (16.26.1). Deduce from (b) that the group rrl(X, e) is commutative. Consider the case in which X is a topological group. 2.
With the same hypotheses on X as in Problem l(c), let (Y, X, p) be an arcwise-connected covering of X,and let e' ~ p - ' ( e ) . (a) Let y l , yz be two paths in X with origin e and the same endpoint. Suppose that these paths lift to two paths yi , y ; in Y with origin e' and the same endpoint. Show that, for each path a in X with origin e, the paths yl * a and y2 * a (resp. a * yl and a * y ~ lift ) to two paths in Y' with origin e' and the same endpoint. (Argue as in Section 16.27, Problem 4.) (b) Suppose in addition that X is locally arcwise-connected. Show that there exists a unique continuous law of composition (x' y') H x' * y' on Y such that p(x' * y') = p(x') * p(y') and e' * e' = e'. In particular, if the law of composition on X makes X a topological group, then the law of composition on Y makes Y a topological group.
30 COVERING SPACES OF A LIE GROUP 227
If we identify p-'(e) with a quotient of nl(X, e) by means of the action of rl(X, e) onp-'(e), thenp-'(e) becomes a subgroup of Y contained in the center of Y. Furthermore, Y is then a Galois covering space of X, whose automorphism group G may be identified with the opposite of the kernel of the homomorphismp : Y -+ X. If X satisfies the condition of Section 16.29, Problem I , we may take Y to be the universal covering of X, andp-'(e) is then isomorphic to r l ( X , e). 3. Let I be the interval [0, I] of R,let X be a separable metrizable topological space, and let a be a point of X. For each n 2 0, let P n ( X ,a) denote the subspace of Wx(ln) (16.27.4.1) consisting of the continuous mappings of 1" into X which are equal to a on the frontier C,-lof I" in R". (For n = 0, we define lo = {O}, and then P o ( X , a) is canonically identified with X.) We have B,(X, a)= Qa, in the notation of (16.27).
(a) For n 2 I , let a. denote the constant mapping 1"+{u}. For each f~ P n ( X ,a), let
f~ 9'l(9'n..l(X, a),a m - l be ) the mapping defined byf(t ) ( x 2 , . . ., x.) = f ( t , x 2 , . . ., x,,) for (t, x 2 , . . ., x.) E I". Show t h a t f ~ f i sa homeomorphism of the space P n ( X ,a)onto
Pl(P,,-l(X, a),a,,-l).Let r n ( X ,a) denote the fundamentalgrouprl(B,-l(X, a),an-l), which corresponds canonically to the set of arcwise-connectedcomponents of 9',,(X, a). For n = 1, the identification of Po(X,a) with X identifies r l ( X , a) with the fundamental group of X,which justifies the notation. The group r m ( X ,a) is called the nth homotopy group of X at the point a. (b) Show that the elements of r,,(X,a) may be identified with the homotopy classes of continuous mappings of S. into X which map el to a (for homotopies leaving the point el fixed). (c) Show that if n 2 2, the group r n ( X ,a) is commutative. (Use Problem 1 above and Section 16.27, Problem 2(c).) (d) Let y : I X be a path in X from a to b, and let f~ P n ( X ,a). Show that there exists a continuous mapping g : I" x I + X such that, if we put f ; ( x l ,. . . ,x.) = g(xl, .., x,, , t), then j"=f and ft E Pm(X, y ( f)) for all t E I. (Use Section 16.26, Problem 3(c).) Moreover, the class of flin r n ( X ,6) depends only on the class offin r,,(X, a) and the class of y in Qa, (same method). Deduce that r n ( X ,a) and r n ( X , b) are isomorphic groups. When X i s arcwise-connected, we denote by n.(X) any of the groups rnW,a). Show that r,(X)= {0} if X is contractible. Show also that if X and Y have the same homotopy type (Section 16.26, Problem 2) and are arcwise.-connected, then r-(X)and rm(Y) are isomorphic. .j
.
4.
With the notation and hypotheses of Problem 3, let A be a closed subset of X containing the point a. Let K.-l be the complement in Cn-lof the set of points (xl, ..., xnw1, 0) such that I x , ~ < 1 for 1 sjsn- 1. Let 9,,(X, A, a) denote the set of continuous mappingsf: I" + X such thatf(C"-,) c A andf(K.-l) = {a), considered as a subspace of gx(1"). Suppose that n 2 2. For each f~ = U X ,A, a), let f e 9'1(.%-1(X, A, a), ~ " - 1 ) be the mapping defined byf(t)(x2, . . ., x.) =f(t, x 2 , . . . , x.). Show thatfwfis a homeomorphism of the space &(X, A, a) onto the space Pl(%l(X, A, a),an-,). Let rn(X, A, a) denote the fundamental group rI(2Ll(X,A, a),a n - l ) ;it is called the nth homotopy group of X modulo A at the point a. We have TAX, {a},a) = r n ( X ,a). (a)
228
XVI
DIFFERENTIAL MANIFOLDS
(b) Show that for n 2 3, the group ?r,,(X, A, a) is isomorphic to n2(4-20(,
A, aLan-2)
and hence is commutative. (c) Let nl(X, A, a) denote the set of arcwise-connected components of the space S,(X, A, a) (consisting of the paths in X with both origin and endpoint in A). Define a canonical bijection of n.(X, A, a) onto w ~ ( ~ ~ a), - ~9m-l(A, ( X , a), an-l). (Remark that A, a) is canonically homeomorphic to .21(9'n-l(X, a),9n-l(A, a), a,,-l .)
-%a,
5. (a) Let X , Y be two separable metrizable spaces, A (resp. B) a closed subset of X (resp. Y), a (resp. 6 ) a point of A (resp. B). For each continuous mappingf: X +Y such thatf(A) c B andf(a) = b, define canonically a mapping
f* :n n ( X ,A, a)
--f
nAY,B, b).
If n 2 2, or if n = 1 and A = {a},B = {b}, the mapping f* is a group homomorphism. (b) Deduce from (a) that if i :X -+ X is the identity mapping and j :A -+ X the canonical injection, then there exist corresponding canonical mappings i, : n.(X, a) -+ nm(X, A, a) for all
n 2 1,
j* : nm(A,a)+7rn(X, a) for all
n 20
(where n o ( X , a) denotes the set of arcwise-connected components of X ) . Define also a canonical mapping of 5.@, A, a) into 9,,-l(A, a) (for n 2 1) by restricting f~ Sn(X,A, a) to l"-' (the set of points (xi,... ,x.) E I" with x. = 0). Show that this mapping induces canonically a mapping a : ?r,,(X,A, a) T"-~(A,a), which is a homomorphism for n 2 2. (c) Show that the mappings defined in (b) form an exact sequence --f
VOW,a) 'J.no@, a) 7 ni(X, A, a) 1 ' .ni(X, a) 7 n i ( A a) C- *
* a
1 ' .
a) '~r nn(A, a) 7 n n
+ i K
A, a) 1 ' .* * *
(the homotopy exact sequence): For the first three mappings, this means that the image of a in no(A, a) is the inverse image under j* of the arcwise-connected component of a in X, and that the image of i* in 7rl(X, A, a) is the inverse image under a of the arcwise-connected component of a in A. (Using Problem 4(c), reduce to proving exactness for the first four mappings.) 6. Let B be a separable, metrizable, locally compact space, and (X, B, p ) a Co-fibration
whose fibers are differential manifolds.
(a) Let 6 E B. Since p ( x b ) = {b}, we have a mapping p* :wm(X,Xb ,a)-+TAB, h) for a E X, (Problem 5(a)). Show that p* is a bijection. (Use the homotopy lifting theorem; cf. Section 16.28, Problem 10.) (b) Deduce from (a) and from Problem 5(c) the homoropy exact sequence of fiber bundles
0 +no(B, b) + n o ( X , a) + T o ( X b , a) + Vi(B, b) + ... * * *
7.
+ nn(B,6 ) + nn(X,a) + n.(Xb,
a) +nn+l(B,b) +
* '
Let X be a connected differential manifold, (Y, X, p ) a connected covering of X. Show
30 COVERING SPACES OF A LIE GROUP
229
that for n 2 2, the groups rm(X) and r"(Y) are isomorphic. (Use the exact homotopy sequence.) In particular, n.(S1) = (0) for n 2 2. 8.
Let G be a connected Lie group, H a connected closed Lie subgroup of G. Show that rl(G/H) is commutative. (Use the exact homotopy sequence.)
9. (a) Let f:S , +S. be a continuous mapping, where rn < n, such that f(el) = el . Show that f i s homotopic to a C"-mapping g : S, + S. such that g(el) = e l , under a homotopy leaving el fixed. (Argue as in (16.27.8).) Deduce that nm(Sn)= (0) for m
to z. (d) If 2 s p 5 n and k < n - p , show that the kth homotopy group rk(Sn,p) of the Stiefel manifold Sn, is zero. (Use the fibration of Sn., defined in (16.14.10), and (a) above.)
10. Show that, for n 2 4, nl(SO(n))is isomorphic to r , ( S O ( n - 1)) (and hence isomorphic to 2/22). (Apply the exact homotopy sequence to the principal bundle SO(n) with structure group SO(n - l), and use Problem 9(a).)
11. (a) Let G be a simply-connected Lie group, H a connected Lie subgroup of G. Show that the homogeneous space G/H is simply-connected (cf. Section 16.29, Problem 6). (b) Let G be a connected Lie group, 6 its universal covering group, p : 6 G the canonical projection. For each connected Lie subgroup H of G, let 8 =p-'(H) and let 8,be the identity component of 8.Then there exists a unique Ca-mappingfof onto G/H such that the diagram --f
e/n,
is commutative, where the vertical arrows are the canonical mappings. Show that (6/B,, G/H, f) is a universal covering of the manifold G/H, and that rl(G/H) is isomorphic to 8/8,.
CHAPTER XVll
DIFFERENTIAL CALCULUS ON A DIFFERENTIAL MANIFOLD: 1. DISTRIBUTIONS AND DIFFERENTIAL 0PERAT0RS
The previous chapter has provided us with the necessary algebraic and topological foundations for analysis on differential manifolds, which is our goal. The task now is to generalize to the context of differential manifolds the classical notions of differential calculus : derivatives, partial derivatives, differential equations, and partial differential equations. This is neither obvious nor simple, since we no longer have at our disposal the underlying vector space structure which served to define the notion of the derivative of a mapping of an open subset R of R” into R“. If we bear in mind the definition of differential manifolds by means of charts, we can of course seek by these means to bring everything back to the classical definitions; but it is essential to verify that in this way we obtain notions which are intrinsic to the differential manifold, that is to say, which do not depend on the choice of charts. Now, once we have acquired the notion of the tangent vector space at a point to a manifold (16.5), the only “infinitesimal” notion which intuitively appears to be intrinsic is that of the tangent linear mapping (16.5.3) and, for realvalued functions on a manifold, the notion of the differential (16.5.7) which is essentially a particular case of the previous notion. A generalization of the notion of a “partial derivative” appears to be more problematical, because in R“this notion is tied to the choice of a particular basis of this vector space. However, if we observe that, for a mappingf of R c R” into R’”,the partial derivative D j f ( x ) (1 5 n) is just the value Df(x) . ej of the total derivative DJ(x) at the particular vector e j , we are naturally led to define the “derivative of a real-valued functionfin the direction of a tangent vector h, at the point x” to be the value OhX * f = (df(x), h,) of the differential df(x) = d, f at the point x, for the vector h,.? From here we pass to the generalization of the
sj
t Some authors define tangent vectors at a point by identifying them with these operators. We have preferred to give a more “geometrical ” definition, which remains close to intuition and distinguishes carefully this intuitive aspect from the operator aspect. 230
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
231
partial derivative, not any more at a point, but as a function of the point, by taking for each point x a tangent vector h, depending on x, that is to say a vectorfield Xon the manifold (16.15.4); the function x H ( O X - f ) ( x )= Ox(,) .f is then the generalization of the notion of a first-order partial derivative. The difficulties appear to be much more serious when it comes to the generalization of higher-order partial derivatives, where our “geometrical” intuition leads us badly astray. We arrive in fact at the right definition by an unexpected detour: a linear combination f w caDaf(x)of partial deriv14 6 m
atives of order 5 m at a point x E CI t R”can be characterized as a linearform on the vector space of m times continuously differentiable functions on R, having the following two properties: (i) it takes the same value for two functions which are equal in some neighborhood of x (in other words, it is a “local” operator); (ii) it is continuous for a topology in which two functions are “close” to each other if all their partial derivatives of order S m are close to each other in the sense of the topology of uniform convergence on compact sets (defined in (12.14.6)). It is clear that we cannot define partial derivatives (for a function on R c R“)in this way without running a vicious circle; but it is perfectly possible to transport this definition to a differential manifold M by means of a chart: for it can be verified that, although the partial derivatives of the local expression (16.3) of a function defined in a neighborhood of a point x E M depend on the choice of chart, nevertheless the notion of an “ m times continuously differentiable function ” and the topology envisaged above do not depend on the particular chart chosen (17.2). The notion at which we arrive in this way is that of a point-distribution, which is a particular case of the notions of distribution and current on a differential manifold. These notions, which embrace simultaneously the two basic concepts of infinitesimal calculus-derivative at a point, and integralhave become fundamental in contemporary analysis, and we therefore begin this chapter with an elementary account of them (Sections 17.3-17.12). In this account we have emphasized the “operator” aspect of distributions, much more than the “generalized function” aspect which some authors put in the foreground. The latter aspect in fact acquires a meaning only when one has available a privileged measure (17.5.3) on the manifold under consideration, and to give it the leading place obscures the fundamental role played by distributions in the theory of linear functional equations (Chapter XXlll). For distributions on a Lie group, the group structure is reflected by the operation of convolution of distributions (17.1I),which generalizes convolution of measures (14.5). The fundamental importance of this operation will appear clearly only when we come to Fourier analysis (Chapter XXII), but already in this chapter its utilization for the regularization of distributions renders it very valuabte. Furthermore, convolution of point-distributions lies at the base of the infinitesimal study of Lie groups (Chapter XIX).
232
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
Just as the notion of a vector field generalizes that of first-order partial differentiation, so the notion which generalizes partial differentiation of arbitrary order is that of a field of point-distributions (which transforms realvalued functions into real-valued functions). However, this notion is not general enough for all applications, because it is necessary to be able to “differentiate” not only real-valued functions but also vector-valued functions, and more generally (in conformity with the very nature of differential manifolds) functions which, at each point x of a manifold M, take their values in a vector space depending on x-in other words, sections of a vector bundle E over M. Guided by the notion of a point-distribution, we arrive thus at the general notion of a diflerential operator on a vector bundle E, which transforms (differentiable) sections of E into sections of another vector bundle F (which may or may not be the same as E) over M (17.13). It is this notion which will enable us to define intrinsically the concept of a linear partial differential equation on a manifold (Chapter XXIII). The rest of the chapter is devoted to the study of certain first-order differential operators which are essential for the following chapters. The fundamental property of symmetry of the (total) second derivative (8.12.2) is reflected, in analysis on manifolds, by the existence of two firstorder differential operators which depend only on the structure of the differential manifold : exterior differentiation of differential forms (17.15), which transforms a pform into a (p + 1)-form; and the Lie derivative ZHOx Z which, for a given vector field X, transforms a tensor field Z of type (r, s) into a tensor field of the same type (17.14). It should be emphasized that the value of the Lie derivative Ox . Z at a point x depends not only on the value X(x) of the vector field Xat the point x , but also on its values in a neighborhood of x . It is not possible to define intrinsically, for a tensor field Z which is not a scalar function, the notion of “derivative at a point x in the direction of a tangent vector h,,” at any rate in terms of the manifold structure alone. The reason for this is that the manifold structure provides no intrinsic way of comparing tangent vectors at two distinct points, strange though this may appear to our “intuition.” In order to be able to make such comparisons, it is necessary to endow the manifold with an additional structure defined by what is called a linear connection.? This notion is introduced in Section 17.16 for an arbitrary vector bundle ; its relation with an analogous notion for principal bundles will be brought out in Chapter XX, where we shall also study in detail the most important type of linear connection, the Levi-Civitu connection on a Riemannian manifold. In this chapter we shall only show how the
-
f It appears intuitively obvious that we should be able to compare tangent vectors at different points, precisely because we think always of a manifold as embedded in some R”, which carries a canonical linear connection so “natural” that we take it for granted.
1 THE SPACES d ( r ) ( U )
233
presence of a linear connection allows us to define the notions of covariant derivative of a tensor jield at a point x in the direction of a tangent vector h, (which generalizes the notion of the derivative of a scalar function at x in the direction of h,) (17.1 7), and of covariant exterior differential of a dzferential form with values in a vector bundle (17.19), which generalizes the notion of the exterior differential of a scalar-valued differential form, and leads (17.20) to the notions of curvature and torsion of a linear connection.
1. T H E SPACES d(')(U) (U O P E N IN
R")
Recall (9.1) that, for each "multi-index" v = (vl, v,, Iv( =
n
C
j= 1
v j (the totaldegree of v) and v! = v , ! v,!
..., v,) E N" we put
v"!. If v =(vj) and v'=
(v;) are multi-indices, we put v + v' = (vj + v;), and the relation v 5 v' means that v j 5 vj for all j; in which case we define v' - v to be the multi-index (v; - vj). For each vector x = (xi) E C",put xv = XI' x? * - . x: E C.Finally let D' denote the partial differentiation operator D11D3 D; (Do being the identity mapping). Let U be an open subset of R".For each integer r' > 0, we denote by b $ ) ( U ) or gCr)(U) the complex vector space of all C'-mappings of U into C . The intersection &,.(U) or &(U)of the decreasing sequence of spaces &(.)(U)is therefore the space of all C"-mappings of U into C. We shall show that &(U)(resp. &(')(U))can be endowed with the structure of a Hausdorff locally convex topological vector space, defined by a sequence of seminorms (hence metrizable (12.14.5)), and having the following property :
(*) A sequence cfk)of functions belonging to B(U)(resp. &Cr)(U))converges to 0 if and onIy iJ; for each compact subset K of U and each multi-index v (resp. each multi-index v such that I v 1 5 r), the sequence of restrictions of the Dv' to K converges uniformly to 0.
It follows immediately from (3.13.14), applied to the identity mapping, that if such a topology exists it is unique. As to its existence, an increasing sequence (K,) of compact subsets of U will be said to be fundamental if U is the union of the K,, and each K, is contained in the interior of K,,,. Such sequences exist (3.18.3). For such a sequence (K,), for each pair of integers s 2 0 and m > 0 and each functionfE c#~)(U),where r 2 s, put
234
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
It is clear that for s S r each of the is a seminorm (12.14) on B(')(U) and that ps,,5 pr, The topology on B(U) (resp. c#~)(U))defined by the seminorms pS,., for m > O and s 2 0 (resp. 0 < = s s r ) is Hausdorff, for if po, ,.Cf) = 0 for all m, thenfvanishes on each K, and hence on U. This topology satisfies the condition (*) (because each compact K c U is contained in some K,, by virtue of the Borel-Lebesgue axiom), and hence it is independent of the fundamental sequence (K,) chosen. We remark also that the topology on B@)(U)is defined already by the seminorms pr,,(m > 0). A subset H of B(U) will be said to be bounded if each of the seminorms p s , , is bounded in H. This property depends only on the topology of &(U) (12.14.12). (1 7.1.2) (i) The spaces gO(U) and 6(U) are separable Frkchet spaces. More precisely, there exists a sequence offunctions in b(U), with compact supports contained in U, which is dense in each of the spaces @(U) and &U). (ii) Every bounded subset in &(U) is relatively compact in &(U).
(i) Let (f,)be a Cauchy sequence in b ( U ) (resp. @)(U)). It follows from the definition of the seminorms p S , , that for each multi-index v (resp. each multi-index v such that I v I 5 r) there exists a continuous mappingf") :U -,C such that the sequence (D'f,) converges uniformly to f'")on each K, (7.2.1). Putting f =f (O), it follows from (8.6.3) that f is indefinitely differentiable (resp. of class Cr)and thatf'') = D'ffor all v (resp. for I v I r). Consequently f i s the limit of the sequence ( f p ) in &(U) (resp. &'(r)(U)),and therefore B(U) (resp. @)(u)) is complete. To prove the second assertion we remark that by virtue of (12.14.6.2) there exists a sequence (u,) of continuous functions with compact support which form a dense set in #O)(U) = UJU). Let A be Lebesgue measure on R",and consider the function g of class C" defined in (16.4.1.4), with support 1". For each k > 0, put gk(X) = k"g(kx),so that g k ( X ) dA(x) = 1. The sequence (gk) is said to be a regularizing sequence. Since the support of gk is k-'l", for fixed q the support of each of the functions vks= gk * us is compact and contained in U for all sufficiently large k ((3.18.2) and (14.5.4)). Let us show that the functions U k , with support contained in U form a dense set in 6(U) (resp. in each cV(U)). Since Ukq(x)= /gk(X - y)u,(y) d ~ ( y )it, follows from (13.8.6) that I)kq is of class C" and that, for each multi-index v ,
s
(17.1.2.1)
1 THE SPACES g(r)(U)
235
Fix a real number E > 0 and two integers m and r. We shall show that for each f~ B(U) it is possible to find a ukq such that I D’(f-ukq!(x) 1 5 e for all
s
x E K, and all v such that I v I r ; this will prove the assertion above. Let F be a C“-function which is equal to f on K,+, and has support contained in K,+, (16.4.3). By (13.8.6) the functionf, = F * gk is of class C“, and we have
It follows therefore from (14.11.1) that for sufficiently large k we have I D’(fk - f ) ( x ) I 5 j e for all x E K, and I v I 5 r. On the other hand, if k is sufficiently large, then for all x E K, we have, by (3.18.2) and (14.5.4)
I D‘(f, - Vkq)(x)I =
Is. +.
Dvgk(x - y)(F(y) - uq(Y)) dA(Y)
5 NI(D”Sk)
SUP
ysKm+i
I
I F(Y) - U,(Y) I.
Now, for a fixed and suitably large k, we can by hypothesis find a large q so that the right-hand side of this inequality is S+Efor 1 v I 5 r. This completes the proof of (i) for b ( U ) ; the proof for @)(U) is similar. (ii) If a sequence (f,)is bounded in b ( U ) , then each of the sequences ( D i f , )(1 6 i S n) is uniformly bounded in each K,, hence it follows from (7.5.1) and the mean value theorem that (f,)is equicontinuous. By definition of the bounded sequences in b(U), it follows that for each multi-index v the sequence (Dvfk)is equicontinuous. By applying Ascoli’s theorem (7.5.7) and the diagonal procedure (cf. 12.5.9) we can therefore find a subsequence (fk,) such that each of the sequences (DYfkk,) converges uniformly on each of the K, . In other words, the sequence (A,) converges in b(U),and this proves (ii). (17.1.3) For each multi-index v, the linear mapping f is continuous.
For pS,,(D”f) 6 ps+ (17.1.4)
H D’fof
b ( U ) into b ( U )
,,(f),from which the assertion follows (12.14.11).
For each function g
E
b ( U ) (resp. g E SCr’(U)),the linear mapping
f~ f g of’E(U) (resp. cV(U)) into itself is continuous.
For each pair of integers s, m (resp. s 5 r and m), let as,,, be the greatest of the least upper bounds of the I D’g I on K, for I v I s. By Leibniz’ formula
236
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(8.13.2), there exists a number cs,,,,independent off and g such that ps, ,,,ug)S cS,,,,aS,,,,ps,,,(f). Hence the result (12.14.11). (17.1.5) Let cp be a mapping of class C" (resp. Cr) of an open set V c R" into U. Then the linear mapping f wf 0 cp of B(U) into B(V) (resp. of &)(LJ) into s(')(V)) is continuous.
Put cp = (ql,cp2, ..., cp,), where the cpj are scalar-valued functions. Let (KL) be a fundamental sequence of compact subsets of V used to define seminorms pi, ,,,on &(V) as in (17.1.I). For each pair of integers s, m, let a:, ,,, be the greatest of the least upper bounds of the functions sup(1, I D"cpjI) on KL for I v I 5 s and 1 sj 5 n. Finally let q be an integer such that cp(Kk) c K,. By repeated application of the formula for the partial derivatives of a 'composite function (8.10.1) we obtain, for each f E B(U) ~ sm ,( f
where c:,
0
CP) 5 4,rn
~ s q ,( f L
,,, is a constant independent off and cp. This proves the proposition.
2. SPACES O F C"- (resp. C'-) SECTIONS O F VECTOR BUNDLES
Let X be a pure differential manifold of dimension n and let (E, X, n) be a complex vector bundle of rank N (16.15) over X. If U is an open subset of X, we recall (16.15) that the complex vector space of Cm-sectionsof E over U is denoted by T(U, E). Likewise we shall denote by T(')(U, E) the vector space of Cr-sections of E over U. We propose to generalise the results of '(17.1) to these spaces: b(U) is the space T(U, E) when X = R" and E = X x C is the trivial complex line bundle over X. Since in general we cannot attach a meaning to the notion of partial derivatives of a section of E, we shall reformulate the problem as follows: we have to show that the space T(U, E) (resp. rcr)(U,E)) can be endowed with the structure of a Hausdorff locally convex topological space, defined by a sequence of seminorms and having the following property: (**) A sequence (uk)of sections of T(U, E) (resp. T(')(U, E)) converges to 0 ifand only if, for each chart (V, cp, n) of X over which E is triuializable, each difeomorphism
zH(cp(n(z)),
vl(z),
vN(z))
of n-'(V) onto q(V) x C N, where the v j are linear on each fiber n-'(x), each compact K c cp(V) and each multi-index v (resp. such that I v I 5 r), the sequence
2 SPACES OF Cm- (resp.Cr-) SECTIONS OF VECTOR BUNDLES
((D'wjk)I K ) k t uj(uk((P-'(t)))for
converges uniformly to 0 for 1 S j S N, where
237
wjk(t)
=
(Pv).
Here again, by virtue of (3.1 3.14), the topology is necessarily unique. To establish existence, consider an at most denumerable family of charts (V,, cp, n) of U such that the V, form a locally finite open covering of U, and such that E is trivializable over each V, (12.6.1). For each u, let z H ( ( P a ( 4 Z ) ) , vla(z),
u,a(z))
* * *9
be a diffeomorphism of n-'(V,) onto q(V,) x CN,the via being linear on each fiber n-'(x). Let (KmO;)mnhl be a fundamental sequence of compact subsets of cp,(V,), and let p i , , , , , be the corresponding seminorms (17.1 .I) on &(cpa(Va)) (resp. &(r)(cpa(Va))). For each section u E T(U, E) (resp. u E T(")(U,E)) we define (17.2.1)
where u, is the restriction of u to V,. It is clear that the ps, are seminorms, and that if p o , ,,,,(u) = 0 for all m and all a, thenu(x) = 0 in eachV, and therefore u = 0. To show that the topology defined by these seminorms satisfies the condition (**), we have to show that if ( u k ) tends to 0 in the topology defined by the seminorms ps,,,,,, then the DYwjkconverge uniformly to 0 on K (in the notation of (**)). The compact set cp-'(K) meets only a finite number of the open sets V, ; by applying the Borel-Lebesgue axiom to the union of these V,, it follows that there exists an integer m and indices uh (1 h 5 q) such that the C~&~(K,,,,) cover q-'(K). If w h j k is the restriction of w j k to cp(V n V,,), then clearly it is enough to show that the restrictions of the D"whjkto K n cp(cp;'(K,,,)) converge uniformly to 0. Now, for each z E n-'(V n V,,), we can write N
=
vj(z)
1
I= 1
cIjh(.rr(z))ula,(z),
where the C l j h are C"-functions on V n V,, . Let $h : dv va,> qah(V be the transition diffeomorphism. Tf we put -+
Uklk
= Olah
then we have
('k
I vah)
vah)
q,',
N whjk(t>
=
CIjh((P-'(f))UhZk(~h(t)), I= 1
and the assertion is now a consequence of (17.1.4) and (17.1.5), because each sequence ( u , l k ) k z 0 converges to 0 in the space d(cp,,(V,,)) (resp. &(r)(qah(~ah))).
238
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
Equivalently, we may say that a sequence (uk) converges to 0 in T(U, E) if and ody if, for each a, the sequence of restrictions ttk I V, converges to 0 in Tor,, E); and T(V,, E) is isomorphic to (8(cp,(Va)))N,so that we are brought back to the spaces &(U).
(17.2.1.1)
The spaces T(U, E) and l?)(U, E) are separable Frichet spaces. Moreprecisely, there exists a sequence of Cm-sectionsof E over U, with compact support contained in U, which is dense in each of the spaces T("(U, E) and T(U, E). Euery bounded subset of T(U, E) is relatively compact in T(U, E). (17.2.2)
Let (Uk) be a Cauchy sequence in T(U, E). By virtue of (17.1.2), there exists for each a a section f, E T(V,, E) which is the limit of the sequence (UkIV,), and for any two indices a, fi the restrictions of f, and fBto V , n V, coincide, so that the fa are restrictions of a section f E T(U, E). By the definition of the topology of T(U, E), the sequence (uk)converges to f. In the same way we may show that T(')(U, E) is complete. To prove the second assertion, we remark that it follows from (17.1.2) that for each a there exists in T(V,, E) a sequence (Uk,) of sections with compact support contained in V,, which is dense in T(V,, E) and in each r@)(V,,E). If (g,) is a (?-partition of unity subordinate to the covering (VJ, then the sections which are linear combinations of the gaUk, are dense in T(U, E) and in each T"'(U, E). For each compact K c U meets only a finite number of sets Supp(g,), say those with indices ah (1 5 h 4 4);for each section u E T(U, E), the restriction of u to K therefore coincides with the restriction to K of 4
g,, u, and by virtue of (17.1.2) and (17.1.4), each section gahu may be
h= 1
approximated arbitrarily closely by the g,, Ukah ; whence the assertion follows. Finally, if (Uk) is a bounded sequence in T(U, E), then it follows from (17.1.2) that for each u there exists a convergent subsequence of the sequence (ukI V,) in T(V,, E). By applying the diagonal procedure, we obtain a convergent subsequence of the sequence (uk). Remark (17.2.2.1) The last part of the proof may be used, more generally, to show that a vector subspace H of T(U, E) is dense in T(U, E) provided that, for each v E H, the sections gav also belong to H; it is eqough to verify that, for each a, the restrictions to V, of the sections belonging to H form a dense set in T(V,, E). (17.2.3) If F is another complex vector bundle over X, u,, an element of T(U, E) and vo an element of T(U, F), then the linear mappings ti Hu 63 vo
2 SPACES OF Cm-(resp.C'-) SECTIONS OF VECTOR BUNDLES
and
VHU~
239
@ v of T(U, E) and T(U, F), respectively, into T(U, E @F)
are continuous. Likewise, if wX
E
T(U,
P
A E*), then
u++i(u)w,* and w * ~ i ( u ~ ) of w *T(U, E) and T(U,
the linear mappings
A E*), respectively, into
r(U,'AE*) are continuous. The same is true of the mappings SHE tHsOA t of T(U,
A
A
A to and
E) and T(U, E) into T(U, T E ) . All these propositions follow froin (17.1.4) by virtue of the definitions of (16.18), by reducing to the case where E is trivial, as we may assume by use of the general principle stated in (17.2.1 .I).
PROBLEMS
Let M,N be two differential manifolds. For each integer r 2 I , let B(')(M; N) denote the set of all C'-mappings of M into N. (a) LetfE B(')(M; N). Then there exists a denumerable locally finite covering of M by compact sets K., such that each K. (resp. each f&)) is contained in the domain of definition of a chart (Ua, 'p., ma)of M (resp. a chart (Ve,&, n.) of N). For each E > 0 and each a,let W(f. K., QJ=, (CI., E ) denote the set of all CI-mappings g : M 4N such that d K J c V. and IIDY& 0 (flId)0 cp;')(x) - DV& (sl K.) 0 p:l)(x)ll 4E for all x E va(K,) and I Y I 5 r . Show that there exists on B(')(M; N) a unique topology Yc (called the coarse C'-topology) such that, for each f~ B(')(M; N), the finite intersections of the sets W(f,K., cp#, #,, 1 ln) (where the K., cpm, &, are fixed and n is variable) form a fundamental system of neighborhoods off(cf. Section 12.3, Problem 3). Show that the topology so defined is independent of the choice (for eachfe B(')(M; N)) of the families (&), (&, ($,) satisfying the stated conditions. (b) If N is a vector bundle over M, show that 37 induces on r(')(M; N) the topology defined in (17.2). (c) If N is a submanifold of a differential manifold P,then B(')(M; N) is a subset of 6(')(M; P). Show that the topology induced on LP(')(M; N) by the coarse Cr-topology of B(')(M; P) is the coarse C'-topology. (d) Let M, N, P be three differential manifolds. Show that the mapping (f,g ) ~ og f of 8(*)(M;N) x 8(')(N; P) into b(')(M; P) is continuous with respect to the coarse CI-topologies on these three spaces. 0
(a) With the notation of Problem 1, for each family (6.) of numbers 6. > 0, let WCf, (6.)) denote the intersection of the sets W(f, K., cpa, $*, 6.) for all a.Show that there exists on B(')(M; N)a unique topology Y f(called the fine CI-topology) such that, , are fixed and the 6, for each f e b(')(M; N), the sets W(f, (6.)) (where the K., g ~ =(6. variable) form a fundamental system of neighborhoods off. Show that the topology so defined is independent of the choice (for each f~ b ( ' ) ( M ; N)) of the families (Id), (v,,), (k)satisfying the stated conditions.
240
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(b) If N is a submanifold of a differential manifold P, then &(')(M; N) is a subset of B")(M; P). Show that the topology induced on B(')(M; N) by the fine C-topology on 8')(M; P) is the fine C'-topology. (c) Give an example of an element f~ B(')(M; N) for which there does not exist a denumerable fundamental system of neighborhoods of f for the fine C1-topology (take M = N = R). (d) Suppose that fo E b(')(M; N) is a homeomorphism of M ontofo(M) and thatf, is of rank equal to dim,(M) at each point x E M. If P is another differential manifold, show that for eachgo E B(')(N; P) the mapping ( f , g ) ~ofof g B(')(M; N) x &(')(N; P) into B(')(M; P) is continuous at the point (fo ,go) with respect to the fine C-topologies. (e) With the notation of (d), give an example in whichfo is of rank equal to dim,(M) at each point x of M and is injective, but is not a homeomorphism of M ontofo(M), and the mapping g H g o f o fails to be continuous at some point go with respect to the fine C'-topologies (cf. (I 6.8.5)). (f) Suppose that fo E @)(M ;N) is of rank equal to dim,(M) at each point x E M. Show that there exists a neighborhood Vo of fo in B(')(M; N) for the fine C-topology such that each g E Vo is also of rank equal to dim,(M) at each x E M. If moreover fo is injective, show that there exists a neighborhood V1 = W(fo ,(6.)) of fo contained in Vo such that the restriction of each g E V, to each K, is injective (argue by contradiction). Give an example in which fo is injective and such that there exist noninjective mappings of M into N in each neighborhood of fo for the fine C'-topology (cf. (1 6.8.5)). If, however, fo is a homeomorphism of M onto fo(M), then there exists a neighborhood V2 of fo contained in V, (for the fine C-topology) such that each g E V, is a homeomorphism of M onto g(M). (Consider first a covering of M by compact sets L. such that L a c K, for all F. If d is a distance defining the topology of N, show that d(fo(L.), fo(M - K.)) > 0, and deduce that for a suitable choice of the family (a:), the functions g E W(fo, (6;)) C V, are injective. Put L(f0) =fo(M) -fo(M). By considering sequences (x,) in M with no cluster values, show that if g E V, and if the sequence g(x.) converges, then its limit must belong to L(fo). Finally remark that the distance from fo(K,) to L(fo) is >0, and use this remark to construct a neighborhood Vz c W(fo ,(6;)) such that g(M) n L(fo) = 0 for all g E V2.) If moreoverfo(M) is closed, then so is g(M) for each g E V2 . Finally, if M and N are connected and iffo(M) = N, theng(M) = N for all g E V, . 3. (a) Let U be an open subset of R",A a compact subset of U, and V a neighborhood of A such that V C U. For each C-function f on U and each E > 0, show that there exists a C-mappingf, : U + R which coincides withfon U - V and is of class C" on a neighborhood of A, and such that I D'fi(x) - D'f(x) I 6 E for all x E U and all v such that 1 Y I 5 r. (Let g be a C'-function equal tofon a compact neighborhood of A contained in V, and zero outside this neighborhood; take a suitable regularization h of g and consider the function f, =f+ (h - g).) (b) Let M, N be two differential manifolds. Show that for each functionf, E Q(')(M; N) and each neighborhood W(fo, (6.)) of fo in the fine C-topology (Problem 2), there exist C"-mappings of M into N in this neighborhood (proceed by induction as in (16.1 2.1 I), using (a) above).
4.
Show that in B(')(M; N)the set of mappings which are transversal over a submanifold
Z of N is a dense open set in the fine C'-topology (cf. Section 16.25, Problem 16).
3 CURRENTS AND DISTRIBUTIONS
241
3. C U R R E N T S AND D I S T R I B U T I O N S
(17.3.1)
Let X be a pure differential manifold of dimension n, and consider
(A
T(X)*)(,, on X, whose global sections are the the complex vector bundles complex-valued dzfferential p-forms on X (1 6.20.1), for 0 5 p 5 n. For p = 0, these are by definition the complex-valued functions. For brevity we shall denote by bz)(X) (resp. b,(X)) the Frtchet space T'"(X,
(
( A T(X)*)(,,)
(resp. T(X, ),T(X)*)o,)) of complex differential p-forms of class C' (resp. Cm) on X. For each compact subset K of X we denote by 9g)(X; K) (resp. g P ( X ; K)) the vector subspace consisting of the complex differential p-forms of class C' (resp. C") with support contained in K; this is clearly a closed subspace of 8g)(X) (resp. b,(X)). We denote by g;)(X), (resp. .9,(X)) the union of the subspaces @)(X; K) (resp. 9JX; K)) as K runs through all the compact subsets of X, i.e., the space of all complex differential p-forms of class C' (resp. Cm) with compact support. When p = 0 we shall drop p from these notations. We can now proceed exactly as in the definition of a measure (1 3.1), except that the Banach spaces X(X; K) are now replaced by FrCchet spaces. A p-current (or a complex pcurrent) (or a current of dimension p ) on X is by definition a linear form T on 9JX) whose restriction to each FrCchet space 9JX; K) is continuous; in other words (3.13.14), in order to verify that a linear form T on g P ( X )is a p-current, it must be shown that for each sequence (ak) of C" differential p-forms, with supports contained in the same compact set K. and which converges to 0 in b,(X), the sequence (T(ak))tends to 0 in C. A 0-current on X is called a distribution. With the notation of (17.2.1), T is a p-current if and only if, for each compact subset K of X, there exist integers s, m and a finite number of indices a l , . .. , a,, together with a constant aK 2 0, such that, for each C" p-form o with support contained in K, we have (17.3.1 .I)
IT(o) I 5 aK
* I
ps,rn,
(17.3.2) Suppose that a p-current T is such that, for each compact subset K of X, the restriction of T to gP(X;K) is continuous with respect to the topology induced by that o f g ; ) ( X ; K). In that case T is said to be a p-current of order I-r . The order of a current is the smallest integer r with this property (when such integers exist). If there exists no integer r with this property, then T is said to be a current of infinite order. It is clear that if T is the restriction to
242
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
gr)(X)
gP@)of a linear form T' on whose restriction to each 9r)(X;K) is continuous, then T is of order s r . Conversely, we shall show that if T is of order S r , then it is the restriction of such a linear form T', which moreover is unique. Let K be a compact subset of X and let K be a compact neighborhood of K in X. Then there exists a C"-function h which is equal to 1 on a compact neighborhood of K and which is 0 on X - K (16.4.2). For each pform b E S%g)(X;K), there exists a sequence ak in gP(X) which converges to /3 with respect to the topology of &$)(X)(17.2.2). The sequence (hak),which belongs to gP@; K'), therefore also converges to /I in @)(X; K') by (17.1.4). In other words, the closure of 9,(X; K ) in Sr)(X) contains 9$')(X; K) and is contained in 9r)(X;K). The existence and uniqueness of T' now follow immediately by applying (12.9.4). We shall often write T in place of T'. Examples of currents (17.3.3)
Let x be a point of X, and let z, be a tangent pvector at x (i.e., an
element of A T,(X)). Then the mapping ct H ( 2 , , a(x)) is a continuous linear form on S$')(X), hence is a p-current of order 0; it is called the Diruc pcurrent defined by z,, and is sometimes denoted by E=*. When p = 0, we obtain the scalar multiples of the Dirac measure (13.1.3). P
(17.3.4) It follows from (17.3.2) that distributions of order 0 on X are continuous linear forms on each of the Banach spaces 9io)(X; K) = X(X; K), and are therefore precisely the (complex) measures on X. (17.3.5) Let T be a p-current and w a C" differential q-form (i.e., an element of S,(X)), with q j p. For each (p - q)-form B E BP-,(X; K), we have w A /I E O p ( X ;K), and it follows therefore from (17.2.3) that the linear form B -T(w A B) is a (p - q)-current, which is denoted by T A w. If T is of order s r , then so is T A U ,and in this case we can also define T A O when w is a differential q-form of class C'. When q = 0, so that w is a complex-valued function g, we write T * g or g T in place of T A w ;if T is a measure, this definition agrees with that of (13.1.5), because of the fact that the closure of 9(X; K')in VC(X) contains X ( X ; K) for each compact neighborhood K of K.
-
(17.3.6) Suppose that p > 0, and let Y be a C" vector field on X.For each pform a E 9,(X; K), we have iy a E 9 p - , ( X ; K) (16.18.4). For each (p - 1)current T, it therefore follows again from (17.2.3) that the linear form a t+T(iy * a) is a pcurrent, which is of order j r if T is of order j r, and which is denoted by 'iy * T.
-
3 CURRENTS AND DISTRIBUTIONS
243
(17.3.7) If X, Y are two locally compact metrizable topological spaces, a continuous mapping u : X -,Y is said to be proper if for each compact subset K of Y,the inverse image u-'(K) is a compact subset of X. It then follows that if F is any closed subset of X, its image u(F) is closed in Y. To see this, let (y,,)be a sequence of points in u(F) converging to a point y E Y. Then the set K consisting of the y,, and y is compact, and therefore F n u-'(K) is compact; choose for each n a point x,, E F n u-'(K) such that u(x,,) = y,,, then the sequence (x,,) has a subsequence (x,,,) converging to a point x E F. Since u is continuous, it follows that u(x) = y, that is to say y E u(F). Now let X, X be differential manifolds and u : X -,X' a mapping of class C', where r 2 1. If a' is any p-form on X' of class C"(s 2 0), then by the formula (16.20.9.3) the inverse image %(a') is defined and is a p-form on X of class C i n f ( r - 1 , s ) . ,moreover it is clear that Supp('u(a')) is contained in u-'(Supp(a')).
If we suppose that the mapping u is proper, it follows that for each compact subset K of X the mapping a'H'u(a') is a linear mapping of 9r-')(X'; K) into 9f-')(X; u-'(K)). Furthermore, this mapping is continuous; this follows immediately from (17.2), the local expression of 'u(a') (16.20.9.2), and (17.1.4). Hence, for each pcurrent T of order S r - 1 on X, the linear form a'HT('u(a')) on 9r-"(X) is a p-current of order S r - 1; it is denoted by u(T) and is called the image of T by u. If u is a proper mapping of class C' of X' into another differential manifold X", then u u is proper of class C', and we have (u 0 u)(T) = u(u(T)) for each p-current T of order 5 r - 1 on X. If u is a diffeomorphism, then u(T) is defined for every current T on X and has the same order as T. If T is a distribution on X, then u(T) is the distribution on X' such that u(T)(g) = T(g u) for each functiong E @'-')(X'). But this formula makes sense also for functions g E &')(X); hence u ( T ) is defined also for distributions of order r. When u is a homeomorphism and T is a measure, we recover the definition of (1 3.1.6). If G is a Lie group which acts differentiably on X on the left (resp. on the right), the image of a current T under the diffeomorphism XHS x (resp. XHX * s) will be denoted by y(s)T (resp. 6(s-')T. When X = G, so that G is acting on itself by left translation (resp. right translation), and T is a measure, then 7(s)T (resp. 6(s)T) coincides with the measure so denoted in (14.1.2). If X is a differential manifold, Y a closed submanifold of X, then the canonical immersion j : Y -,X is proper; hence, for each p-current T on Y, the imagej(T) is defined and is a p-current on X. By considering the local expressions it is immediately verified that j(T) has the same order as T. For measures, this notion agrees with that defined in (13.1.7). 0
0
(17.3.8) The set of all p-currents on X forms a vector space, which we denote by 93b(X). The subspace of currents of order r is denoted by g;(')(X). When p = 0 we suppress p from the notation, so that 93'(X) denotes the space of
244
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
distributions on X, and 9'(')(X) the subspace of distributions of order r. If a is a differential p-form and T a pcurrent, we shall often write or in place of T(a). For example, for a Dirac p-current (17.3.3), we have <Ex,
9
>
a = (z,
9
44).
If T is a distribution, we write (T,f) or (f, T ) or even
(1 7.3.8.1)
in place of T(f), for a functionfe 9(X), whenever there is no risk of ambiguity. 4. LOCAL DEFINITION O F A CURRENT. SUPPORT O F A CURRENT
(17.4.1) Let U be an open subset of a differential manifold X. For each compact subset K of U, it is clear that the mapping a Ha I U is an isomorphism of QP(X; K) onto QP(U;K) (resp. of 9:)(X; K) onto 9$')(U; K)); the inverse isomorphism sends a differential p-form jE gP(U;K) to the p-form j"which is equal to fi in U and is zero in X - U. (By abuse of notation we shall often write a in place of a I U when u is a differential p-form on X with support contained in U.) For each p-current T on X, the mapping jHT(fl') is therefore a p-current on U; it is said to be the p-current induced by T on U, or the restriction of T to U, and is denoted by T, . The order of T, is at most equal to that of T, but it may be strictly less than that of T. It should be noted that a current on U is not necessarily the restriction of a current on X (Section 17.5, Problem 2), and that when it is the restriction of a current on X, the latter need not be unique. However, there is the following result, which generalizes (1 3.1.9): (17.4.2) Let (U,),EL be an open covering of X. For each 1 E L let T, be a p-current on U,, such that for each pair of indices 1,p the restrictions of T, and T, to U, n U, are equal. Then there exists a unique p-current T on X whose restriction to U, is equal to T, for each 1 E L.
We shall not write out the proof in full detail; it is based, step by step, on the proof of (1 3.1.9), with the obvious modifications. We begin by writing a p-form u E gP(X)as C ai , where ui = hi a and Supp(hi) c U,, for a suitable i
li; for this it is sufficient to invoke (16.4.2) instead of (12.6.4). It follows that T is unique and necessarily given by T ( a ) = T,,(ai), and the proof of the i
fact that this formula does define a linear form on BP(X)(in other words, that the number T(a)does not depend on the particular decompositionof a chosen)
4 LOCAL DEFINITION OF A CURRENT
245
goes over without change. It remains to show that if a sequence (ak)tends to 0 in QP(X; K), then T(ak)+ 0. We may take the same finite sequence (hi)for all the a k , and each of the sequences (TA,(hiak))kbl then tends to zero by virtue of (17.1.4); this proves our assertion. (17.4.3) It follows in particular from (17.4.2) that if the restriction of a current T to each member of a family of open sets U, is zero, then the restriction of T to the union of the U, is also zero. Hence there is a largest open subset V of X such that the restriction of T to V is zero; the complement S = V is called the support of T, and is denoted by Supp(T). A point x E X belongs to the support of a p-current T if and only if, for each neighborhood V of x, there exists a p-form a E gP(X)with support contained in V and such that T(a) # 0. If T,, T2 are two p-currents, it is clear that
c
SUPP(T1 + T2) = SUPP(T1)
“ Supp(T2),
and that if o E &,(X) is a q-form with q 5 p, then Supp(T A o)c Supp(T) n Supp(w). If n : X + X’ is a proper mapping of class C‘, then for any current T on X of order S r - 1 , we have Supp(n(T)) c n(Supp(T)). If n is a diffeomorphism of X onto X‘, then Supp(n(T)) = n(Supp(T)). If Y is a closed submanifold of X, and j : Y -+ X the canonical immersion, then Supp(j(T)) = Supp(T) for any current T on Y. When p = 0, we recover the definition of the support of a measure on X (1 3.19), by virtue of the fact that each space X ( X ; K) is contained in the closure of 9 ( X ; K’) in X ( X ; K’), whenever K’ is a compact neighborhood of the compact set K. (17.4.4) Let X, X‘ be two pure differential manifolds of the same dimension n, and let n : X’ + X be a local diffeomorphism (16.5.6). Then for each current
T on X there exists a unique current T’ on X’ with the following property: for each open subset U’ of X‘ such that the restriction nu, : U’ + n(U‘) is a diffeomorphism,we have nu,(TLt) = Tn(,,.). For there exists an open covering (U;) of X such that each of the restrictions nu;. is a diffeomorphism; if n ~ :is the inverse diffeomorphism, put Ti = X;:(T~(~;)). For any two indices I, p, the mappings nui and nu; agree by definition on U; n UL, hence n;; and n;: agree on n(U;) n n(UL). This implies that T i and TI have the same restriction to Ui n U;, and the existence and uniqueness of T’ therefore follows from (1 7.4.2). The current T’ is called the inverse image of T by n, and is denoted by ‘n(T). For example, if X‘ is a universal covering of X (1 6.29), then it follows from the definitions that the fundamental group of X leaves invariant the inverse image on X’ of every current on X. Conversely, every current T’ on X’ having this property is the inverse image of a current T on X. To see this, we take a
246
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
covering (U,) of X by connected open sets over which X' is trivializable, and we define T, to be the image by the canonical projection n of the restriction of T' to any one of the connected components of x-'(U,). More particularly, taking X = R" and X = T",the inverse images on R" of currents on T" are precisely those which are invariant under the group Z" (acting on R" by translations); such currents are said to be periodic with Z" as group of periods. (17.4.5) Let A be the support of a pcurrent T. We shall show that it is possible to attach a meaning to T(a)for some pforms a E &,(X) which are not compactly supported: it is enough that A n Supp(a) should be compact (which will always be the case if A = S u p p Q is compact). For if h : X + [0, 11 is a C"-mapping which is equal to 1 on some compact neighborhood of A n Supp(a), and is compactly supported (16.4.2), then T(hcr) is defined because ha has compact support. Furthermore, if h, is another function having the same properties as h, we have T(h,a) = T(ha),because there exists an open neighborhood V of A n Supp(a) on which h(x) = h,(x), and the support of (h - h,)a is contained in Supp(a) n 6 V, and therefore does not intersect A. We may therefore define T(a)to be T(ha)for any function h with the properties stated above. It is immediate that the set of pforms a E &,(X) such that A n Supp(a) is compact is a vector subspace of &,(X), and that a H T ( a ) is a linearform on this vector space. Next, consider a sequence (a,,) of pforms in &,(X), such that (i) all the sets A n Supp(a,) are contained in ajixedcompact set K; (ii) the sequence (a,,) tends to 0 in &,(X). Then T(a,,)+ 0. For if we choose as above a function h which is equal to 1 on some compact neighborhood of K, then the sequence (ha,,) tends to 0 in g,(X; V) (17.1.4); since T(a,) = T(ha,,) for each n, our assertion follows immediately.
5. CURRENTS ON A N O R I E N T E D M A N I F O L D . D I S T R I B U T I O N S ON
R"
(17.5.1) Consider now an oriented pure differential manifold X of dimension n. We have defined in (16.24.2) the notion of an integrable differential n-form v on X,and its integral, denoted by v or v. Now consider a locally integrable differential (n - p)-form p, where 0 5 p 5 n; for each pform a E 9$'"(X; K), the n-form /3 A a is locally integrable and has support contained in K, hence is integrable. We shall show that the linear form a H /3 A a on g?)(X) is a pcurrent (or order 0). The proof reduces immediately to the situation where X is an open set U in R",and then we have
I Ix
s
5 CURRENTS ON AN ORIENTED MANIFOLD
247
where the bH(resp. the u1-H) are the coefficients of B (resp. a) relative to the canonical basis of the %‘(U)-module &‘$?,,(U) (resp. &‘r)(U)),and in the summation I = {1,2, .. .,n} and H runs through all subsets of n - p elements of I. Then wehavetoshowthateachofthelinearmappingsoI-H-/ bH(x)a,-H(x) dx is continuous on each of the Banach spaces X(U; K), where K is any compact subset of U; and this follows from (13.13) because each of the functions bH is locally integrable. Let Ts be the pcurrent so defined. If we denote by &,.-,, ,oc(X)the vector space of locally integrable differential (n - p)-forms on X, then we have a linear mapping B H T ~of 8,,-p,loc(X)into 9:’)(X). From (13.14.4) it follows immediately that the kernel of this mapping is the subspace of negligible (n - p)-forms. Since the support of a Lebesgue measure on X is the whole of X, the restriction of the mapping B-Ts to the space &‘$?,,(X)of continuous differential (n - p)-forms is injective, so that such a form may be identified with a pcurrent of order 0. Under this identification, the notions of support are the same for the continuous (n - p)-form fl and the pcurrent T, with which we have identified it. For, by reducing as above to the case where X is an open subset U of R”,if xo E Supp(fl), then there is an index H such that bH(xO) # 0; we can then choose a 1 - H such that the integral
s
b,(x)a[-H(x) dt’ dt2 * ’ ’dtn
is #O, and such that Supp(a1-H) is contained in an arbitrarily small neighborhood V of xo ; defining a1-H’ to be 0 for H’ # H, we obtain a form a with support contained in V and such that B A a # 0, which proves the assertion.
I
In particular, for each locally integrable n-form u, the mapping f w / f u is a meusure T, on X, which is positive if and only if u(x) 2 0 almost everywhere (relative to the orientation of X) (13.15.3). (1 7.5.1 .I) Again, iff is any locally integrable complex function on X, then the mapping U H fu is an n-current T, on X, of order 0. If U is any open is the characteristicfunction of subset of X, the n-current Tq, on X, where qPu U, is called an open n-chain element on X, and linear combinations of open n-chain elements are called open n-chains on X. By abuse of notation, we shall often write U in place of T,, , and C Aj Ui in place of AjTPu,.
s
i
i
(17.5.2) We retain the notation and assumptions of (17.5.1). Let y be a continuous differential q-form, where q S p ; then the (n - p + q)-form B A y is locally integrable, and it follows immediately from the definitions that (17.5.2.1)
Tshy = TB A Y.
248
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
The left-hand side of this formula is meaningful if we suppose only that y is measurable and locally bounded, or, on the other hand, that fl is measurable and locally bounded and that y is locally integrable. Under these conditions we dkfine the right-hand side of (17.5.2.1) by this formula, and then we have (1 7.5.2.2)
T,, A
p = (-
1)4(n--p)T,
h
y.
Next, let z : X + X be an orientation-preserving diffeomorphism. Then the pcurrent n(T& is the linear form u ‘ w J X f l ~ ‘ z ( a ’ )on 9$‘)(X’). By (16.24.5.1) we have
and therefore (1 7.5.2.3)
11(Tg) = Ttn-i(B).
If fl is a locally integrable differential (m - p)-form on an oriented submanifold Y of X, of dimension m, then clearly we can form the imagej(T,) of T, under the canonical immersion j : Y + X, and j(T,) is a p-current. However, when m < n, this current is not in general expressible in the form T, for some locally integrable (n - p)-form y on X, even if j? is of class C”. For example, if p = 0, the support of the measure j(T,) is the submanifold Y, which is negligible with respect to any Lebesgue measure on X, so that a measure of the form T, with support Y is necessarily zero, whereasj(T,) # 0 in general (cf. (17.10.7)). Finally, let z : X + X be aproper mapping of class C‘ with r 2 1 (17.3.7), the manifold X’ being not necessarily orientable. Then for each locally integrable complex-valued functionfon X, n(Tf) is an n-current of order 0 on X‘ (which therefore vanishes if dim X‘ n), defined by the formula
-=
(17.5.2.4)
where CL’ is any continuous, compactly-supported differential n-form on X’. In particular, if we take f to be (px, the constant function equal to 1 at all points of X, then x(T,+,,) is called the n-chain element without boundary on X (cf. (17.15.5)) defined by the proper mapping z, and we write J n cd in place of
IX
%(a’). When n = 1, this is an integral along a particular type of
unending path ” (1 6.27). If X is a closed submanifold of X’ and z is the canonical injec“
5 CURRENTS ON AN ORIENTED MANIFOLD
tion, we shall sometimes write X in place of n(Tqx),and
IX
249
u' in place of
Jx %(a'); but it must be borne in mind that this number depends not only on the manifold X but also on its orientation, and changes sign when the orientation is reversed. (17.5.3) If we fix a C" differential n-form uo belonging to the orientation of X, then every differential n-form on X can be written uniquely asfuo, wheref is a complex-valued function on X. The formfu, is locally integrable if and only iffis locally integrable; in other words, the mappingf-fu, is a linear bijection of the space 2,0c(X)of locally integrable complex-valued functions on X onto the space &,, Ioc(X).We shall write Tf in place of Tfuoand identifv the function f with the corresponding distribution T, . (17.5.3.1) The choice of uo allows us to identifv n-currents and distributions (i.e., 0-currents), because gt+guo is an isomorphism of the FrCchet space 9(X; K) = .9,(X; K) onto 9,,(X; K), for each compact subset K of X; this follows immediately from (17.1.4). Hence every n-current is uniquely expressible as gu, -T(g), where T is a distribution. We shall denote this n-current by Tluowhen it is necessary to avoid ambiguity; in this notation, we have (T,>,uo= T,,, for a l l f e 2Ioc(X). (17.5.3.2) In future we shall make these identifications only when X is an open subset U of R",endowed with the canonical orientation and the canonical n-form uo = dtl A dt2A * * A dt" restricted to U. Then no risk of confusion arises except as regards the image of a current under a diffeomorphism x of U onto an open subset U' of R".If T is a distribution on U and n(T) its image on U', then the image x(Tlvo)is given by
-
(17.5.3.3)
where J(x-l) is the Jucobiun of the inverse diffeomorphism n-l, by virtue of (1 6.20.9.4). In particular, for each functionfE 210c(U)we have (17.5.3.4)
4Tf)
= T,,
Y
where f' is the mapping x' ~ f ( n - ' ( x ' ) ) J ( n - ' ) ( x ' ) . (17.5.3.5) It is clear that the kernel of the linear mappingfHT, of 210c(U) into W(U) is formed by the negligible complex-valued functions on U (relative to Lebesgue measure). Passing to the quotient, it follows that the space Lloc(U) of classes of locally integrable functions on U (13.13.4) may be
250
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
canonically identified with a subspace of the space of distributions on U. A fortiori, we may identify the spaces L'(U), L2(U),and L"(U) with subspaces See (17.8) for questions relating to the topologies of these spaces. of 9'0. (17.5.4) Questions of a local nature concerning currents reduce to the case where the manifold X under consideration is an open subset U of R",and it is this case that we shall be mainly concerned with in the remainder of this chapter. Each of the spaces SF)(U) (resp. Sp(U))is then an S(r)(U)-module
(resp. an b(U)-module) which is free of rank basis for the p-forms: (H = d<" A
,ti'
(where H is the set of integers il < i2 < *
A
*
(3.
*
A
Relative to the canonical
drip
- - < ip), each pcurrent is of the form
where the TH are distributions on U. Hence the study of currents on an open subset U of R" reduces to the study of distributions on U, and it is therefore the latter that we shall mainly consider. (17.5.5) The only concrete examples of distributions that we have given so far have been distributionsof order 0 (i.e., measures). We shall now show that there exist on U c R" distributions of all orders. For this we observe that, for each compact K c U and each multi-index v, the mapping f~+D"f is a continuous linear mapping of 9(U; K) into itself, by virtue of (17.1.3) and the fact that Supp(Dvf)c Supp(f). Consequently, if T E 9'(U) is any distribution on 9 0 is a distribution, which we denote on U, the linear form f HT(D'' by (- 1)I"ID"T; the distribution DVTis called the deriuative of multi-index v of the distribution T. For each functionf E 9(U) we have (17.5.5.1)
< D T , f ) = (- l)"'(T, D'f),
from which it follows that
for any two multi-indices v, v'. We shall also write 8"T/8x;l 8x7 *
8x2 in place of D'T. Clearly we have
5 CURRENTS ON AN ORIENTED MANIFOLD
251
In the particular case of a distribution of the form Tg, where g E B(')(U) and Iv I 5 r (with r 2 l), we have D'T, = TDvg
(17.5.5.4)
(which justifies the.factor (-l)lvl introduced in the delinition of DT). By virtue of (17.5.5.2), it is enough to establish the formula (17.5.5.4) for I v I = 1. By definition, iff€ 9(U),we have
I...
...&i-'@i+
dt"
g(x)Dif(x) dt' JR
(the integrals being extended to the whole of R" by extending the function gDif by zero outside Supp(f)); but because Difis compactly supported, we obtain by integrating by parts (8.7.5) n
ll
g(x)Dif(x) dt' = -
JR
f(x)Dig(x) dt'7 JR
from which it immediately follows that (Tg, D i f ) = -(TDlg, f). We remark also that for each function g E &(')(U)and each distribution T E 9'(U) we have (17.5.5.5)
Di(g * T) = (Di g) * T + g * Di T ;
this follows immediately from the definition (17.5.5.1) and the rule for differentiating a product. From the definition of the topology of B(U) (resp. &'(')(U)) by seminorms (17.1 .I), it follows that if T is a distribution of order sq, then D"T is of order at most q + r, where r = sup v i . We shall see that it can be exactly of this order.
1S i S n
Examples (17.5.6) The derivatives of the Dirac measure E, at a point a = (ai)E R" are given by (17.5.6.1)
(D"E,,f> = ( - 1)'"1 D"f(a)
252
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
for f E 9(R"). This distribution is of order exactly r = sup vi . To see this, let j be an index such that vj
S(t1, t z
3 * * * 9
tn)
lsign
and let f E 9(Rn) be a function of the form
= r,
= g(tj)
n (ti - ai)"b(t1,
i# j
* * * 3
tn),
where u E 9CR") is equal to 1 throughout some neighborhood of a, and g E &(R). Now replace g by a sequence of functions gk E &(R) such that the functions DSgkare uniformly bounded on a compact neighborhood of aj for s < r, and such that D'gk(aj) tends to 00 as k --f co. In other words, we may assume that n = 1 and, by translation, that aj = 0. If ho E 9 ( R ) is the function defined in (16.4.1.4), such that ho(0) # 0, then we may take
+
(cf. (8.14.2)). We have then
for s < r ; these functions are uniformly bounded for Ix I 5 1 by the number
independent of k, whereas Drgk(0)= kho(0). This also enables us to give an example of a distribution of infinite order m
on R,namely the linear form f H
Dkf(k) on 9(R).
k=O
(17.5.7) The Heavisidefunction is by definition the characteristic function of the interval [0, co [ of R and is denoted by Y or Y,. For f E 9(R) we have
+
or, identifying Y and Ty , (17.5.7.1)
DY
= c0,
the Dirac measure at the origin. Here the order of T, is equal to that of its derivative. (17.5.8) The function g which is equal to log x for x > 0 and is zero for 4 0 is locally integrable on R (because in a neighborhood of 0 we have
x
5 CURRENTS ON AN ORIENTED MANIFOLD
253
IIogx( < I / x p f o r O < p < l ) . Wehave
(DT, , f ) =
- jomf’(x)log x dx
for f E 9(R). If a > 0, we obtain, on integrating by parts,
Sincef is continuously differentiable, we may write f(x) -f ( 0 ) = X h ( 4 wheref, is continuous on R. Since ct log a+ 0 as a 40, we obtain therefore (17.5.8.1)
The restriction of DT, to the open interval U = 10, + a[coincideswithT,, and therefore may be identified with the function l/x on U. It can be shown that the order of DT, is 1 (Problem 2), whereas its restriction to U is of order 0. (17.5.9) Consider the function g(x) = x sin(l/x). Here the function Dg, defined for x # 0, is not integrable on any neighborhood of the origin, but the limits of g’(x) ak and Jam g’(x) dx as a + 0 exist, and are denoted by
j:
I-“ -m
lom
g’(x) dx and g’(x) dx (they are improper integrals). The same argu00 ment as in (17.5.8) this time gives
(improper integrals), but it can be shown that the distribution DT, is again of order 1 (Problem 4). (17.5.10) Let U be an open subset of R”,and let F E B(U) be a function satisfying the conditions of (1 6.24.11). With the notation used there, consider the (positive) measure on the oriented manifold E, defined by the positive (n - 1)-form 0, (16.24.2), and let p, be the image of this measure under the canonical immersion E, -+ U (1 3.1.7). The elementary version of Stokes’ formula (16.24.11.1) can then be interpreted as giving the derivative of the characteristic function (identified with a distribution on U) :
254
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(17.5.11) In future, whenever we identify a function f E .YlOc(lJ)with the distribution Tf, we shall interpret the expressions Dj f (1 $ j 5 n) as meaning the distributions DjTf = aTf/axj,except when f is continuously diflerentiable, in which case, as we have seen, the distribution DjTf can be identified with the continuousfunction DjJ: Whenever we say that a functionf E U,,(lJ) has a derivative which ‘‘is a function” D j f e U,(U), we shall mean that there exists a locally integrable function g such that DjTf = T, ,and we shall denote this function by Djf; but, unless the contrary is explicitly stated, we shall never attempt to define this function g in terms of differencesf ( x + h) -f (x). In any case, the function g is defined only to within a negligiblefunction, and it is only when there exists a continuous function (necessarily unique) in the class of g that it is appropriate to choose this function as a privileged representative of the class. (17.5.12) Consider in particular the case n = 1, and letf be a locally integrable complex-valued function on R. For each compact interval [a, b] c R, the f dA (which by abuse of notation we shall write as f ( t ) dt integral (1 3.9.1 6))has a meaning. Put
I[#,*,
1;
if x (17.5.12.1)
2 0,
F(x) = if x < 0.
-lxoj(t)dt
Then the function F is continuous on R, by the dominated convergence , f q c a ,x,), theorem (1 3.8.4) applied to the sequence of functions f q [ x , x k(resp. where (xk)is a decreasing (resp. increasing) sequence with x as limit. We shall show that (17.5.1 2.2)
DTF = Tf
(or DF =f,with the conventions described above). To prove this, we must show that, for all u E O(R), +m
(17.5.12.3)
F(t)u’(t)d t = - J - m f ( t ) ~ ( r )dt.
Since u has compact support, contained in some interval [a, b], the left-hand side of (17.5.12.3) does not change when F(t) is replaced by F(t) - F(a), and therefore is equal to
5 CURRENTS ON AN ORIENTED MANIFOLD
(17.5.12.4)
l c u'(t) dt
255
f(s) ds.
However, since the function f@ u' is locally integrable with respect to the Lebesgue measure A @ 1on R2 (13.21.16), the same is true of its product by the characteristic function of the closed set {(s, t ) : s 5 t } . Applying the Lebesgue-Fubini theorem (13.21.7) to this product, we see that (1 7.5.12.4) is equal to Jabj'(s) ds j s b d ( t )dt = - Jabf(s)u(s) ds,
and (17.5.12.3) now follows. We may therefore say that the distribution T, is a primitive of T,. Every other primitive is of the form TF+,, where c is a constant. For in order that a function o ~ 9 ( R should ) have a primitive belonging to Q(R), it is necessary and sufficient that u ( t ) dt = 0 ; the functions satisfying this relation form a hyperplane H in the vector space 9 ( R ) (A.4.15), and by definition (17.5.5) the distributions T such that DT = 0 are those which vanish on H; hence (A.4.15) for such a distribution T, there exists a constant c such that T(u) = c / +" u(t) dt for all v E 9 ( R ) , in other -W wordsT = T,.
1'" -a)
Remark (17.5.13) Suppose that, for two indicesj, k between 1 and n, the continuous functionfonu c R" is such that the four derivatives Djf, Dkf, Dj(Dkf), and Dk(Djf) (in the sense of (8.9)) exist and are continuous on U. Then we have Dj(Dkf) = Dk(Djf), because the relation Dj(DkT) = Dk(DjT) is true for all distributions T, and in particular for T, ; the remarks of (17.5.11) therefore apply. (17.5.14) For each distribution T E 9'(U), there exists a largest open subset V of U such that the restriction of T to V is a function belonging to 8(V). The complement of V in U is called the singular support of T and is denoted by Supp sing(T). In examples (17.5.6)-(17.5.9), the singular support of the distributions considered consists of a single point. Clearly we have
SUPPsing(T) 3 Supp sing(D'T), for g E b(U).
-
Supp sing(T) 3 Supp sing(g T)
256
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
PROBLEMS
1. Let K be a compact interval in R such that 0 E 8, and let 2 be the hyperplane in Om;K) consisting of the functions which vanish at 0.
For each f~ dE“ let 3be the function t H f ( t ) / t . Then the mapping f ~ is anf isomorphism of the subspace 2 onto Om;K). (Argue as in (16.8.9.1)J (b) Deduce from (a) that for each distribution T on R there exists a distribution S such that lR* S = T, where lRis the identity mapping t H t. If T is of order s r , then S is of order s r 1. (a)
+
2.
Let p be apositive measure on the open interval 10, cot in R.Show that p is the restriction of a distribution on R if and only if there exists an integer k 2 0 and a constant c > 0 such that p ( [ ~1J) , 5 c . E - ~for E €10, 1 [, and that if this condition is satisfied there exists a distribution of order 6 k 1 on R whose restriction to 10, co[ is equal to p. (To show that the condition is necessary, apply (17.3.1 .l) and consider a function 0 belonging to 9ck)(R)which is equal to 0 for t 5 48 and to 1 for t E [ E , 11. To show that the condition is sufficient, show that the measure with base p and density tk+’ on 10, m[ is bounded, and apply the result of Problem 1.) If h is the least integer k satisfying the above condition, no distribution which extends p is of order
+
+
fz
+
3. Let K c R be a compact interval, not consisting of a single point.
I
(a) Show that the mappingfw Dfis an isomorphism of 9(R; K) onto the subspace ;cP of
Om;K)consisting of the functions f such that
+ f ( t ) dt = 0. --OD
(b) Deduce from (a) that for each distribution T on R there exists a distribution S such that DS = T (i.e., a “primitive” of T). (c) For each positive measure p on R, show that if 8 is the increasing function, continuous on the right, which corresponds to p (and which is defined only up to a constant; cf. Section 13.18, Problem 6), then DTe = p. 4.
Letfbe a locally integrable function with respect to Lebesgue measure on 10, a], where a > 0. Suppose that the integral / i f ( t ) dt is convergent, i.e., that the limit of / : f ( t ) dt
exists and is finite as
E
+O.
Show that, for each function g E 9(R), the integral
Jif ( t ) g ( t )dt converges, and that T
:g~
Jb”
f ( t ) g ( t )dt is a distribution of order
5 1 on
R.This distribution T is of order 0 if and only iff is integrable over 10,a]. (If (t.) is a decreasing sequence with limit 0, apply the formulas (1 3.20.1) and (1 3.20.3) on each of the intervalsJ., t m - , [ . )Iffis of class Cm,then the singular support of T consists only of the point 0. 5. If a distribution TEO’(R) is such that all the derivatives DkT are measures, then T = T, for somefe 801).(Remark that if Dk+*Tis a measure, then D 1 = T,, where g is a continuous function, by using Problem 3(c).)
5 CURRENTS ON AN ORIENTED MANIFOLD
257
6. Let P : X -+ X , P’ :X ax” be two proper mappings of class Cm. Then for each current T on X we have P’(P(T)) = (P‘ 0 P)(T).
7. Let P :X -+ X be a proper mapping of class C “. If T is a pcurrent on X and a’is a C “ q-form on X’,show that P(T A %(a’)) = T(T)A a‘. 8. (a) Let P :X -+X be a C “-mapping, and let T’ be ap-current on X with the following property; if A = Supp(T’), every point x E P - ~ ( A )has an open neighborhood V, in X such that P IV, is a diffeomorphism onto an open subset of X . Let p , be the inverse of this diffeomorphism, and consider thep-current p,(T&,)) on V, for each x E ?r-’(A). Show that thesep-currents are the restrictions of ap-current T on X with support contained in P-I(A). Thisp-current T is called the inverse image of T‘ by P and is denoted by %r(T’). (b) If X,X’are oriented manifolds of dimension n and if /?’ is a locally integrable differential (n - p)-form on X , whose support A satisfies the condition in (a), show that if we put p = %r@‘), then ‘P(T,.) coincides with TBin a neighborhood of each point of P - ~ ( A )at which P preserves the orientation, and with -T, in a neighborhood of each point of P-~(A’) at which P reverses the orientation. (c) Take X = X = R,and P to be the mapping t H t z - a’, where a > 0. Then the inverse image by P of the Dirac measure eo is E. E-,, . The inverse image by P of the Dirac l-current ceca) (in the notation of (18.1)) is the 1-current (1/2u)(~~(., -E-~(-.,).
+
9. Let X,Y be two oriented pure differential manifolds, of dimensions, respectively, n and m ; let P :X -+ Y be a submersion and a a C differential (n - m &form on X (where k 5 m) with compact support. Show that the image P(TJ of the (m - k)current T,. is equal to Tab, where a b is the integral of a along the fibers of P (Section 16.24, Problem 11). (Reduce to the case of (16.7.4).)
+
For eachp-current S on Y (withp m),we define the inverse image of S by P to be the (n - m +¤t %(S) such that <‘.rr(S), a> = <S, ab)
for every compactly supported C “ differential (n - m +&form a on X. If S = T,. , where /?’ is a differential (m - p)-form on Y, then ‘P(T~,)= T4, where p =‘P(/?’). If y’ is any C“ differentialq-form on Y, then ‘P(S A y’) =%(S) A ‘~(y’). In particular, for each point y E Y,‘P(E$ is the closed (n - m)-chain element rr-’(y), endowed with the orientation induced by P from the orientations of X and Y. 10. Let G be a Lie group of dimension n, and let uo be a left-invariant C “ differential
n-form on G (19.16.4), so that the linear form f~ (1 6.24.2).
f
fuo is a Haar measure
Po on G
(a) The image of Po under the diagonal mapping x H ( x , x ) of G into G x G is called the (left) trace measure on G x G corresponding to U O , and is denoted by tr. For
each function f E X ( G x G) we have therefore tr(f)
=s f ( x , x )
dP0(x).
(b) If G = R,show that the measure tr on R2is equal to D, T,, , where U is the set t2)E R2 such that t12 t 2 . of points (c) Let P : G x G -+ G be the mapping (x, y) H x y - ’ , which is a submersion. For
(t,,
258
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
each z E G , r - I ( z ) is the left coset (z, e)D, where D is the diagonal subgroup of G x G; so we may identify G with (G x G)/D and r with the canonical mapping
G x G +(G x G)/D. Let u = 'prl(uo) A'pr2(uO),which is the left-invariant b-form on G X G corresponding to the Haar measure P = Po @Po. For each differential b-formfu on G x G, where f~ 9 ( G x G), show that (fu)b =fbuo, wherefb is the function on G (identilied with
s,
(G x G)/D) defined in Section 14.4, Problem 2, such that fb(z) = f(zw,w) dPo(w). Deduce that, for each ncurrent T on G, we have <'TO, fu> = . If the Dirac measure E, on G is identified with an n-current, then %r(ee) is identilied with the measure tr defined in (a).
6. REAL DISTRIBUTIONS. POSITIVE DISTRIBUTIONS
(17.6.1)
Let X be a differential manifold. For each p the vector bundle
P
AT(X)* may be identified with a subbundle of the real vector bundle
( A T(X)*)o,,
P
P
which is the direct sum A T(X)* 63 i A T(X)* (16.18.5). It follows that C?~,<X) (resp. C%'r)(X)) is equal to B,,,(X) iC%'p,R(X) (resp. &'gb(X) i&'gL(X)),where C%'p, ,(X) (resp. &z'!(X)) is the space of real differential p-forms of class C" (resp. Cr) on X. A pcurrent T on X is then ,(X) is real-valued, and a real pcurrent said to be real if its restriction to dP, is often identified with its restriction to C%'p,R(X). As in (13.2), for each current T the conjugate current T is defined; the real currents 9 T = +(T + T) and 9 T = (1/2i)(T - T) are called, respectively, the real and imaginary parts of T.
+
+
(17.6.2) Every positive distribution T on a diflerential manifold X (i.e., every distribution T such that T(f)2 0 for all f 2 0 in 9(X)) is a positive memure on X.
We may assume that X is an open subset U of R", by virtue of (13.1.9). Let K be a compact subset of U, and let h : U + [0, I ] be a C"-mapping with compact support and equal to 1 on K. For each real-valued function f E 9 ( U ; K), we have
f s Ilf II h 11f 11 T(h). Hence, if now
- I l f IIh 5
and therefore - 11f 11 T(h) S Tcf) f=fi
+if2€9(U;K)
fi real-valued, we have IT(f)l 5 2T(h) 11f 11. This shows that T is a with fi, distribution of order 0, hence a measure. Moreover, with the same notation as
7 DISTRIBUTIONS WITH COMPACT SUPPORT
259
in the proof of (17.1.2), for each compact neighborhood K' of K contained in U, and for each function f E s ( U ; K), the functions gk * f belong to 9 ( U ; K') and converge uniformly to f on K' as k + 00, hence T(f) = limT(g, * f ) ; but iff 20, then gk * f 2 0, hence T(g, *f)2 0 by hypothk+ m
esis, and so finally T(f ) 2 0. Hence T is a positive measure.
7. DlSTRlB UTI0NS WITH COMPACT SUPPORT. P OINT-DISTRI BUTlO N S
(17.7.1) A distribution whose support is compact is offinite order. To prove this we need only consider the case of a distribution T on an open subset U of R". Let A be the (compact) support of T and let V be a compact neighborhood of A contained in U. If h : U + [0,1] is a C"-mapping which is equal to 1 on V and whose support B is compact, then we have T(f) = T(hf) for all f E 9(U). Define the seminorms (17.1.1) on &(U) by taking a fundamental sequence (K,,,) of compact subsets of U, and choose m large enough so that B = Supp(h) c K,,,. Then, for each functionf E 9 ( U ) , the function hf belongs to 9 ( U ; K,,,) and therefore, by virtue of (12.14.11) and the definition of a distribution, there exists an integer r and a constant a > 0 such that, for all
f E WJ),
where c is a constant independent off (17.1.4). This proves that T is of order S r . Also, for each function f~ @(U), we can define T(f) to be equal to T(hf), because hf E &"'(U). The inequality (17.7.1.1) shows that the linear form T thus extended is continuous on &(')(U). In the same way we may show that, if a p-current T has compact support, then its value can be defined for all p-forms u E &JX). If T has order s r - 1, this allows us to define the image n(T) of T by an arbitrary mapping n of class C' (where r 2 1 ) from X to X'. For T('n(a'))is defined for all differentialp-forms a' E c?F-')(X'); moreover, for each compact subset K' of X', the linear mapping a'H h . 'Tc(a') of 9r-')(X'; K') into 9 r - ' ) ( X ; B) is continuous, by virtue of (17.2), the local expression of %(a'), and (17.1.4). Hence the assertion follows. The relation Supp(n(T)) c n(Supp(T)) is still valid. We denote by S'(U) the subspace of 9'(U) consisting of the distributions with compact support. (17.7.2) Let T E &'(U) be a distribution of order r with compact support A c R".Then T(f) = 0for allfunctions f E'&(U)whose derivatives of order s r vanish at all points of A.
260
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
For each E > 0, let A, be the compact subset of R" consisting of the points whose distance from A (relative to the norm llxll = sup Iti 1 on R") is 5 s . Then there exists e0 > 0 such that A, c U for all E 5 E,, (3.18.2). Since the derivatives of order r + 1 off are bounded in absolute value on A,, by a number independent of E , it follows from the hypothesis on f and from Taylor's formula (8.14.3) that, for e S +e0 and any x E A,, we have If(x) I 5 b * E'", where b is a constant independent of E ; for there exists a vector t with norm 5 2 8 such that x - t E A, and the derivatives of order -
for all IvI 5 r and all E S +cO. Next, consider the functions gkintroduced in (17.1.2), and put u&= gk
* (PAzlr
for each integer k 2 3.5;'. Since the support of gk is contained in the ball with center 0 and radius Ilk, it follows immediately (14.5.4) that uk(x) = 1 for x E AlIkand that Supp(u,) c A,, . Moreover, by (13.8.6), for all x E R" and all v we have DVuk(x) = klvI+" lA2,Dvg(k(x - t ) ) dA(t), and therefore IDvUk(X)( klvi+"
ID'g(kt)l dA(t) = k"'
in other words
6.
ID'g(t)l dA(t);
(17.7.2.2)
for all v such that 1 v I g r, where c is a constant independent of k. This being so, since uk(x) = 1 in some neighborhood of A, we have T(f ) = T(ukf)for all k, and by hypothesis we have
for some constant a independent of k. However, the definition (17.1.1) and Leibniz' rule (8.13.2) show that there exists a constant M such that Pr, nt(l(k
f)5 M '
sup
IPl+lrl6r.
XEA3/k
IDPUk(x)I I D"f(x) I '
7 DISTRIBUTIONS WITH COMPACT SUPPORT
261
for all k. Using these inequalities, (17.7.2.1), and (17.7.2.2), we obtain Pr, m(Ukf)
5 c/k,
where C is a constant. Hence, by (17.7.2.3), we have
Q.E.D.
T ( f ) = lim T ( u k f )= 0. k+
OD
We remark that the conditionf(x) = 0 for all x E A does not ensure that T ( f ) = 0, as is shown by the example in which n = 1,f ( x ) = x and T = D E ~ . From (17.7.2) we obtain the following corollary: (17.7.3) Every distribution T E S'(U) whose support consists of a single point
a is a linear combination of a$nite number of derivatives D'E, of the Dirac measure at the point a. Let r be the order of T. For each f E b(U),we can write
c
1
f(x) = I v I s r V. 7 D'f(a>(x- a)Y + g(x), where all the derivatives of g of order I r vanish at the point a. By virtue of (17.7.2), we have T(g) = 0 and therefore, if cv is the value of T on the function (x - a)'/v!, we obtain
T d f )=
c cvD'f(a).
I4 S r
PROBLEMS
srn.
For each open 1. Let T E $I(R") be a distribution with compact support K, of order of a finite neighborhood U of K, show that T is a sum of derivatives of order number of measures with supports contained in U. (Let N be the number of multiindices Y such that 1 v 1 m. Also let V be a relatively compact open neighborhood of K such that V c U. Show that the mapping r :f-(Dvf)lv15,,,is an isomorphism of the 9 )onto a closed subspace F of the Banach space (V(V))N, and Banach space W")(R"; hence that T 0 7r-I is a continuous linear form on F. Use the Hahn-Banach theorem (Section 12.15, Problem 4) to extend this linear form to a continuous linear form on
srn
(wv)N.)
2.
(a) For each function f~ am), show that the limit
262
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
exists and is finite. T is a distribution on R of order 2, whose support A consists of the points l/n (n 2 1) and 0. Iff1 A = 0, then T(f)= 0. (b) Show that T is not expressible as a finite sum of derivatives of order 5 2 of measures with support contained in A. (c) For each integer k 2 1, let ft E 9(R) be a function which takes its values in the interval [0, k-1'2], is equal to k for t 2 Ilk, and is zero for t l/(k 1). All the derivatives offt are zero at the points of A, and the sequence (fk) converges uniformly to 0; but the sequence T(ft) tends to co. Hence, in order that T(ft) should tend to 0, it is not sufficient that (&) and each of the sequences (D'ft) should tend to 0 uniformly on A.
+
+
8. T H E WEAK T O P O L O G Y ON SPACES O F DISTRIBUTIONS
The space 9b(X) of p-currents on a differential manifold X of dimension n is a vector space of linear forms on 9,(X), and therefore can naturally be equipped with the weak topology (12.15.2) defined by the seminorms (17.8.1)
T H Il,
where a runs through 9,(X). Whenever we use topological notions in the spaces 9b(X), it is always the weak topology that is meant, unless the contrary is expressly stated. Example (17.8.1 .I) Let (gk)be a regularizing sequence of functions belonging to 9(R"), and identify the g k with distributions. Then the sequence (gk)converges weakly t o the Dirac measure go a t the origin.
It follows directly from the definition of the weak topology that the linear mappings T H T A ~Twn(T), , and T H ~ ( T )defined in (17.3.5) and (17.3.7) are continuous. Likewise, it follows from the definition (17.5.5.1) that the differentiations T H D'T on 9'(U) (where U is open in R")are continuous. Remark
The derivatives D,T of a distribution can also be defined as limits in the space 9'(U): if ( e j ) is the canonical basis of R",then we have (17.8.2)
(17.8.2.1)
Di(T) = lim
r-0. i # 0
(y(tei)T - T)/t.
8 THE WEAK TOPOLOGY ON SPACES OF DISTRIBUTIONS
For iff
E 9(U), the
263
function ( 7 ( t e i ) f - f ) / t is the mapping x I+
( f (x - te3 - f (x))lt,
and Taylor’s formula applied to f and its derivatives shows that, as t+ 0, the function (7(tei)f -f ) / t tends to -Di f in the space 9(U;K), where K is any compact neighborhood of the support off.
Let Z be a metric space, A a subset of Z, and let ZH T, be a mapping of A into 9b(X), and zo a point of A.Suppose that, for each p-form a E gP(X), thefunction z w tends to a limit as z tends to zo whilst remaining in A. Then the linear form at+ lim is a pcurrent, equal to lim T,
(17.8.3)
z
-
20, Z E
A
z
4
zo, z E A
in 9;(X). In particular, fi a sequence (Tk) of p-currents on X is such that, for the sequence converges in C, then the sequence each p-form a E gP(X), (Tk) has a limit in 9i(X). This is an immediate consequence of the Banach-Steinhaus theorem (12.16.5) applied to each of the FrCchet spaces 9 J X ; K). Likewise, if Z is an open interval in R (resp. an open set in C), and if ZHT, is weakly differentiable (resp. weakly analytic) in Z, then the weak derivative Ti is a p-current (12.1 6.6). Finally, suppose that Z is locally compact, and let p be a positive measure on Z. If for each form a E Edp(X) the function ZH is p-integrable, then by applying (13.10.4) to each of the FrCchet spaces g P ( X ;K)it follows that the mapping a-j dp(z) is a p-current, which is denoted by Tz dp(z) and is called the weak integral of ZH T, with respect to p. (17.8.4) For each integer r, the topology induced on 9$)(X) by the weak topology of 9;(X) is coarser than the weak topology of 9$)(X) considered as a space of linear forms on 9 g ) ( X ) , because O,(X) c gg)(X). I n particular, on the space M(X) = 9”O’(X) of meusures on X, the topology induced by the weak topology of 9’(X) is coarser than the vague topology. More particularly, since the spaces L’(U), L*(U), and Lm(U)(where U is an open set in R”)are identified (algebraically) with subspaces of 9’(U), and since the norm topologies on these spaces are in all cases finer than the vague topology, they are a fortiori finer than the topologies induced by the weak topology of 9‘(U). In fact, the topologies induced by the weak topology of 9’(U) on these spaces are strictly coarser than the other topologies mentioned above. For example, if (gk) is the “regularizing sequence” of functions introduced in (17.1.2), which converges weakly to E,, ,then it follows from the continuity of
264
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
differentiation relative to the weak topology that each of the sequences (D'g,) converges weakly to D"tO,which is not a measure for v # 0. We shall see later (17.12.3) that in fact every distribution on U is the weak limit of a sequence of functions belonging to 9(U). (17.8.5)
Let H be a bounded subset of 9i(X) (12.15).
(i) The weak closure of H in 9i(X) is compact and metrizable relative to the weak topology. (ii) IfU is open in R" and H a bounded subset of 9'(LJ), then (with the notation of (17.1)) for each integer m there exists an integer r and a number cr,, such that I Tcf) I 5 cr, p,, ,cf) for all f E 9 ( U ; K,) and all T E H . Assertion (ii) follows from the Banach-Steinhaus theorem (12.16.4) applied to the Frkchet spaces 9 J X ; K,). Assertion (i) then follows by using (17.2.2), by an argument analogous to that of (13.4.2), which we shall not repeat here. In particular, the restrictions of the distributions T E H to a relatively compact open set have bounded orders. (17.8.6) I f a sequence (Tk) of currents belonging to 9i(X) converges weakly to T , thenfor each compact subset K of X and each boundedsubset B of 9 J X ; K ) (17.1), the sequence () converges to uniformly on B. We reduce to the case where X is an open subset of R", and then the proposition is an immediate consequence of (17.8.5(ii)). be a sequence of p-currents on X , and (ak) a sequence of (17.8.7) Let (Tk) q-forms belonging to b , ( X ) , where q 5 p . Then the sequence (TkA uk)converges to zero in 9;-&X) in each of the following two cases: (1) The sequence (ak) is bounded in &,(X) (17.1) and the sequence (Tk) converges to 0 in 9i(X). (2) The sequence (ak)converges to 0 in the FrPchet space b , ( X ) and the sequence (T,) is bounded in 9i(X).
We have to show that, for each ( p - 9)-form /3 E the sequence () tends to 0. The question reduces immediately to the case where X is an open subset U of R",and p = q = 0. Let K be the support of p. In case (l), the sequence (a& is bounded in 9(U;K) by virtue of (17.1.4), hence the result follows from (17.8.6). In case (2), the sequence (upp) converges to 0 in 9 ( U ; K) by virtue of (17.1.4), and the result follows from (17.8.5(ii)).
8 THE WEAK TOPOLOGY ON SPACES OF DISTRIBUTIONS
265
Example (17.8.8) Let (A,,) be an increasing sequence of strictly positive real numbers, and let (c,,) be a sequence of complex numbers; suppose that there exist two numbers p, 0 > 0 such that A,, 2 np and I c,, I S nu for all n 2 1. Then the series m
(17.8.8.1)
C c,,eianx n=O
is convergent in 9'(R). There exists an integer k > 0 such that I c,,/Atl 5 l/n2, namely any integer k such that kp - 0 2 2. Then it is enough to show that the series (1 7.8.8.2)
converges in 9'(R), because if T is its sum, then the series (17.8.8.1) will converge to DkT,by virtue of the continuity of differentiation; but the series of continuous functions (17.8.8.2) is normally convergent in R, hence converges also in 9'(R) if its terms are regarded as distributions (17.8.4).
PROBLEMS
Let z, be a nonzero tangent n-vector at a point x E R",and let (V,) be a fundamental system of bounded open neighborhoods of x in R".Show that the sequence of n-currents (open n-chain elements (17.5.1.1)) (&(V,)-'VJ tends to the limit cz,, where c - l = , uo being the canonical n-form on R". Let (V,) be a fundamental system of bounded open neighborhoods of the origin in R"-l. Let uk be the locally integrable differential (n - 1)-form on R"= R x R"-' which is equal to &-l(Vk) -I d p A ... Ad@ on R x V, and is 0 elsewhere. Show that, as k + 00, the sequence of 1-currents T,, (17.5.1) tends to the 1-chain element without boundary R x {0}(17.5.2), R being canonically oriented.
+
Let T be a distribution on R.For each h # 0 in R, put A,, T = y(h)T - T, and A n = Ah(A;- 'T) for all integers p > 1. (a) Show that as h+O the distribution (I/hp)At;T tends to DPT. (b) Let f be a continuous function on R. The function f is said to be conipletely monotone if, for each integer p 2 1, A p f ( x ;h, h, . . ., h) has the same sign as hp for all h # 0 (Section 8.12, Problem 4). Show that f is then analytic. (Use (a), Problem 5 of Section 17.5, and Problem 7(c) of Section 9.9.) (S. Bernstein's theorem.)
266
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(c) Suppose that, for some integer p 2 1, Apf(x; h, h, ...,h ) / P +0 as h +0, for all x E R.Show that there exists a dense open subset U of R such that, on each connected component of U, f is equal to a polynomial of degree s p - 1. (Use(a), and observe that if a sequence (fJ of continuous functions converges pointwise to 0, then sup (1 f.I ) I
is finite at each point; then apply (12.16.2).) If also there exists a Lebesgue-integrable function g 2 0 such that 1 Apf(x; h, h, ..., h)/hp1 6 g(x) for all h # 0, then f is a polynomial of degree at most p - 1. Show that a distribution on R ' which is invariant under all translations of R" (i.e., is equal to its image under each translation) is a constant function (or, more accurately, a distribution T,, where c is a constant function on R").(Use(17.8.2.1)J What are the translation-invariant n-currents on R"? (b) A l-current on lo, co[ which is invariant under all homotheties hA:x-hr (where > 0) is of the form Tcu0,where c is a constant. A distribution on 10, co[ invariant under all hAis of the form T,, wheref(x) =cx-I, c being a constant. (Usethe isomorphism XH log x of lo, co [ onto R,and (a) above.)
4. (a)
+
+
+
5. (a) Letfbe a holomorphic function on an open set U c C, and let zo E U. Let S c U be a closed annulus with center zo , defined by rl 5 Iz - zo1 5 rz . Also let u be a Cm-
function on R, everywhere LO, with support contained in [rl,rzl, and such that
fu(t) dt
= 1. Show that
+
where r = Iz - zo I = ((x - XO)' 0,- YO)^)"^. (b) Let U be an open disk in C with center xo E R,let U+be the intersection of U with the half-plane fz > 0, and let f be a holomorphic function on U+,regarded as a distribution belonging to P ( U + ) . Show that f is the restriction to U+ of a distribution belonging to 9'(U) if and only if, for each compact interval K C R n U with center xo,there exists an integer k > 0 and an interval J = 10, c[ in R,such that SUP XCK.)~J
lY'f(x+iy)l <+a.
(To show that the condition is sufficient, consider an iterated primitive of f in U +. To show that it is necessary, observe that the hypothesis implies that, if J is any interval
-
lo,c[ in R such that K x J c U+u R, there exists a constant A > 0 and a multi-index a
(al,az)such that
+
for all u E P(U). Let zo = xo iyo E U+,and let S be a closed annulus with center zo contained in U+ . For each z = x iy E U+ such that the line passing through z and zo meets the real axis at a point t E K, let S . be the annulus with center z which is the image of S under the homothety with center t and ratio y/yo. Then by (a) abovewe have
+
9 EXAMPLE: FINITE PARTS OF DIVERGENT INTEGRALS
267
where
Then use the inequality (l).) (c) Show that the extension property in (b) is equivalent to the following: in every open interval I c U n R with center x o , the functions X H ~ ( X iy) converge to a distribution in o'(1) as y tends to 0 through positive values. (For the necessity of the condition, use Section 17.12, Problem 6.) (d) Show that, for each compact interval K in R and each function f e &(R), there exists a sequence (f.)of polynomials such that, for each integer p 2 0, the restrictions to K of the functions DpLconverge uniformly on K to Dpf(cf. (14.11.3)). If T E B'(R)
+
s
has support contained in K,show that the function u defined by u(z) = (x - z) dT(x) for z E C - K is analytic on this open set and that for each polynomial f, we have
where y is a suitably chosen circuit in C - K. Conclude that T is the limit in &(R) of a family of functions of the form X H F(x iy), where F is holomorphic in the half-plane 4 z > 0, and y tends to 0 through positive values.
+
9. EXAMPLE: FINITE PARTS O F DIVERGENT INTEGRALS
(1 7.9.1) Let X be an oriented pure differential manifold of dimension n, and let F be a redvalued continuous function on X. Suppose that the open set Uo= (x E X : F(x) > 0} is not empty and that the frontier P of U, (where F(x) = 0) is negligible with respect to Lebesgue measure on X. The function F-'cpuo, which is equal to 0 in X - Uo and coincides with F-' on Uo,is not in general locally integrable in a neighborhood of P, because F-' is not in general bounded in such a neighborhood. If uo is a differential n-form on X belonging to the orientation of X, then the mapping f~ ~uoF-'fvois a measure on X - P, zero on X (P u Uo),which in general cannot be extended to a measure on X. In this section we shall indicate methods of wide applicability of constructing an extension which is a distribution on X (more precisely, this distribution will extend to 9(X)the restriction of the measure fwsuoF-tfo, to 9 ( X - P)).
-
juo
A first method consists of considering the integral Ffi, (where as usual t c means eclog' for real numbers t 0), which is an analytic function of 5 in the half-plane Eo :95 > 0 in C, for all functions f E X(X)(13.8.6). In other words, on restricting to 9(X), we obtain a distribution T( FJf.o
=-
:f-lU
268
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
on X, which is a weakly analytic function (12.1 6.6.1) of 1; on E, ,and which we seek to analytically continue to a larger open subset of C. It may happen that such an analytic continuation exists on an open set containing the point ( = - 1, for each f E X(X),in which case its value at 1; = - 1 will still be a measure on X. Another case which occurs frequently (see the examples below) is that in which an analytic continuation [ H Tp) exists on E, - { - l}, where E, is a half-plane BY > a,with u < - 1, and TI")has a simplepole at the point - 1;in other words, for each function f E 9(X), we have
where A is a linear form on 9 ( X ) and, for all in a neighborhood V c E, of - 1, B, is a distribution on X and the function 1 ; B, ~is weakly analytic in V. It follows then (12.1 6.6.1) that A is a distribution. Moreover, if Suppcf) n P = @, then we have A(f) = 0; for [H T&f) is then an entire function of 1;, hence coincides with 1 ; ~ T ? ) ( f )in E, - {- l}; consequently, as 1; + - 1, Tr(f) and Bscf) both tend to finite limits, whence the assertion follows. Hence we have Supp(A) c P. Consequently B-l is an extension to 9(X) of the distribution f-/uo F-lfu, defined on 9(X - P), and is called the finite part of this integral. (Of course, there are infinitely many such extensions, obtained by adding to B-, any distribution with support P.) Examples (17.9.2) SO
Take X
= R" (n
2 l), and let F be the function r(x) =
(;I,
(5')'
,
that Uo = R" - (0) and P = (0). Take uo to be the candhical n-form
d t ' A d t Z A -*.Adc;".
If t~ is the "solid angle" differential (n - 1)-form on the unit sphere S,,-', then by (16.24.9) we may write (17.9.2.1)
TC(f) =
for 91;> 0 and any f E X(R"), and the function (1 7.9.2.2)
M,(d
=
Is.-.
f(PZ)tJ(Z)
is continuous for p 2 0, compactly supported, and of class C" on 10, + a[ (1 3.8.6). Hence we see already that, forfE X(R"), the function ryis integrable
9
EXAMPLE: FINITE PARTS OF DIVERGENT INTEGRALS
269
not only for Wt:> 0, but for Wt: > -n (13.21.10), and Tc is therefore a measure on R" for all t: in this half-plane. For each integer m > 0 and for f E 9(R"), we shall define an analytic continuation [H TY)(f) of the function t:t+TS(f), as follows. Replacefin (17.9.2.2) by its Taylor series up to order 2m (8.14.3):
+
where ( p , z) ~ g ( pz), is continuous on [0, co[ x Sn-l and has support contained in a set of the form [0, po[ x Sn-l.Hence, by splitting the integral into two parts, we have f o r at:> - n
where
1
(17.9.2.5)
= Jsn-*
- zVa(z). v!
Now, on the right-hand side of (17.9.2.4), the last integral is an entire function of (13.8.6); the first is an analytic function of in the half-plane at:> - n - 2m; and therefore the right-hand side of (17.9.2.4) is a meromorphic function of in the half-plane Wc > - n - 2m, having at most simple poles at the points of the form - n - k,where 0 5 k 6 2m. It is this function which is the desired analytic continuation T:")(f). Since T:"'+')(f) and T:")(f) coincide with TC(f) for W c > -n, they coincide throughout the domain of definition of TY)(f) (9.4.2). We shall therefore denote by TC(f)the function, meromorphic in the whole complex plane C , which coincides with each T',"')(f) in the domain of definition of the latter. We shall now determine the residues of T,(f) at its poles. Notice first that the symmetry Z H -z multiplies the form CJ by (- 1)" and preserves (resp. reverses) the orientation of Sn-l if n is even (resp. odd) (16.21.10). It follows therefore from (17.9.2.5) and (16.24.5.1) that c, = 0 for
c
270
XVH DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
all multi-indices v of odd total degree. On the other hand, if Iv I = 2k is even, then the residue of TS(f) at the point -n 2k has the value
-
as follows from substituting the expansion (1 7.9.2.3) in the expression for
M,(p). To calculate this number, we introduce the Lapluciun, which is the following differential operator on R":
(17.9.2.7)
A = D:
+ D; + - .. + D: .
Using the formulas
we obtain (17.9.2.8)
Art = c([
+ n - 2)r5-2
for all [ E C and x E R" - {O}. This implies, by (17.5.5.1), that
for W[ > -n first of all; but since both sides of this relation are meromorphic functions of [,it follows that it remains valid for all [ which are not poles of Tc,., i.e., [ # - n - 2k. By iteration we obtain, under the same conditions,
and since none of the linear factors on the right-hand side vanishes when [ = -n - 2k, the residue of T,(f) at this pole is given by res-n- 2kT&f) = ((-2)(-4)
- - .(-2k)(-n)(-n
- 2)
.-.(-n
- 2k + 2))-' res_,,Tc(AkS).
Now apply the formula (17.9.2.6),replacingfby A'ffandk by 0; since
9
EXAMPLE: FINITE PARTS OF DIVERGENT INTEGRALS
271
by (17.9.2.2), we obtain finally (17.9.2.11)
1 res-n-ZkT,(f) = -DZkM,(O) (2k)!
(when k = 0, the denominator is to be replaced by 1). Let C, denote the constant which appears in the right-hand side of this formula. We can now, following the method described in (17.9.1), define the finite part Pf(rr) for all c E C.For values of c other than the poles of Tc, we define Pf(rr) = Tr, and for [ = - n - 2k (17.9.2.12)
(Pf(r-"-2k),f) = lim
{+-n-Zk
(T,(f)
- ck A"f(O)(C+ n + 2k)-').
The formula (17.9.2.9) gives the Laplacian of Pf(rs) for [ not equal to a pole of TC.To obtain its values at the poles, we may proceed as follows. For not equal to a pole of T, , write
<
so that the distribution Bk-l,c tends to Pf(r-"-zk+2) as [ + -n Replacingfby Af in this formula, and using (17.9.2.9), we obtain
and therefore, letting [ -,- n - 2k (17.9.2.1 3)
- 2k + 2.
+ 2 , the formula
A(Pf(r-n-2k+Z )> = 2k(n
+ 2k - 2)Pf(r-"-Zk) - 2,k! n(n(n++2)4k. -. (n2 )+a n2k - 2) Ake0 *
and, in particular, for k = 0, (1 7.9.2.14)
A(Pf(ra-")) = -(n - 2)!2.q,.
A variant of this method consists in remarking that, for We > -n, Tr(f) is the limit as a + 0 of the integral
272
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
where U, is the exterior of the ball of radius a > 0. For each a > 0, this integral is defined for all E C (assuming that f has compact support); for WC > -n - 2m,the right-hand side of (1 7.9.2.4) is therefore the limit of
when ( is distinct from the poles of T,, and for 5 = -n - 2m the finite part
T,(f) =
/o+m
+ a[
x!f<-x)dx.
The same method as before, but with much simpler calculations, gives us now an analytic continuation of T,(f) to a meromorphic function on C, with simple poles at the points -k - 1 (kE N), and residues
We define Pf(x5) to be T, when C is not a pole, and (17.9.3.1)
Pf(xTk-') = lim [-+-k-l
(Ti -
(- Ok k!(C k 1)
+ +
We find this time by the same method (17.9.3.2)
D(Pf(xYk-')
=
-(k
(- l ) k + + 1)Pf(Xyk-') + (k -Dk+'e0. + l)! 1
It is convenient to introduce at this point the distribution (17.9.3.3)
by standard properties of the gamma-function, Y c ( f )is not only meromorphic on C but is an entire function of [, with values (17.9.3.4)
for k
E N.
Y-k = Dk&0
9 EXAMPLE: FINITE PARTS OF DIVERGENT INTEGRALS
273
We observe that Y1is the Heaviside function already considered in (17.5.7). The formula (17.5.7.1) generalizes to
DY, = Yc-1
(17.9.3.5)
for all 5 E C.The support of Y , is the half-line [0, -k (kE N), but the support of Y-, is {O}.
+ a[if C is not equal to
(17.9.4) We shall sketch one last example, without entering into the details of the calculations. Take X = Rn and uo to be the canonical n-form. The coordinates in R" will be denoted by t', t', ...,Y-l, and F will be the function given by
when 5' 2 0 and integral
n-1
i= 1
(tj)' 5 (to)2,and s(x)
= 0 otherwise. We consider the
(17.9.4.1)
as a function of the complex variable C. Sincefhas compact support, the range of integration may be replaced by the subset of U, for which 5 a for some suitable a > 0. Put
<,
(
x = ar,ar-l - t z ) E ~ x ~ n - l , l+t
where z E S n - 2 , 0 5 r 5 1, and 0 5 t 5 1; then we obtain (first of all for
WC > n) (17.9.4.2)
T,(f)
= l0*rr-
d r ~olt(i-"'i.
G(t, r, C) dt,
- ty-2
f ( a r , ar -z)o(z) l+t
where (17.9.4.3)
G(t, r, I) =
25-n+laC(l (1
+ t)'i
1-t
(0 denoting as before the solid angle, but this time in Sn-2). It is immediate that for fixed 5 the function (t, r ) - G(t,r, C) is of class C" for r E R and
274
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
t > - 1. By taking the Taylor series of this function in a neighborhood of (0, 0), with an arbitrarily large number of terms, it is easily verified that the function of two complex variables (initially defined for Wcr > 0 and Wg > 0)
extends to an entire function. Finally, by taking the Taylor series of the function r w G ( t , r, [) in a neighborhood of r = 0, and using properties of the gamma-function, it can be shown that (17.9.4.4)
We now introduce the distribution (17.9.4.5)
which is an analytic function of [ in the half-plane W[ > 0; the constants are chosen so that, by (17.9.4.4) and properties of the gamma-function, we have (17.9.4.6)
Next we introduce the d’dlembertian, which is the differential operator
It is easily verified that
ass =[([ + n - 2)Sr-z
for all [ E C and x such that s(x) # 0. From this formula we obtain (17.9.4.8)
O(Z,) = z, - 2
at any rate for W[ > 2. We shall deduce from this that Z, is an entire function of [. From the properties of I(a, P), and the Gauss-Legendre formula
r(c)= 25-1~-1’zr(+5)r(3(5+ i)),
+
it follows that the only possible poles of Z, are those of r(t([ I)), that is to say the negative odd integers. By analytic continuation, the formula (17.9.4.8) is valid for all [ except possibly the negative odd integers; but
O(Z,>(f)= z,(of)
9 EXAMPLE: FINITE PARTS OF DIVERGENT INTEGRALS
275
has no pole at [ = 1, hence the same is true of Z C b Z ( f ) in ; other words, Z , has no pole at 5 = - 1; applying the same argument repeatedly, we see that Z, has no pole at any negative odd integer, which proves our assertion. By induction, it follows in particular that
Remark (17.9.5) By iterating the formula (1 7.9.2.8) and using (1 7.9.2.14), we obtain for odd n the analog of (17.9.4.9) for the iterated Laplacian:
(1 7.9.5.1) Ak((Pf(r2"-")) = r - l ( k - 1) ! ( 2 k - n)(2k - 2 - n) * * * (2 - n)Qn 80
(k a positive integer). For n even, say n = 2p, there is an analogous formula for the function rZk-" log r, with k 2 p (so that the function is integrable): (17.9.5.2)
Ak(r2k-nlog r) = (- l ) p - ' 2 2 k - 2 (k - I ) ! @
- p ) ! ( p - l)!Qn&EO.
PROBLEMS
1. (a) For each functionf6 W)(R), show that as E
> 0 tends to 0, the sum
1'
tends to a limit. This limit is called the Cauchy principal value of the (in general nonconvergent) integral mapping f HP.V.
I'
+ f ( r ) / r dr and is denoted by P.V.
-m
-m
-m
f ( t ) / t dt. Show that the
f ( t ) / tdt is a distribution of order 1 on
R.This distribution is
denoted by P.V.(l/x). Show that a primitive Qf P.V.(l/x) is the integrable function log 1x1. Calculate the successive derivatives of P.V.(l/x). (b) More generally, for any function g of the form
where A is a constant and h is continuous, the distribution P.V.(g) (or P.V.(g(x))) is defined as in (a), by replacing the function I / x by g(x). Show that if T is an increasing diffeomorphism of R onto R such that ~ ( 0= )0, then ~ - ~ ( P . v . ( l / x )= ) P.V.(n'(x)/n(x)).
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
276
2.
Let rn be an integer 22. Show that as E > 0 tends to 0, the double integral
+
(where z = x iy) tends to a limit, for each function f~ 9(R2). (Reduce to the case where f (x, y) = z P B and change to polar coordinates.) Show that the mapping
is a distribution on RZ of order m - 1. This distribution is denoted by P.V.(l/z"). Prove the formulas
3. Show that for 5 = n - 2 - 2k which is not of the form -2rn (where rn is an integer ZO), the support of Zc(17.9.4.5) is the cone defined by to20,s(x) = 0. 4. Let f be a holomorphic function on an open set in C containing the closed unit disk D : 111 5 1, and let fD be the function which is equal to f in D and vanishes outside D.
Show that the distribution fD on R*has derivative afD/aZ equal to the distribution
where
E
is the circuit tt-+e'* ( O $ t $ 2 n ) .
(Use the elementary Stokes' formula
(16.24.11).) 5.
Express in terms of Cauchy principal values (Problem 1) the distributions on R defined by the formulas
and
6. For
> 0 and 9 p > 0, consider the distribution TA,,, on 10, 1[ defined by
Show that (~,p)nT,,,,(f)\extends to an analytic function except for A = - n or p = -n, where n EN,and determine the form of this function near these singular points. Outside the singular points, TAspis a distribution on 10, 11, denoted by P f ( X y ' ( 1 - x)";1).
10 TENSOR PRODUCT OF DISTRIBUTIONS
277
10. T E N S O R P R O D U C T O F D I S T R I B U T I O N S
(17.10.1) Let U be an open subset of R",let T be a distribution of order s m on U , let E be a metric space and f a mapping of U x E into C.
(i) Suppose that there exists a compact set K c U and a neighborhood V of a point zo E E such that (1) f( ,z ) E 9(m)(U;K ) for all z E V ; (2) (x, Z)H DVf(x, z ) is continuous on U x V ,for each multi-index v such that I v I 6 m. Then the function 2- F(z) =
-
is continuous on U x V . Then F is direrentiable on V (in the usual sense) and we have (1 7.10.1.l)
= (T,
a f(.
7
2)).
aZ
(i) Let (zn)be a sequence in V converging to zo . By virtue of (3.1 6.5) and the continuity hypothesis, each of the sequences of functions (D"f(x,z,,)) converges unvormly to DVf(x, zo)in K . Hence the result, having regard to the definition of distributions of order S m (17.3.2), and to (3.13.14). (ii) By virtue of the hypothesis and (17.5.13), we have
The same reasoning shows that each of the sequences of functions
converges uniformly to 0 in K. Hence the sequence
converges to 0, which proves (ii).
278
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(17.10.2) Let X , Y be two diTerentia1manifolds, K (resp. L) a compact subset of X (resp. Y). Let K' (resp. L') be a compact neighborhood of K (resp. L). Then as u (resp. v ) runs through 9 ( X ; K') (resp. 9(Y; L')), tbe closure in 9 ( X x Y ; K' x L') of the set of linear combinations of functions of the form u @ v (13.21.14) contains 9 ( X x Y; K x L).
The proof reduces immediately to the situation in which X (resp. Y) is an open subset of R" (resp. R"). By a translation and a homothety, we may assume that K' x L' is contained in the cube I = [-+, +]"+" in Rm+".With the notation of (14.11.3), put
n
m+n
Gh(Z) = ak("+")
gk(li)
i=1
..., ["+") E Rm+".If h E 9 ( X x Y; K x L), put Ph = h * G,; then (13.8.6) we have D'P, = D'h * Gk= h * D'G, . Hence (14.11.3) in the cube I , the function P, coincides with a polynomial in the coordinates and for each multi-index v, the D'P, converge uniformZy on I to D'h. Let p (resp. a) be a C"-function on X (resp. Y) with support contained in K' (resp. L') and equal to 1 on K (resp. L). Then the sequence of functions (p @ a)Pk,which belong to 9 ( X x Y;K x L'), converges in this space to h, by virtue of (17.1.4), and (p@a)P, is a linear combination of functions of the form u @ v with u E 9 ( X ; K') and u E 9(Y; L'). for z = ([I,
c',
We may now follow the procedure of (13.21 .I) for defining the product of two measures, to define the product S @I T of a distribution S E 9'(X) and a distribution T E 9 ' 0 . (17.10.3) Let X,Y be two differential manifolds, S a distribution in X , and T a distribution on Y.Then there exists a unique distribution R on X x Y such that, for all f E 9(X) and g E 9(Y),
Furthermore, for each function h G 9 ( X x Y), the function XHH(X) = (T, h(x, *)> belongs to 9(X), and we have ( R , h ) = ( S , H> or equivalently (by abuse of notation) (17.10.3.2)
The uniqueness of the distribution R clearly follows from (17.10.2). If h =f 0 g , we may write H(x) =f(x)(T, g>, and (S, H > = ( S , f >(T, g>. Hence we have only to show that the mapping h~ <S, H), which is a linear
10 TENSOR PRODUCT OF DISTRIBUTIONS
279
form on each of the spaces 9 ( X x Y; M) (M a compact subset of X x Y), is continuous on each of these spaces. Now, it follows already from (17.10.1) that H E 9 ( X ; prl(M)), for each h E 9 ( X x Y; M). Next, we need only prove that h H ( S , H ) is continuous when M is contained in a chart of X x Y; in other words, we may assume that X and Yare open sets i n R“ and R”,respectively. Let D‘”’ (resp. D”””) denote partial differentiations with respect to the coordinates in X (resp. Y). Let (hj) be a sequence of functions in 9 ( X x Y; M), converging to 0 in this space. For each multi-index v”, the derivatives D”””hj(x, y ) converge to 0 uniformly in M ; since hj(x, y ) = 0 for y q! pr,(M), it follows from (17.3.1.1) that the functions Hj(x) = (T, hj(x, .)) tend to 0 uniformly in pr,(M). The same is true of the derivatives D’”’Hj(x) for each multi-index v’, because, by virtue of (17.10.1.1), D’”’Hj(x) = (T, D’”’hj(x, -)) and by hypothesis, for each multi-index v”, D”””(D’”‘hj(x,y ) ) tends to 0 uniformly in M, so that the argument above applies. The definition of distribuQ.E.D. tions shows therefore that (S, Hj> + 0. This theorem therefore gives simultaneously a proof of existence and a method of calculation by successive application of the distributions S and T. Evidently we may invert the order of application and thus obtain the formula
The distribution R defined in (17.10.3) is called the product (or tensor product) of S and T, and is denoted by S @ T. It is clear that when S and T are measures on X and Y, respectively, the distribution S @ T is the product measure defined i n (13.21), having regard to the uniqueness property of (17.10.3) and to (17.3.2). Moreover, with the notation introduced above, the tensor product of distributions has the following properties : (17.10.4)
(i) Supp(S @ T ) = Supp(S) x Supp(T).
(ii) If S has order r and T has order 5 s, then S @ T has order 2 r + s. Furthermore, i f u E &(X) and u E &‘(Y),then (1 7.10.4.1)
(U
@ U) . (S @ T) = ( U . S ) @ ( v . T).
(iii) I f ( S , ) is a sequence in 9 ’ ( X ) and (T,,) a sequence in 9’(Y), and ifone of these two sequences is weakly bounded, and the other cohverges weakly to 0, then the sequence (S, @ T,) converges weakly to 0.
280
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(iv)
If X is an open subset of Rm and Y an open subset of R”,then
(17.10.4.2)
D’v’D”v“(S@ T) = (D”‘S) @ (D””’T).
(i) It is clear that Supp(S @ T) c Supp(S) x Supp(T). Conversely, if (a, b) E Supp(S) x Supp(T) and if U (resp. V) is a compact neighborhood of a (resp. b), then there exists a functionfe 9 ( X ; U) and a function g E 9(Y;V) such that (S, f) # 0 and (T, g) # 0, so t h a t f 8 g E 9 ( X x Y; U x V) and <S 8 T , f @ g) # 0 by (17.10.3.1). (ii) If T is of order s, then the derivatives D”’H of order I v‘I I r are by seminormsinvolvingonly the derivatives majorized (by virtue of (17.10.1 .I)) of h of total order 5 r + s, whence the first assertion follows. Also the values of the two sides of (17.10.4.1) for h E 9 ( X x Y) are equal, by virtue of (17.10.3.1), when h is of the formf@ g, and therefore in general by (17.10.2). The same argument also proves (iv). (iii) Suppose, to fix the ideas, that the sequence (T,) is weakly bounded and that the sequence (S,) converges weakly to 0, and let h E 9 ( X x Y; M). Then, as x runs through prl(M), the functions h(x, * ) : y~ h(x, y ) have their supports contained in pr,(M), and for each multi-index (v’, v”) the functions y~ D””D”v“h(x, y) are uniformly bounded. It follows therefore from (17.8.5(ii)) that for each multi-index v’ the sequence of functions D”’H, , where H,(x) = (T,, h(x, .)), is uniformly bounded in pr,(M); but then this sequence is relatively compact in 9 ( X ; pr,(M)) (17.2.2); and since the sequence (S,) is equicontinuous in this space, by virtue of (17.8.5), the fact that it converges weakly implies that it converges uniformly on each compact subset of the metrizable space 9 ( X ; prl(M)) (7.5.6). Hence the sequence (S,, H,) converges to 0. Q.E.D. Examples (17.10.5) If EL (resp. E:) denotes the Dirac measure at the point a E X (resp. b E Y), then it is immediately seen that the tensor product 6.: 0 is the Dirac measure Hence, (17.10.4.2) = (D“kL) @ (D’””E;). D’v’D’”‘‘~~,,b)
(I7.10.6) Suppose that X is an open subset of R”and that S is the distribution defined by a locally integrable functionf(with respect to Lebesgue measure on X). Then, for each function h E 9 ( X x Y), we have
(17.10.6.1)
s
(f0 T, h> = U‘,h(x, *>>f(4 W4
10 TENSOR PRODUCT OF DISTRIBUTIONS
281
(17.10.7) Let S be a distribution on X, let b be a point of Y ,and let s b denote the distribution on the submanifold X x {b} of X x Y which is the image of S under the diffeomorphism XH (x, b). Then it is immediately verified that
where j : X x {b} --t X x Y is the canonical immersion. When Y = R",this by virtue of (17.10.4.2). When gives the derivatives D"'"(j(S,)) = S 63 D""(E~), Xis an open subset of Rmand S is a locally integrable function, the distribution j@b) is sometimes said to be a singlet or simple layer on x x {b}, andits derivatives D"'"(j(Sb))to be multiplet luyers on X x (6). (17.10.8) In the same way we define the tensor product
-
T = T1@ T, where, for 1
@ * *@I
rn
Trn= @ Tk , k= 1
k 5 m yTkis a distribution on a manifold X,; T is a distribution
on the product manifold X =
n Xk,and we leave to the reader the task of m
k= 1
extending the results of this section to these multiple tensor products.
PROBLEMS
1. Let U be a bounded open subset of R"and let (gm).r be a total orthonormal sequence in 9i(U)(relative to the measure induced on U by Lebesgue measure) (Section 13.11, Problem 7). (a) Show that the functionsg, @gkform a total orthonormal sequence in g&(Ux U). (Usethe fact that the functions u 0u with u, u E X,(U) form a total set in 9 i ( U x U), cf. (13.21.1).) m (b) Show that in the distribution space 9 ( U x U) the series C(g.@gn)converges to .=1
the measure induced on U x U by the trace measure tr (Section 17.6, Problem 10) on
R" x R".(Consider first the series
c m
(w Ig. @ g"), where w
"=I
is of the form u @ u, with
9(U); then use (17.10.2).) (c) Conversely, let ( g J m r l be an orthonormal sequence in 9 i ( U ) such that the series
u, v E m
c(g.09,)converges in o'(U x U) to the measure induced by the trace measure tr.
"=I
Show that the sequence (9") is total. (Use (6.5.2) and the fact that 9(U)is dense in
%W).)
2. Let U be an open subset of R",let V be an open subset of R", let u, (I s j 6 r ) be linearly independent continuous functions on U, and let uk ( I 5 k 5 s) be linearly
282
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
independent continuous functions on V. If T, (1 5 j 5 r) are distributions belonging to 9'(V), and Sk (1 5 k 5 s) distributions belonging to 9'(U), such that the sum
is a continuous function on U x V, show that the T, are continuous functions on V and the Sk continuous functions on U. (Remark that there exist functions vjE 9(U) such that
C S,-@D"""Eo, 1U"l
SP
where the Seware distributions belonging to
em").(Consider the distributions
f *
11. C O N V O L U T I O N OF DISTRIBUTIONS ON A LIE GROUP
(17.11.1) Let G be a Lie group, n an integer > 1 and let m : G" + G be the mapping ( x l ,x 2 , . . . ,x,,)I-+ xlx2 ... x, . A sequence (TI, . . . , T,) of n distributions on G is said to be strictly convolvable if the supports A, = Supp(T,) of these distributions have the following property: for every compact subset K
of G, the set m-'(K) n
fl A, is compact in G. Let T = @ T , be the product n
n
k= 1
distribution on G", with support A
n A k .For n
=
k= 1
k= 1
each function f E 9(G), it
follows that (T, f 0 m ) is defined; for if K is the support off, then f m E &'(G") and the support off m is contained in m-'(K), hence our assertion follows from (17.4.5). Moreover, for each compact subset K of G, if a sequence of functions fp E 9 ( G ; K) converges to 0 in this space, then for each compact neighborhood V of m-'(K) n A the restrictions of the functionsf, m to V converge to 0 in &'(V) (17.1.5), and hence we deduce from (17.4.5) that the sequence ((T, f, 0 m ) ) tends to 0. Consequently the mappingf H ( T , f m ) is a distribution on G, called the convolution (or convolution product) of the sequence (TI, . . ., T,) and denoted by T , * T2 * * . . * T, . Equivalently, we may write (17.10.3) 0
0
0
0
(17.11.1.1)
(Ti
* T2 * * . . * T , , f ) =
.J!
* *
s f ( x , x , .. x,) dTl(xl).. * dT,(x,).
11 CONVOLUTION OF DISTRIBUTIONS ON A LIE GROUP
283
When the Tkare strictly convolvable measures, then by virtue of (17.3.4) and (1 7.10.4) their convolution product as distributions is identical with their convolution product as measures, in the sense defined in (14.5). Examples (1 7.1 1.2) If all the supports Ahwith at most one exception are compact, then the sequence (T,, T, , . . . , T,,)is strictly convolvable. The proof is the same as in (14.5.4). Let P be the set of points x = (ti)E R" such that ti 2 0 for allj. If there exists a = ( a j ) E R" such that the supports A, all belong to the set P + a,then the sequence (Tk) is strictly convolvable. For, by expressing the fact that a sequence of n points xk = ( t k j ) E R" is such that the xk belong to P + a and that x , + x, + * - - + x, belongs to a compact subset K of R", one sees that there exists a constant C such that for 1 S j 5 m,
c (tkj fl
- crj)
k= 1
S
c-
nccj,
and since by hypothesis all the t k j - ccj are 20,we have 0 5 t k j - aj 5 c - ncrj for each index k , so that the point ( x , , ~ , , ..., x,) belongs to a compact subset of R"". A finite sequence of measures on G may be convolvable (in the sense of (14.5)) without being strictly convolvable. In certain situations the convolution of distributions (other than measures) which are not strictly convolvable can be defined (Problem I). (17.11.3) Suppose that the sequence (TI, T, , . . . , T,,) is strictly convolvable. IfAk = Supp(T,) for 1 5 k 5 n, then
(17.1 1.3.1)
The proof is the same as (14.5.4), using the fact that the support of T I @T, @ @ T,, is A, x A, x ... x A,,. 1 . .
The following is a corollary of (1 7.11.3): (17.11.4) Ifthe distributions S andT are strictly convoluable andifU c Gisan open set such that (Supp(S))-'U n Supp(T) = @, then U n Supp(S * T) = @.
Let A = Supp(S), B = Supp(T). Since A-'U n B =@, we have AB n U =@ and therefore (as U is open) n U = @, whence the result.
AB
Convolution of distributions has algebraic properties analogous to those of convolution of measures:
284
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(17.11.5) If a finite sequence of nonzero distributions (Tl, T2,.. . ,T,,) is strictly conuolvable, then for each h = 1,2, . . . ,n the sequences (Tl, ...,Th), (Th+ 1, . . . , T,,) are strictly convolvable; the distributions T, * * * * T, and Th+l* * * * * T,, are strictly convolvable; and we have
-
(17.11.5.1) (Tl *
* * *
* Th)* (Th+,*
* T,) = T1* T2 *
- * *
* T,,
(" associativity of convolution ").
Let Ak= Supp(Tk)for 1 5 k 5 n ; by hypothesis, these sets are nonempty. For h 1 6 k 6 n let z& be a point of A&. Then the set of points (xl, . .., xh) E G" such that xlx2 * * xh E K and x, 8 Ak for 1 5 k 5 h is also the set of points (xl, ..., xh) E Ghsuch that
+
-
x1x2'*'xhzh+l" ' Z n E I ( Z h + l " ' z , , ; in other words, it is the section at the point
rn-l(Kzh+l *
-
*
z,,) n
n
k= 1
(zh+l,
..., z,,) of the
set
Ak considered as a subset of the product G" x G"-h.
Hence it is compact, which shows that the sequence (Tl, . .., Th) is strictly convolvable. Similarly the sequence (Th+l,..., TJ is strictly convolvable. Now put B = A1A2 * * . Ah, C = Ah+1* A,,. We shall show that for each compact subset K of G the set of pairs (y, z) E B x C such that y z E K is compact; by virtue of (17.11.3), this will prove the second assertion. Let then V be a compact neighborhood of K in G; there exists in each Aka sequence of points (xip))such that the sequence of products xiP)xkp)* xip) tends to y , the sequence of products xi$l * xip)tends to z , and such that
--
.'p'
... xp)xIp!l ...x'R' E v
n
L = rn-'(V) n Ak is compact, k= 1 the same is true of its projections M and N on G" and WPh,respectively. The for all p. Since by hypothesis the set
first part of the proof shows that y belongs to the (compact) image M' of M under the mapping (xl, . . . ,x , , ) ~ x ~ xx,, ~ and that z belongs to the (compact) image N' of N under the mapping (xh+1, . .. ,x,,)H xh + * * * x,,. Hence (y, z) E M' x N', which completes the proof of the second assertion. Finally, put R = T, * T, * * Th and S = T,+, * * T,; then, by definition, for each function f E 9(G) we have
11 CONVOLUTION OF DISTRIBUTIONS ON A LIE GROUP
285
whence the formula (17.1 1.5.1) follows. (17.11.6) For each point s E G and each distribution T E W(G)we have (17.1 1.6.1)
E,
* T = ~(s)T,
T * E~ = 6(s-')T
The proof is the same as in (14.6.1 .I).
In particular, if e is the identity element of G, then
E,
* T = T * E,
= T.
(17.11.7) For each distribution T on G, let T be the image of T under the diffeomorphism xt-i x-l of G onto G. If the sequence (T1, ... ,T,) is strictly convolvable, then so is the sequence (T, , . .., Tl) and we have (1 7.11.7.1)
* T1= (T, *
Tn* T,,-, *
* T,)"
(1 7.11.8) Suppose that G is commutative. Ifthe distributions S and T on G are strictly convolvable (in that order), then so are T and S , and
T * S = S * T.
(17.11.8.1)
The proofs are immediate.
Let (S,), (T,) be two sequences of distributions on a Lie group G. Suppose that the supports of the S, are contained in aJixed compact subset A of G. I f one of the two sequences is weakly bounded and the other converges weakly to 0, then the sequence (S, * T,) converges weakly to 0. (17.11.9)
Let f E 9(G) and let K = Supp(f). I f m : G2+ G is the mapping ( y , z ) yz,~ then the set m-'(K) n (A x G ) is compact, and if h E 9(G2) is equal to 1 on a compact neighborhood of this set, we have (Sn
* T n ,f>= ( S n 0 T n
9
h(f 0 m>>
for all n. Now apply (17.10.4(iii)). It should be remarked that the conclusion of (17.11.9) may be false if the supports of the distribution S, are not all contained in a fixed compact set. For an example we may take G = R and S, to be the Dirac measure E - , at the point -n, and for T the measure
nE, defined by a mass n at each integer n=l
point n > 0. Then the sequence (S,) converges weakly to 0, but the measure S, * T has mass n at the point 0, and therefore does not converge weaklytoo.
286
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
Let n : G + G be a homomorphism of Lie groups, and let S , T be two distributions on G. Suppose that either (1) rc is proper (17.3.7) and S, T are strictly convolvable, or (2) rc is arbitrary and S , T have compact support. Then n(S) and n(T) are strictly convolvable, and we have (17.11 .lo)
~17.11.10.1)
n(S * T)
= n(S)
* rc(T).
In case (2), n(S) and n(T) are compactly supported (17.4.3), hence are strictly convolvable. In case (1) it is enough, by virtue of (17.4.3), to show that for each compact subset K’ of G’ the relations x E Supp(S), y E Supp(T), and n(x)n(y)E K’ imply that the pair (n(x), n(y)) belongs to a compact subset of G x G . However, since n(x)n(y)= n(xy) and since n-’(K’) is compact by hypothesis, the point (x, y ) belongs to a compact subset of G x G, where the result follows. For each function f E Q(G’),we have then J f ( z 7 4 4 s * T))(z’) = =
sssfo) f
(n(z))4 s * T)(z) dSW
= Ssf(n(x)n(y))W x ) = SdT(Y) =
jf ( X ’ 4 Y ) )
d(n(S))(x’)
Sd(n(SNx7 sf(”.y.) d(n(T))(y’)
which proves (17.11.I0.1). Remark (17.11.10.2) If S and T are distributions with support {e},then n(S) and n(T) have support {e’}, and the formula (17.11.10.1) remains valid, with the same proof, when n is a local homomorphism (16.9.9.4): for we need only consider functions f whose support is contained i n a neighborhood V of e such that n(xy) is defined and equal to n(x)n(y) for all x, y E V.
In the important case where G = R”,convolution of distributions behaves as follows relative to the operation of differentiation :
11 CONVOLUTION OF DISTRIBUTIONS ON A LIE GROUP
287
(17.11.11) If S and T are strictly convolvable distributions on R", then for 1 5 k n the distributions D, S and T (resp. S and DkT)are strictly convolvable,
and we have
Dk(s * T) = (Dk s) * T = s * (DkT).
(17.11.11.I)
=sf(.
For each function f E 9(R"), put g(x) + y) dT(y), so that by (17.10.1) the function g is indefinitely differentiable, and
s
Dk).(g
=
Since by definition (Dk(S * T),f)
=
-(s,
Dkf
=
Dkg)
(x + y ) dT(y).
* T, Dkf), we have = (DkS, g ) = ((DkS) * T,f),
-(S
from which the first of the equations (17.11.11.1) follows. The second is proved in the same way. It follows by induction that, for any two multi-indices p, v, we have
D"'(S
(17.11.11.2)
* T) = (D'S) * (D'T).
In particular, the derivatives of a distribution on R" can be expressed as convolutions:
* T, D'T = (D"E~)
(17.1 1.I 1.3)
where to is the Dirac measure at the origin.
PROBLEMS 1. Let U be an open subset of R".For each function f~ B(U) and each integer r >= 0, put p , ( f ) = sup IDvf(x)l, which is a real number or +a.Let F ( U ) be the subI v I 6 r.
XE
U
space of &(U) consisting of the functions f for which all the p , ( n are finite. The restrictions of the p , to F ( U ) are norms on this vector space, with respect to which it is a Frkchet space. (a) We have 9(U) c S ( U ) . A distribution T on U is said to be summuble if it is continuous relative to the topology on 9 ( U ) defined by the restrictions of the norms p,. Such a distribution is necessarily of finite order. Let (K,) be a fundamental sequence of compact subsets of U (17.1); for each distribution T E 9'(U) and each integer r 2 0, let pm.,Q denote the least upper bound of the numbers 1 T(f) I where Supp(f) c U - K, and ( f ) 5 I . Show that a distribution T is summable if and only if, for some integer
288
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
r >= 0, the sequence (p,, ,(T)),? converges to 0. In particular, the summable distributions of order 0 are precisely the bounded measures (13.20). (Argue by contradiction to show that the condition is necessary.) Every derivative DYTof a summable distribution is summable. (b) Suppose that U = R“and that K, is the ball llxll m. Let hl :R”+ [0,1]be a C”-mapping which is equal to 1 on K1and is 0 outside K2, and put hm(x)= hl(x/m)for all m 2 2. If T is a summable distribution, show that for each function f E P(R”) the sequence (T(h,f)),L1 tends to a limit, which we denote by T(f). In this way T is extended to a continuous linear form on the Frkchet space S(R”). (c) Show with the help of (b) that the convolution of two summable distributions on R“can be defined, and that this convolution is also a summable distribution. 2.
(a) With the notation of (17.9.3) show that Y . * Y a= Y o + afor any two complex numbers a,p. (Show that it is sufficient to prove the result for Ba > 0 and Bfi > 0.) If T is a distribution on R whose support is bounded below, then Y-k * T = D‘T for all integers k > 0, and Yk * T is the kth primitive of T whose support is bounded below. By extension, for each complex number 5, the distribution Y, * T is called the primitive of order 5 of T, and the distribution Y-4 * T is called the derivative of order 5 of T. (b) Let a,p, y be complex numbers such that 9 p > 0 and 9~> 0. The hypergeometric function F(a, fi, y ; x ) is defined on the interval 1- 1, 1[ of R by the formula
Show that, for y complex and # -n, where n f N, the function can be extended to all values of fi E C in such a way that
(change the variable to w = tx). In particular, for j3 = - k (where k E N) we have
Deduce that
(Jacobipolynomiaf).(Expand the distribution (1 - x ) ”Dkeo as a sum of point-distributions with support lo}.) (c) For each p E C such that 9 p > -4, the Bessel function of order p , defined for x E R, is given by the formula
Show that the function can be extended to all complex values of p in such a way that, for u > 0, we have
*
2 ~ 7 r ” ~ U ” ~ J P ( U ’= ~ ~Yp+1,2 ) (u -It2
cos u1’2).
12 REGULARIZATION OF DISTRIBUTIONS
289
Deduce Sonine's formula
(Convolve with Yq+l.) 3. With the notation of (17.9.4), show that 2. * Z, = Z.+# for all complex numbers a,8. (Same method as in Problem 2.) 4.
Consider B'(R") as a vector space of linear forms on B(R"), and endow b'(R")with the corresponding weak topology (1 2.15.2). For each distribution S E O'(R"), show that the mapping T H S * T of 8'(R") into O'(R") is continuous.
5.
Let u be a continuous linear mapping of B'(R") into
a'@).
(a) Show that the following two properties are equivalent: (1) y(h)u(T) = u(y(h)T) for all T E B'(R") and all h E R"; (2) D, u ( T ) = u(D, T) for all T E B'(R") and 1 5 j In. (Use formula (17.8.2.1) and consider, for each f ~ 9 ( R " ) , the function h ++ ; calculate its partial derivatives.) (b) If u satisfies the equivalent conditions of (a), show that u is necessarily of the form T- S * T, where S E W W ) .(Consider the linear mapping R Hu(R * T) - R * u(T) of b'(R") into O'(R"), for a fixed distribution T E B'(R"), and show that its kernel is the whole of B'(R"). For this purpose, observe that the Dirac measures E~ (x E R")form a total set (12.13) in B'(R"), by using Problem 13 of Section 12.15; then remark that the E* belong to the kernel of the linear mapping in question.) 6. Let G, G' be Lie groups and let S,T (resp. S', T') be strictly convolvable distributions on G (resp. G'). Show that S @ S' and T OT'are strictly convolvable distribution on
G x G' and that
(S @ S')
* (TOT') = (S * T) @(S' * T').
12. REGULARIZATION OF DISTRIBUTIONS
(17.12.1) Let p , m be two integers 20 such that p 2 m. IfT E W("')(R")and (resp. i f T E B'("')(R")andf E C~'(~)(R")), then the distribution T * f may be identijied (17.5.3) with a function in L?(P-m)(R") such that, for each x E R",
f E @')(R")
(17.12.1.1)
(T * f ) ( ~=)( T , j ( ~ ) y )=
s
f(X
- y ) dT(y).
The 'fact that the function x w J f ( x - y) dT(y) belongs to C?(~-"')@V') follows from the hypotheses and from (17.10.1). Next, if S = T *ft then we
290
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
have by definition, for any u E 9(R"),
s s
<S, U > = dT(x)
f ( M x + Y ) d&y)
= (T €3 1,g ) ,
where A is Lebesgue measure on R", and g(x, y ) =f (y)u(x+ y). However,
-
hence (S, u ) = (T €3 (u A), h), where h(x, y ) =f(y
<s,u> = which proves (17.12.1.I).
s s UCv)
d4Y)
- x ) ; and therefore
f (Y - 4dT(x),
In particular: (17.12.2) If T E C'(R") and f E B(R"), or if T E 9'(R") and f E 9(R"), then T * f E b(R").If T E b'(R") and if (f,) is a sequence of functions in b(R") converging to 0 in this space, then the sequence (T * f,) tends to 0 in &(R").
Let A be the (compact) support of T, let V be a compact neighborhood of A, and let K be any compact subset of R". If we put g,(x, y) =f,(x - y), then the sequence of elements g,(x, .) of b(R") converges t o 0 in this space, uniformly with respect to x E K. For the partial derivative with multi-index v of the function y~ g,(x, y) is (- l)lvlDvfp(x- y). Since the sequence (Dvf,(z)) converges uniformly to 0 in the compact set K + (-V), the sequence of functions (x, y) H D'f,(x - y ) converges uniformly to 0 on K x V; hence, by the definition of distributions, the sequence of functions X H h,(x) = /&(x - y ) dT(y) converges uniformly to 0 in K, and the same is true of the sequence of partial derivatives X H Dvh,(x) for any multi-index v , by virtue of (17.10.1).
For each open subset U of R", the set 9 ( U ) of Cm-functions on U with compact support, ident$ed with a space of distributions on U (17.5.3), is weakly dense in 9'(U).
(17.12.3)
Let (K,) be a fundamental sequence of compact subsets of U, and let h, be a function belonging to 9(U) which is equal to 1 on K, . Since for each u E 9(U) the reexists an integer m such that K, is a neighborhood of the support of u, it follows that for each distribution T E 9'(U) we have
12 REGULARIZATION OF DISTRIBUTIONS
291
and therefore the sequence (h, . T) converges weakly to T. We are therefore reduced to the case where T has compact support A. Let W be a neighborhood of 0 in R"such that A + w c U.If (gk) is a regularizing sequence (17.1.2), then Supp(g,) c W for large k . Since the sequence of distributions (gk) converges weakly to the Dirac measure e0 , it follows from (17.11.9) that the sequence (T * gk) converges weakly to T * c0 = T. This completes the proof. This proof shows moreover that a distribution T with compact support can be approximated by functions belonging to 9(U)whose support is contained in an arbitrary neighborhood of the support of T (cf. Problem 13). (17.12.4) Let T be a distribution on R",with compact support A. Thenfor each neighborhood V of A there exists afinite number of continuousfunctionsf, on R*, with supports contained in V,such that T is equal to a sum of partial derivatives D v S(in which some of the v& may be zero). k
Let m be the order of T (17.7.1) and let p be an integer such that 2 p - n 2 m + 1. Then it follows from (17.9.5) that there exists a function E E C'"'(R") such that APE= e 0 , or equivalently, such that (Apeo)* E = e 0 . We then have
T = c0
* T = (Apeo)* (E * T) = AP(E* T);
the distribution E * T is in fact a continuousfunction on R",by reason of the choice of p and (17.12.1), but its support is not in general contained in V. However, let W be a compact neighborhood of 0 in R" such that A + W c V, and let g E Q(R") be a function with support contained in W, and equal to 1 in some neighborhood of 0. Since the function 1 - g vanishes on a neighborhood of 0, the function (1 - g)E belongs to I(R")(17.9.5), and hence the same is true of (Ap((l - g)E)) * T (17.12.2). Moreover, the support of gE is contained in W, and hence so also is the support of Ap(gE), and therefore also the support of the function u = Ap((1 - g)E) = t o - Ap(gE). Consequently Supp(u*T) c V . If also we p u t u = g E , then Supp(u) c W and so Supp(v * T) c V. Since T = Ap(v * T) + (u * T), the proof is complete. Remark
(17.12.5)
The formula (17.12.1.1) gives in particular
for all f E G9(p)(Rn)and all T E W(""(R")with p 2 m.In particular, this shows that if T * f = 0 for all f E G9(R"), then T = 0.
292
XVll
DISTRIBUTIONS A N D DIFFERENTIAL OPERATORS
Since s f ( x + y ) dT(y) = / f ( x - y ) dT(y), the definition of convolution (17.11.1) shows that if S, T are strictly convolvable distributions on R",then for allfe 9(R") we have (17.1 2.5.2)
'
(S
* T,f)= ( S , T *f)= (T, S *f).
PROBLEMS
1. Let S, T be two distributions belonging to 8'(R"). Show that
Supp. sing@ T ) c Supp. sing(S)
+ Supp. sing(T).
(Decompose each of S, T into the sum of a distribution whose support is contained in an arbitrarily small neighborhood of the singular support and a function belonging to
.ww.)
2.
8'w)
What can be said about the convolution of an arbitrary distribution T E with a polynomial? Deduce that every T E b'(R") is the limit of a sequence of polynomials, with respect to the weak topology of O'(R").
3. What can be said about the convolution of an arbitrary distribution T E b'(R")with the product of a polynomial and an exponential exp(<x, x')), where x' is a linear form on R"? 4.
(a) Let T E O'(R") be such that, for eachfc X(R"), the distribution T *fis a locally bounded function. Let K be a compact subset of R" and let H be a compact neighborhood of 0 in R". Show that the mappingf -(T *f)I H is a continuous mapping of the Banach space X(Rn; K) into P*(H, AH), where AH is Lebesgue measure on H. (Show first that for each function g E 9(R") the mappingf-((T * f ) g)lH is continuous (14.10.6); then replace g by the functions belonging to a regularizing sequence, and use the BanachSteinhaus theorem.) (b) Show that, for each function f~ X(R"), the distribution T *f is a continuous function. (Use(a) and the fact that O(Rn;K) is dense in X(R";K).) (c) Deduce from (a) and (b) that T is a measure on R". (Use.the formula (17.12.5.1).) (d) Show likewise that if T * f i s locally bounded for eachfE P ( h ) (1 $ p < a) with compact support (resp. f~ .GJcr)(Rn)),then T is a function belonging to -Y*(A), where ( l / q )+ (l/p) = 1 (resp. a distribution belonging to 9'(')(R")). (e) Deduce from (c) that if T * f i s a measure for eachfE P1(h) with compact support, then T is a measure. (Consider the convolution (T *f) * g,where g E X(R").)
+
5.
Let T E 9'(R") be a distribution such that T * f~ &(R") for all f~ O(')(R"). Show that T E &(R"). (Argue as in (17.12.4)J
6. A distribution all of whose derivatives are of order =<mbelongs to B(R")(use (17.9.5)).
12 REGULARIZATION OF DISTRIBUTIONS
293
7. Let H be a subset of 9'(Rn). Show that the following conditions are equivalent: (a) H is bounded in W(R") (1 2.1 5).
Gg) For eachfe 9(R") and each compact K c R",the set of restrictions to K of the functions T * f,where T E H, is bounded in V(K). ( y ) For eachfs OW),the set of functions T * f,where T E H, is bounded in W(R"). (8) There exists an integer m 2 0 such that, for each relatively compact open subset U of R", the restrictions to U of the T E H are sums of derivatives of order s m of continuous uniformly bounded functions on U. (Toshow that (y)implies remark first that if V is a relativelycompact open neighborhood of 0 in R"and i f f € g(R"; V), then the set of mappings uc, ((T *f)* u ) 1 U of O(R";V) into - P ( U , A"), where T runs through H,is equicontinuous. Apply Section 12.16, Problem 10 to deduce that there exists an integer m and a neighborhood W of 0 in OCm)(R"; V) such that, for all u, v E W and T E H, the restriction of T * u * v to U is a function bounded above by 1 in absolute value. Using the fact that 9(R"; V) is dense in 9('")(V), show that for all I(, v E W"(V), the restrictions to U of the distributions T * u * v are continuous functions on U which are uniformly bounded as T runs through H. Finally, use the formula
(a),
T = Afp(gE* g E * T)-2Ap(gE
* u * T)+ u * u * T,
in the notation of the proof of (17.12.4).) 8.
Let (Tk) be a sequence of distributions belonging to 9'(R"). Show that the following conditions are equivalent: (a)
(fl
Tk+ 0 in W(R").
For eachfe L@(R")the sequence of functions (Tk*f)converges uniformly to 0 on each compact subset of R". (y) For eachfe L@(R") the sequence of distributions (Tk*f)converges to 0 in O'(R"). (Argue as in Problem 7.) 9. Let L : 9(R")49'(R") be a linear mapping such that, for each compact subset K of R", the restriction of L to 9(R"; K) is continuous and L(f* g) = L(f) * g for all f, g E O(R"). Show that there exists a distribution S E 9'(R") such that L(f) = S *f.
(Observe that if (&) is a regularizing sequence, the sequence (L(gk) *f)converges to L(f) in o'(R"), and use Problem 8.)
10. Let T be a distribution on R".Show that, for T to be an analytic function on R", it is necessary and sufficient that, for eachfc 9(R"), T * fshould be an analytic function on R". (Toshow that the condition is necessary, use Cauchy's inequalities to majorize the derivatives D'(T *f) on a compact set K. Conversely, if T * f is analytic for each f~ 9(Rn), observe that for each relatively compact open set U and each relatively compact open neighborhood V of 0 in R",the mapping
u,:f-sup
I(DT *f)(x)/v!ll/IvI
is finite and continuous on 9(R"; 0)for each multi-index v ; furthermore, the set of is bounded above as v runs through N",for each f~ 9(R"; V). Using Baire's theorep (12.16.2), deduce that there exists a constant c > 0 such that, for each f~ 9(R"; the set of functions D'(T * f ) / ( w ! d V 1 ) (for Y E N")is uniformly bounded on U. Using Problem 7, show that there exists an integer m 2 0 such that the same property holds for eachfE 9("')(V), and takefof the form gE,in the notation of the proof of (17.1 2.4).)
u.(n
v),
294
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
11. Let T E I'(R") be such that S = BT/axl ax2 ... ax. is a measure (resp. a locally integrable function). Show that T is then a bounded (resp. continuous) function. Convolve S with the function Y(x1)Y(x2). . . Y(x.), where Y is the Heaviside function, and use (14.10.6).)
12. (a) Show that every distribution T E &(R? can be written in the form D'f, wheref is a continuous function on R" (but not necessarily of compact support). (b) Show that there is no compactly supported distribution T on R such that Dco = D T for some p > 0. (c) Let B be the disk llxll 1 in RZ.Show that there exists no compactly supported distribution T on Rz such that P)B = D T for some multi-index Y # (0, 0). c0
+
13. Give an example of a distribution on R whose support is a compact interval I, not consisting of a single point, and which is not the limit in W(R) of functionsfe O(R) with support contained in I. 14. Let S, T be two distributions belonging to O'(R), whose supports are boundedbelow in R; then they are strictly convolvable (17.11.2). Show that S * T = 0 implies that either S = 0 or T = 0. (Remark first that for all f, g E 9(R) we have (S *f) * (T * 9) = 0,
and use Titchmarsh's theorem (Section 11.6, Problem 11) to deduce that for example S *f= 0 (withfnot identically zero), and hence (S * u) *f= 0 for all u E O(R); then use (1 7.1 2.5).) 15. Give an example of a sequence of functions (fn)in O(R), with supports contained in a fixed compact set, which converges in b'(R) but which is such that the norms llf.ll (in V"(R)) and N1Un) (in L1(R)) are not bounded.
16. Let H be a bounded set of distributions belonging to B'(R), whose supports are all contained in a fixed compact set K. If I is a compact interval which is a neighborhood of K, then there exists an integer r such that the distributions T E H are of the form D'F, where the functions F are continuous on 1 (Section 17.7, Problem 1 and Section 17.5, Problem 3). Show that the set H. of functions Y, * T, where T E H,is bounded in (use (17.8.5)). If H,+,is the set of functions Y,,, * T, then H,,, is bounded in %(I).Deduce that if (TJ is a sequence of distributions belonging to H which converges to T E b'(R), then the sequence of primitives (Y,+2 * Tn)is a sequence of continuous functions which converges uniformly on I to Y,+** T.
17. Let T E b'(R). For each A > 0, let TA denote the distribution defined by TA(f) = A-' f ( x / A )dT(x) (in other words, ( T A )is~ the ~ ~image of TI,, under the homothety
f
XHXIA).
Show that if TAconverges to a distribution To+ in b'(R) as A + O , then this distribution must be a constant function on R.(Observe that, for each functionfE 9 ( R ) and each h # 0, if tends to the same limit, by expanding the difference f ( x hh) - f ( x ) by Taylor's formula, stopping at the order of T; then apply Problem 4 of Section 17.8.)
+
13
DIFFERENTIAL OPERATORS
295
18. (a) Let f be a real-valued function defined on an open interval I = 10,c[. Suppose that there exists a compact interval J = [a, b] c I such that a < b, n functions a,(&, .., ~ , , - ~ ( defined h) on a sufficiently small interval 10, y [ , and an increasing function &(A) defined on this interval and tending to 0 as h+O, such that, for all x E [a, b] and all h E 10, y [ , we have
.
I f ( h ) - ao(h) - al(X)hx - ...- a.-l(h)(hx)"
I 5 &(h)P.
Then each of the functions a,(& tends to a limit b, as h +0, and we have f ( x ) = b o + b 1 ~ + . . . +bn-lx"-l+o(Y)
as x+O in I. (Let a < xo < xi < ... < x . - ~< b be fixed points of J and let 8 E ]0,1[ be such that a < 0x0. If t, t' are such that 0 < Bt 5 t' 5 t < y, show that
I (ao(t9- ao(t))+ (4 -' al(t))t'xi ) + ... + (a,-,(t') - an-l(t))(t'xiY-l I 6 W ) t " for 0 5 i 5 n - 1, and deduce that there exists a constant A such that under these hypotheses
la,(t) - a,(t')I
5 AB-'&(t)"-'
( 0 6j 5 n - 1). Now consider the sequence (63 tending to 0, and apply Cauchy's criterion.) (b) With the notation of Problem 17, show that the limit To+ exists if and only if there exists an integer n 2 0 and a continuous function F on an open interval I = 10, c[ such that lim F(x)/x"= A exists and such that the restriction of T to I is equal to x-0.
x>o
D"F; in which case To+= n!A. (To show that the condition is necessary, consider a compact interval J = [Ba, a], where 0 < 0 < 1, contained in 1; by the result of Problem 16, there exists an integer n, an open neighborhood V of J and, for each sufficiently small h, a continuous function GAon V such that the restriction to V of Tnis of the form D"G, , and such that the functions Gnconverge uniformly to n !Ax" inV. Deduce that there exists a continuous function G on 10, b[ such that the restriction of T to 10, b [ is equal to D"G. For all sufficiently small h, we can write G(h)- hlGn(x)= wA(x),where wA is a polynomial of degree - 1, with coefficients depending on A. Now apply (a).) (c) Deduce from (b) that if S is a distribution such that DS = T, and if the limit To+ exists, then so does the limit SO+ . Also, for each function f~ 9 ( R ) , the limit (f.T)o+ exists and is equal tof(0)To+ . (d) If a > 0 and ji?> 0, show that the function x* sin(x-9, which is defined and continuous for x > 0, extends to a distribution T on R for which the limit To+ exists and is equal to 0.
sn
13. DIFFERENTIAL OPERATORS A N D FIELDS O F POINT-DISTRIBUTIONS
(17.13.1) Let X be a differential manifold and E, F two complex vector bundles over X. We have seen that the vector spaces T(X, E) and T(X, F) are canonically endowed with structures of separable complex Frtchet spaces (17.2.2). A C" linear diyerential operator from E to F (or simply a differential
296
XVll
DISTRIBUTIONS A N D DIFFERENTIAL OPERATORS
operator, or even an operator if there is no risk of ambiguity) is by definition a continuous linear mapping f++P * f of the Frkhet space T(X, E) into the FrCchet space T(X, F) which satisfies the following condition:
(L) For each open subset U of X and each section
f 1 U = 0, we have P - fI
U = 0.
fE
T(X, E ) such that
In other words, if two sections f , g of E over X are equal on an open set U, then so are their images P * f and P * g. An equivalent way of stating this is to say that P is an operator of local character. Let (V, cp, n) be a chart of X such that E and F are trivializable over V. If n', d'are the projections of the bundles E, F and if N', N" are their respective ranks over V, then there exist diffeomorphisms ZH (q(n'(z)),v(z)) of n'-'(V) onto q ( V ) x C" and zt+(cp(n"(z)),w(z)) of n"-'(V) onto q(V) x C"' such that v (resp. w) is a linear isomorphism of each fiber n'-'(x) (resp. d""(x)) onto C" (resp. C"'). The mapping f w v f o cp-' (resp. f H w f o cp-') is then an isomorphism of T(V, E) onto (&'(cp(V)))" (resp. of T(V, F) onto (tP(q(v)))"'). If P is a differential operator from E to F, then for each section f E T(X, E) the value P f IV depends only on f 1 V, and there is therefore a well-defined continuous linear mapping gt+ Q * g of (&'(cp(V)))" into (&(cp(V)))"' such that 0
0
-
The linear mapping Q is said to be the local expression of the operator P corresponding to the chart (V, cp, n) and the mappings v and w. (17.13.3) In order that a linear mapping P ofT(X, E ) into T(X, F ) should be a differential operator, it is necessary and suflcient that for each x E X there should exist a chart (V,cp, n) of X at the point x , such that E and F are trivializable over V and such that the corresponding local expression of P should be of the form (17.13.3.1)
where, for each multi-index v such that I v I 5 p , the mapping y w A , ( y ) is a C"-mapping of cp(V) into the vector space Hom,(C", C"') (which can be identified with the space of N" x N' matrices over C).
The condition is sufficient. First, it is clear that it implies the condition (L). Second, since each compact subset of X admits a finite covering by
13
DIFFERENTIAL OPERATORS
297
domains of definition of charts which satisfy the condition of the statement of the proposition, it is enough by virtue of (17.2) and (3.13.14) to verify that the mapping (17.13.3.1) is continuous, and this is a direct consequence of (17.1.3) and (17.1.4). To show that the condition is necessary, we may clearly assume that X is an open subset of R” and that E = X x C”, F = X x C”’, so that T(X, E) = (&(X))“ and T(X, F) = (Cp(X))”’. Replacing P by p P o j , where j is a canonical injection of one of the factors of (&(X))” into this product, and p is a canonical projection of the product (Cp(X))”’ onto one of its factors, we reduce further to the case where N’ = N” = 1. Replacing X if necessary by a relatively compact open set, we may suppose, by virtue of the definition of the topology of &(X) (17.1), that there exists a constant c and an integer p such that for all x E X and all f E &(X) we have 0
(17.13.3.2)
This shows that for each x E X the linear form f H( P .f ) ( x ) is a distribution of order S p on X; furthermore, if x $ Supp(f), then by hypothesis we have ( P .f ) ( x ) = 0, so that the support of this distribution is {x). Hence (17.7.3) it is of the form f H
c a,(x>D>vf(x),
IVlSP
where the a,(x) are scalars. Replacing f successiveiy by monomials xu, we see that for functions
S p the
JtlJ
are of class C“. It follows easily by induction on I vI that all the a, are of class C“, and the proof is complete. For each differential operator P from E to F, and each point x E X, the order ofP at x is defined to be the largest of the integers I v I such that A v ( x )# 0 in a local expression of P in a neighborhood of x . It follows immediately from the rule for differentiating composite functions and from Leibniz’s formula that this number cannot increase when we pass from one local expression to another, and hence it is independent of the particular local expression chosen. By virtue of (17.13.3), a differential operator of order 0 may be written (1 7.1 3.3.3)
fHA.f,
298
XVll
DISTRIBUTIONS A N D DIFFERENTIAL OPERATORS
where, for each x E X, A(x) is a linear mapping of the fiber Ex into the fiber F, , in other words an element of Hom,(E, , F,), and A :x ~ A ( xis) a section (of class C") of the vector bundle Hom(E, F) (which may be identified with the tensor product E* @ F). Such a section may also be identified with a linear X-morphism A : E + F ; for each section f of E, A * f is the section A f of F (16.16.4). 0
(17.13.4) If P is a differential operator from E to F, then for each open subset U of X we may define the restriction of P to U, which is a differential operator PI U from E I U to F I U, as follows : for each section f of E over U and each point x E U, there exists a C"-function h on X with support contained in U, which is equal to 1 in a neighborhood of x; hfextended by 0 outside Supp(h) is a C"-section of E over X; hence P (hf) is defined. The value (P* hf)(x) is independent of the function h chosen, and if we denote thisvalue by (P f)(x), it is immediate from (17.13.3) (or directly from the condition (L)) that P f i s a section of F over U and that f H P * f is a differential operator from El U to F J U . If now (U,) is an open covering of X and if, for each 1,we are given a differential operator P, from E I UAto F I U, , so that for each pair of indices 1,p the restrictions of PAand P,,to U, n U,, are equal, then it follows immediately from (1 7.13.3) that there exists one and only one differential operator P from E to F such that PI U, = PAfor each 1.For each section f E r(X, E) and each x E X, we define (P . f ) ( x ) to be the commonvalue of (PA ( f l U,))(x) for all indices 1 such that x E U, .
-
-
-
-
(17.13.5) If P is a differential operator from E to F and if h is a C"-function on X, it is clear that the mapping f H h(P f ) is also a differential operator from E to F, which we denote by h P (17.1.4). The set of differential operators from E to F is therefore an g(X)-module. Let El, E, , E, be complex vector bundles on X, and let PI : El -,E, and P , : E, + E3 be differential operators. Then it is clear that P , Pl is a differential operator from El to E, . Furthermore, from the local expressions of PI and P, it is immediately seen that if P, is of order p and P , of order q at a point x E X, then Pz PI is of order $ p q at x. 0
0
+
(17.13.6) An important particular case is that in which E and F are both equal to the trivial complex line bundle X x C,so that T(X, E) = T(X, F) = b(X). The local expression of a differential operator P from X x C to X x C is then of the form (17.13.6.1)
13 DIFFERENTIAL OPERATORS
299
where the mappings y w a,(y) are complex-valued C"-functions defined on an open subset cp(V) of R".The operator P is of order p at the point x E V if and only if at least one of the numbers a,(cp(x)) for J v I = p is nonzero. The differential operators from X x C to X x C clearly form a C-algebra with respect to the composition defined in (17.13.5); we denote this algebra by Diff-). We have already seen (17.13.3) that for each x E X the mappingfw (P . f ) ( x ) is a distribution with support contained in { X I ; this distribution is denoted by P(x), so that we have
Thus an operator P E Diff(X) is a C"-jield of point-distributions. (17.13.7) Let u : X + Y be a diffeomorphism. For each differential operator PeDiff(X) we transport P by means of u to a differential operator u*(P)E Diff(Y) as follows: for each functionfe S o , we have (17.13.7.1)
u*(P) * f = (P'( f o
or, in other words, for each x
24))
0
u-l
E X,
which shows immediately, bearing in mind the definition of the image under u of a distribution on X (17.3.7). that for each x E X we have
From (17.13.7.1) it follows immediately that if P,,Pz E Diff(X), then
in the algebra Diff(Y). Further, if u : Y + Z is another diffeomorphism, (17.1 3.7.5)
(u 0 u)* = u*
0
u*
,
which shows that u* is an isomorphism of the algebra Diff(X) onto the algebra Diff(Y). With this notation, if (V, cp, n) is a chart of X, it is clear that the local expression of an operator P E Diff(X) is cp*(PI V).
300
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DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(17.13.8) Let P E Diff(X). For each compact subset K of X, the image under P of 9(X; K) is contained in 9 ( X ; K). If T is a distribution on X, we may therefore consider, for each functionfe 9 ( X ; K), the value (T, P f), and it is clear that the linear mapping ft-+(T, P * f ) of 9 ( X ; K) into C is continuous. Hence we have a distribution 'P * T, defined by the relation
-
for all f E 9 ( X ) (12.15.3). It follows from this definition that Tt-+'P * T is a linear mapping of 9'(X) into itself, and that if the restriction of T to an open set U c Xis zero, then also the restriction of 'P * T to U is zero. In other words, we have Supp('P * T) = Supp(T), and for each open U c X and each T E 9'(X) we have
'(PIU) * (TIU) = ('P * T)I U. If (V, cp, n) is a chart of X for which P has the local expression (17.13.6.1), then, for each function f E 9(cp(V)), (17.13.8.2)
('P'T, f
0
cp) =
C
I4 5 P
(-l)lv'(Dv(avcp(TIV)),f)
having regard to the definition of the derivative of a distribution on an open subset of R" (17.5.5). Remark (17.13.9) Let P : E F be a differential operator. From the local expression (17.13.3.1) and from Leibniz's rule it follows that, for each section f E T(X, E), the mapping ot-+P(of), where u runs through the set of C" scalar-valued
functions on X, is a differential operator from X x C to F ; moreover, if P is of order p at a point x , and if f(x) # 0, then at-+P(uf) is also of order p at x , by virtue of Leibniz's formula. In particular, in order to verify that a differential operator P is of order 0, it is enough to show that P(af) = uP(f) for all u E &(X) and all f E T(X, E). The notion of a real differential operator is defined in exactly the same way, by replacing in (17.13.1) complex vector bundles by real vector bundles, so that T(X, E) and T(X, F) are real Frechet spaces; we have only to replace C by R throughout in the developments of this section. In particular, to say that P E Diff(X) is a real differential operator signifies that, for each (17.13.10)
13 DIFFERENTIAL OPERATORS
301
C" real-valued functionf on X, the function P .f is also real-valued. For each real distribution T on X, 'P * T is then also a real distribution. If E, F are real vector bundles on X and E(,), F(,) their complexifications, then every real differentialoperator P from E to F extends uniquely to a complex differential operator P(,,from E(,) to F(,), because T(X, E,,,) = T(X, E) BRC. (17.13.11) The results of this section are easily extended to continuous linear mappings of I"')(X, E) into T("(X, F) (17.2), where r, s are integers 2 0 . For such an operator one obtains a local expression (17.13.3.1), in which necessarily p S r - s and the A, are assumed only to be of class C'. We leave it to the reader to modify appropriately the other results of this section.
PROBLEMS
1. With the notation of (17.13.1), let P be a linear mapping (not assumed to be continuous) of r(X, E) into r(X, F), satisfying the condition (L). Then P is continuous, and hence is a differential operator (Peetre's theoremLBegin by showing that, for each open subset U of X, the restriction of P to U may be defined as in (17.13.4). This allows us to reduce to the case in which X is an open set in R"and E = X x C", F = X x C"'; we may then assume that N = N" = 1, so that r(X, E) = r ( X , F) = B(X). Then proceed as follows: (1) For each x E X, if f e 8(X) is such that D"f(x) = 0 for each multi-index v, then also Dv(P . f ) ( x )= 0 for all v. (Argue by contradiction, using Section 16.4, Problem 1: There exists a function such that vfis of class C", equal to 0 for all y E X such that 9 - '5 5 Ofor 1 sj 5 n, and equal tofforall y E Xsuch that $ - 5' 2 Ofor 1 5j 5 n. Derive a contradiction by considering P . ( 2 ) A point x e X is said to be regular for P if there exists an integer k,> 0 with the following property: For each function f e B(X) such that D'f(x) = 0 for I v I < k,, we have (P. f ) ( x ) = 0. Show that if U is an open set in X, all points of which are regular for P , then P I U is a differential operator. (Prove that the k , are bounded on each compact K c U ; for this, argue by contradiction and use Problem 2 of Section 16.4. In each relatively compact open subset V of U we may then write ( P . f ) ( x ) = a,(x)D.f(x); prove as in (17.13.3) that the a, are of class Cm.)Deduce that the set
v
(vf).)
IVlSP
S of nonregular points contains no isolated points. (3) Show that the set S is empty. (Prove that there cannot exist a sequence (xk)of distinct points of S, converging to a point x, by using Section 16.4, Problem 2 again.)
2. If P : r(X, E) --f F(X, F) (notation of (17.1 3.1)) is a linear mapping, show that the following conditions are equivalent:
(a) P is a differential operator of order 5 m. tb) For each functionfE cf(X), the linear mapping S"P.(fS)-fP.
is a differential operator of order
5 m - 1.
s
302
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(c) For each family (fi)ldtdm+l of rn we have
+ 1 functions in 8(X),and each S E r(X,E),
where H runs through all subsets of {1,2,
...,rn + 1). (Useinduction on rn.)
3. Let X be a differential manifold, E and F two complex vector bundles over X. Show that the differential operators of order s r from E to F may be identified with the linear X-morphisms of the bundle F a , E ) of jets of global sections of E into the bundle F.
4. Let X be a differential manifold. For each point x E X and each integer r >= 0 and let TP(X) denote the vector space of real distributions of order s r with support contained in {x}. For each real-valued C"-function f on X and each real point-distribution S, E T$), the value <S,, f) E R depends only on the jet J?)(f) of order r. If we denote the value by <S,, J$)(f)>, we have in this way defined a bilinear form on
T$)(X) x P!.)(X), which identifies "$)(X) with the dual of E(X). The space T$)(X) is called the tangent space of order r of X at the point x. The disjoint union T(')(X) of the spaces T$)(X) as x runs through X is canonically endowed with a structure of a vector bundle over X; this bundle is canonically isormorphicto the dual of the bundle F(X)of jets of X into R,and is called the tangent bundle of order r of X. Its rank is
n+
(";')+...+
r+;-1)
if dim(X) = n. The bundle T(')(X) may be canonically identified with the tangent bundle T(X) (whence the terminology). The bundle T('-l)(X) is canonically isomorphic to a subbundle of T(')(W. Define a canonical isomorphism
S,(T(X)) +Fr)(X)/T('-I)(X), where S,(T(X)) is the rth symmetric power (A.17.4) of the bundle T(X). (To a sequence (Zt, .. ,2,)of r vector fields on X corresponds the image in T(r)(X)/T(r-l)(X)of the section O,, 0 Bzz 0 0 B,, of TCr)(X);use (17.14.3)J For each C"-mapping u :X -+Y, define canonically a bundle morphism T(')(u) : T(')(X) +T(')(Y). The diagram
.
is commutative.
14 VECTOR FIELDS AS DIFFERENTIAL OPERATORS
303
14. VECTOR FIELDS AS DIFFERENTIAL OPERATORS
(17.14.1) Let M be a differential manifold,x a point of M, and h, a tangent vector at the point x. The mapping fH (d,J h,) of b ( M ) into R is a distribution of order 1 and support {x}, if h, # 0. For if c = (U, cp, n) is a chart of M such that x E U, and if F =f 'p-' is the local expression off, and v =O,(h,) (16.5.1), then we have (d,. h,) = DF(cp(x)) . v, whence the assertion follows (17.3.1.1). This distribution is denoted by Ohx, and ohx * f = (d,J h,) is called the derivative o f f at x in the direction of the tangent vector h, . Now let X be a C" vectorfield on M. With X we associate canonically a differential operator 8, E Diff(M) by putting 0
for all f E b ( M ) . By (17.13.3), to show that 8, is a differential operator, we may assume that M is an open subset of R",so that T(M) = M x R" and the vector field Xis a mapping XH(X, v(x)) of M into M x R", with v of class C"; the cotangent bundle T(M)* is also identified with M x R",the covector d, f with (x, Df (x)), and we have
The criterion (17.13.3) therefore shows that 8, is a real differential operator of order 4 1, which annihilates the constants. (17.14.2) Conversely, we shall show that every differential operator P E Diff(M) with these properties is of the form 8, for some uniquely determined C" vector field X on M. The hypothesis on P implies (17.13.3) that relative to a chart (V, cp, n) of M the local expression of P is of the type
biDig, where the bi are C" real-valued functions on cp(V). Clearly we may associate with this local expression the vector field X , on V defined by <X,<X),
CP') = bi(cp(x>>
for I 5 i 5 n, and it has to be shown that, if (V, $, n) is another chart of M with the same domain of definition V, then the vector fields X , and X* are the same. Now we have $ = p cp, where p : cp(V) + $(V) is a diffeomorphism; 0
304
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
the assertion then follows from the formulas
for z E q(V) and h a C"-function on $0. We shall often speak of the vector field X as a differential operator, and we shall write X f in place of Ox f when there is no risk of confusion.
-
-
(17.14.2.1) For each C" vectorjeld X on My ex is a real derivation of the algebra &@I) : in other words, f o r f , g E d(M) we have
This is clear from the local expression (17.14.1.2). Conversely, it can be shown that every real derivation of the algebra 8(M) is of the form Ox (Problem 1). (17.14.3) If X , Yare two C" vectorjelds on Mythere exists a unique C" vector Jield on Mydenoted by [ X , Y ] ,such that
ecx,y ,
(17.14.3.1)
=
ex ey - ey ex.
It is enough to show that the differential operator on the right-hand side of (17.14.3.1) is of order 1, since clearly it annihilates constants. Since the question is local, we may assume that M is an open subset of R"and that X, Y are the mappings XH(X, u(x)) and XH(X, v(x)). It follows then from (17.14.1.2) that (0,
*
(4* f > > ( x= ) DZf(x) (v(x),u(xN + Df (4 (Du(x), v(x>>. *
Hence the proposition follows from the symmetry of the bilinear form D2f(x) (8.12.2), and the vector field [ X , Y ] is given by (17.14.3.2)
XH
(x, Dv(x)
*
U(X)
- Du(x)* v(x)).
The vector field [ X , Y ] is called the Lie bracket of the fields X, Y.The following relations are immediately verified : (17.14.3.3)
[ X , XI = 0,
14 VECTOR FIELDS AS DIFFERENTIAL OPERATORS
305
These relations show that the real vector space 6,(M) = I'(T(M)) of C" vector fields on M becomes a real Lie algebra under the bracket operation. (17.14.3.5) Let N be a submanifold of M andlet X , Y be C" vectorfields on M. If X and Y are tangent to N at all points of N (i.e.yif X(x) and Y(x) are in T,(N) for all x E N), then the same is true of [ X , r ] .
Since the question is local, we may assume that M is an open subset of R" and N = M n R", so that T(M) is identified with M x R" and T(N) with N x R" c N x R".With the notation of (17.14.3.2), the hypothesis signifies that for each x E M n R" the last n - m components uj(x), vj(x) of u(x), v(x) are zero (m 1 5 j 5 n). Since the jth component of Dv(x) u(x) is
-
+ fl
(Dvj(x), u(x)) =
DkVj(X)Uk(X),
k= 1
it is zero for j 2 m
+ 1. Similarly the jth
component of Du(x) * v(x) is zero for j 2 m + 1 , and the proposition is proved. (17.14.4) For each C" vector field X on My the mapping YI+ [ X , YJ is a real differential operator of order 5 1 from the tangent bundle T(M) to T(M). If we denote this operator by O x , thenfor any two C" vectorfields Y , 2 on M and any real-valued Cm-functionf on M we have (17.14.4.1) (17.14.4.2)
ex. [ Y ,ZI = [ e x . Y , ZI + [ Y ,ex - ZI, ex . C ~ Y=) (ex
a j - 1
Y +f(eX
. Y).
The first assertion follows from (17.14.3.2), for we may assumethat M is an open subset of R".The formula (17.14.4.1) is simply another way of writing (17.14.3.4). To prove (17.14.4.2) it is enough to show that, for each function 9 E &M),
306
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(17.14.5) For each vector field X on M we have now defined two real differential operators of order 6 1 , both denoted by 8,: one from the trivial line bundle M x R = Tg(M) to itself, and the other from the tangent bundle T(M) = TA(M) to itself. We shall now show that for each pair (r, s) of integers 2 0 there is a unique canonically defined differential operator of order 5 1, again denoted by O x , of the bundle T:(M) of tensors of type (r, s) to itself, such that for any two tensor fields Z E Fi:(M) = r(q:(M)) and Z"E Y:I(M) = rnr(M))we have
In view of (17.14.4.2) this assertion is a consequence of the following more general proposition : (17.14.6) Let X be a C" vector field on M and let Q be a real direrential operator of order 5 1 from Ti(M) = T(M) to itself, such that, for eachfunction f E b(M) and each C" vectorfield Y o n M, we have
Thenfor each pair of integers r 2 0, s 2 0, there exists a unique real dizerential operator D: of order S 1, from T:(M) to itself, such that: (i) 08 = Ox and DA = Q ; (ii) for each C" diflerential 1-form a and each C" vectorfield Y o n M, (17.14.6.2)
8,
*
( Y ,a)
= (Q *
(iii) for any two tensorfields Z' (17.14.6.3)
Y,a)
+ ( Y , Dy
E F::(M),
*
a);
Z"E F$(M),
~:~':=:~*(Z'@Z'Z ) =' )( @ D ~Z :+*Z ' @ ( D $ * Z ) .
Any tensor field Z on M belonging to FZ(M), where r 2 1 (resp.s 2 1) can always be expressed locally, in the domain of definition of a chart of M, as a sum of fields of the form Y @ Z (resp. Z 0 a), where Z EF;-'(M) and Y is avector field (resp. Z E FI- ,(M) and a is a differential form). Hence the uniqueness of the D: follows by induction from (17.14.6.3), once the uniqueness of 0: has been established; but the latter follows immediately from (17.14.6.2), which determines ( Y , 0:. a ) as a function of Y and a. Hence, in order to establish the existence of the 0:satisfying the stated conditions, we may assume that M is an open subset of R" (17.13.4). Let Xibe the constant vector field equal to ei at each point (where e,, .. ., enis the canonical basis
14 VECTOR FIELDS AS DIFFERENTIAL OPERATORS
307
of R")and let cli denote the differential form dt' (1 2 i 2 n). Then Ti(M) is a free b(M)-module having as basis the nr+s tensor fields
xi,Q . . . Q X i , Q c l j l Q ' . . 0 ~ j S .
(17.14.6.4)
If Q . X i =
n
j= 1
a i j X j , then the relations (Xi, a j ) = aij show, by virtue of
(17.14.6.2), that for a forrnfq, where f E b ( M ) , we must have
+2 n
( X i , DY'(fcli))
f a j k ( x k , ai) = a i j e X
k= 1
*f
for all j , and consequently DP-(fcli)=(eX*f)cli-f
n
ujiaj.
j= 1
It is clear that DY,so defined, is a differential operator of order 21 from T(M)* = TY(M) to itself. To define the other D: we may assume that either r 2 1, or that r = 0 and s 2 2. In the first case, by induction it is sufficient to define Dl * UXiQ Z), where Z E Y;-'(M). In conformity with (17.14.6.3) we Put (17.14.6.5)
-
Di * ( f X , 0 Z) = Q ( f X i )0 Z + f X i 0 Di-'
*
Z.
In the second case, by induction it is sufficient to define D: * (Z Qf a i ) , where Z E Y:- l(M), and this time we put (17.14.6.6)
D:
*
(Z @fai) = D:-
. Z 0 ( f a i ) + Z Q DY
-
(foci).
To prove (17.14.6.3), suppose first that r' 2 1, in which case we may assume that Z' = f X i Q Z , with Z' E F::-'(M), and the verification is then a trivial consequence of (17.14.6.5) and (17.14.6.1), using induction. If r' = 0 and r" 2 1, we may assume that Z = f X i 0 Z with Z E F;;-'(M); the multiplication law for mixed tensors (16.18.3.6) then gives Z Q Z = fXi Q Z Q Z, and we use (17.14.6.5) again. Finally, if r' = r" = 0, we use (17.14.6.6), the definition of D:, and the fact that Ox is a derivation. (1 7.14.7) The operator Ox defined on each tensor bundle Ti(M) by applying (17.14.6) (in view of (17.14.4.2)) is called the Lie deriuutioe relative to the
vector field X.
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
308
By induction onp, we obtain from (17.14.5.1) the formula
for any C" vector fields Yl, (17.14.7.2)
-
Ox (a1 @ aZ@
...,Y, on M, and the formula
- - @ a,) ..
for any C" differential 1-forms a,, ., ap on M. Since locally a contravariant (resp. covariant) tensor field of order p is a sum of tensor products ofp vector fields (resp. p differential forms), it follows that, for any permutation c in the symmetric group 6, and any tensor field Z E Y;(M) (resp. Z E Si(M)), we have (17.14.7.3)
O(e,.
z)= ex . (c(z)).
In particular, the operator Ox commutes with the symmetrization and antisymmetrization operators on Yg(M) and S:(M). Since locally a p-vector field (resp. a differential p-form) is obtained by antisymmetrizing a contravariant (resp. covariant) tensor field of order p, it follows that Ox acts on the p-vector fields (resp. the p-forms) of class C". Moreover, it follows from (17.14.7.3) and the definition of the exterior product by antisymmetrization (A.13.2) that if Z is a p-vector field and Z a q-vector field (resp. if a is a p-form and /? a q-form) of class C", then we have (17.14.7.4)
ex . (z' A Z
) = (ex . Z ) A Z" + Z'
A
(ex . z),
(resp.
Next, let Z E Fg(M) = T(Tg(M)) be a contravariant tensor field of order p, and let Z* E F:(M) = r(Ti(M)) be a covariant tensor field of the same order. Then
14 VECTOR FIELDS AS DIFFERENTIAL OPERATORS
309
It is enough to verify this relation locally, when we may assume that Z = Xl €3 X2€3 * * * €3 X p and Z* = al €3 a2 €3 * * * €3 ap ,the X j being vector fields and the aj differential forms; application of (17.14.2.1) then gives the result. By antisymmetrization, it follows that for each differential p-form a and p vector fields XI, . .. X p , (17.14.7.7)
ex
(a,
xl A x, A . P
+
(a, XI
e .
A
A *.*
x,> = (ex
a, X ,
A xj-1 A
[ X , xj] A
A
*
-
*
A
XJ
xj+l A ' * * A
j= 1
xp).
More generally, the relations (1 7.14.5.1) and (1 7.14.6.2) show that the operator r, j 5 s, and Z E g:(M) we have
ex commutes with contractions: for i
ex c;(z)= c;(ex z).
(17.14.7.8)
In particular, we recall that when r = s = 1, a tensor field A E Y:(M) may be identified with an M-morphism of the tangent bundle T(M) into itself, and that if Y is any vector field on M, then c:(A €3 Y) is equal to the vector field A * Y. Hence, from (17.14.5.1) and (17.14.7.8), we deduce that (1 7.14.7.9)
8,
*
(A * Y ) = (0,
*
-
A) * Y + A (0,
*
Y).
Remarks With the notation of (17.14.1), let n : M + N be a C"-mapping of M into a differential manifold N. Then we have
(17.14.8)
(1 7.14.8.1)
by virtue of the definition of the image of a point-distribution (17.7.1) and the formula (1 6.5.8.5) for the differential of a composite function. (17.14.9) With the notation of (17.14.1), let f be a Coo-function,defined in a neighborhood of x , with values in a finite-dimensional real vector space E. Then we may define ohx - f , by replacing (d,f, h,) in (17.14.1) by d , f - h, (16.5.7.1). If (aj)ijj5m is a basis of E and if f = Cfjaj, where the fi are i
real-valued functions, then oh,
.
m
=
(ehx j= 1
-fjlaj
*
310
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
The definition of 8, . f for a C” vector field X on M follows immediately from this. (17.14.10) The formula (17.14.7.5) shows that 8, is a derivation of degree 0 on the anticommutative graded algebra d ( M ) = T(M, A T(M)*) of C“ differential forms on M. Since locally a differential p-form a on M is expressible as a finite sum of forms of the type f dgl A dg2 A * * * A dgp, it follows from (17.14.7.5) that 8,. a is determined by the values of 8, * f and 8, * df for functions f E €(M). Now 8, f is given by (17.14.1.1), and we shall show that
8, * df = d(8, * f).
(17.14.10.1)
For by applying (17.14.7.7) withp
= 1,
we obtain
(8,. df, y > + (df, [ X Yl> = e x * (df, y > = e x * (6, * f ) for any C” vector field Yon M ; and since
(df, [ X , Yl> = Orx, Y] * f= e x * (6, . f ) - 4 . (6, . f ) by virtue of (17.14.3), we have ( 6 , . df, y > = 8, * (6,
.f)
= (d(&
*f),y > ,
which proves (1 7.14.10.1). If X , Yare C“ vector fields on M, then (1 7.14.10.2)
8,
0
i , - i y 0 8, = i r x ,
,,,
both sides being considered as operators on d ( M ) . For since i , is an antiderivation of degree - 1 of d ( M ) (1 6.18.4), the same is true (A.18.7) of the left-hand side of (17.14.10.2). Since the question is local, it follows as above that it is enough to verify that the two sides of (17.14.10.2) take the same values for functions f E €(M) and their differentials df E €,(M); but the operators i y vanish on tP(M), and by (16.18.4.6) we have i , * df = (df, Y ) = 8 , .f.Hence, bearing in mind (17.14.10.1), the verification that the two sides of(17.14.10.2) take the same value for df reduces to (17.14.3). (17.14.11) Although, for a function f E €(M), the value of 8, * f a t a point x E M depends only on the value of X ( x ) , the same is not true for the value of 8, * Y if Y is a vector field. It may be shown that it is not possible to
define “intrinsically” a vector in T,(M) which should be the “derivative” of Y at the point x in the direction of a given tangent vector h, (Problem 2; cf. (1 8.2.14)).
14 VECTOR FIELDS AS DIFFERENTIAL OPERATORS
311
(17.14.12) The results of this section may be generalized without difficulty to vector and tensor fields of class C', where r is an integer 2 0. If X is a vector field of class C', then 8, * Z is defined for tensor fields Z of class C', where s 2 1, and is a tensor field of class Cinf(r,s-l).All the formulas proved in (1 7.14.4)-(I 7.14.7') remain unchanged. (17.14.13) We shall also leave to the reader the task of transposing the definitions and results of this section and the preceding one to the context of complex-analytic manifolds. Here we remark only that differential operators can no longer be defined by a local property; it is necessary to define them by means of their local expressions (relative of course to the charts of a complex-analytic atlas), and C"-functions and sections are replaced everywhere by holomorphic functions and sections. PROBLEMS
1. Let D be a derivation of the ring b(M) of C"-functions on a differential manifold M. Show that there exists a unique Cmvector field X on M such that D = 8,. (First show that the condition (L) of (17.1 3.1) is satisfied. Then either use Problem 1 of Section 17.13, or else give a direct proof with the help of Section 8.14, Problem 7(b).) 2. Show that there exists no linear mapping h,wDh, of the tangent space T,(M) into Hom(fb(M), T,(M)) which is not identically zero and satisfies the following conditions:
(1) for each vector field Y E fA(M) and each functionfE b(M),
Dhx. ( f y ) =f(X)Dhx
'
y + (ohx 'f)y(X);
(2) for each diffeomorphism u of M onto M, DTx(u).hx' (T(u) ' y ) = TdU) ' (Dhx. y). (Use Problem 11 of Section 16.26.)
3. Consider the nz vector fields t'Dk (1 5 j , k 5 n) on R".Show that the vector space they generate is a Lie algebra o and that [e, 01 # g (cf. (19.4.2.2)). 4.
Let X be a C" vector field on a differential manifold M. Show that if X(x) # 0,there exists a chart c of M at the point x such that X i s equal to the vector field XIin the domain of definition of c, where the notation is that of (16.15.4.2).
5.
Let M be a differential manifold and let JE" be a subset of the algebra &)(M), where r is an integer > 0.Show that the subalgebra generated by JE" is dense in b c r ) ( M )if and only if the following three conditions are satisfied: for each x E M there existsf€ 2 such thatf(x) # 0; (2) for each pair of distinct points x , y E M there exists f~ 2 such that f(x) #fO.); (3) for each tangent vector h, # 0 in T(M), there exists f~ % . ? such that Ohx. f# 0. (1)
31 2
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(To show that the conditions are sufficient, consider a compact subset K of M and a relatively compact open neighborhood U of K. If dimx(M) = n, there exist n tangent vectors h, at x and n functions f, E 2' such that Oh, .f,= a,, . We can therefore cover 0 by a finite number of open neighborhoods V, and define on each V, functions gi, ( I 5 j 6 n) belonging to S and such that the mapping x++(gll(x), . . ,gl.(x)) is a homeomorphism of V, onto an open subset of R",which together with its inverse is of class C'. By considering the compact set L which is the complement of the union of the sets (V, n 0) x (V,n 0 ) in fi x 0,show that there exist a finite number of functions hk E 2 ' such that, for each (x, y) E L, we have hr(x) # hk(y) for some index k. Finally, there exist a finite number of functions fi E &' such that, for each x E 0,we have fr(x) # 0 for some index 1. Deduce that there exist N functions F, E 2' such that the mapping CP : x-(F,(x), ... , FN(x)) is a homeomorphism of U onto an open subset @(U) of R" whose closure does not contain 0, and such that both CP and CP -'are of class C'. Hence, for each function f e B(')(M), there exists a function p E &(.)(RN)such that p(0) = 0 andf(x) = p(F,(x), .. , FN(x))for all x E K. Finally, use the Weierstrass approximation theorem.)
.
.
6. Let 0, be the Lie algebra of compactly supported C" vector fields on a differential manifold M.
(a) If a # aCis an ideal in 0 =show , that there exists a point xo E M such that X(xo) = 0 for all X E a. (Argue by contradiction: If X E a is such that X(x) # 0, then for each field Z E (Ye there exists a field Y E such that 2 and [X,Y] coincide in a neighborhood of x, by using Problem 4. Then cover the support K of Z by a finite number of suitably l of class chosen open neighborhoods V, (1 5 i 5 m);take a partition of unity ( j J O s5m C" subordinate to the covering of M formed by M - K and the V1,and by considering successively the vector fields Z1=f1Z9
Zz=f2(Z-Z1),
Z3=h(Z-Z1-Z2),
-..,
prove that 2 can be written in the f o r m z [ X , , Y,],where XiE a and Y rE (3, .) i
(b) Let xo E M be such that X(xo) = 0 for all XEa, and let e = (U, p, n) be any chart of M at xo . Show that the local expressions of the vector fields X Ea relative to the chart c have all their derivatives of all orders zero at the point p(xo). (Suppose that the , X I , where X,are result is false for some field X E a, and consider the Lie brackets [X, the fields associated with the chart c (16.15.4.2)). (c) Deduce from (a) and (b) that the maximal ideals of the Lie algebra Bcare the & o , where for each point xo e M the ideal 3xo consists of all X EBcsuch that, for some chart c = (U, p, n) of M at xo , all derivatives of X of all orders vanish at the point p(xo)(in other words, such that X and the zero vector field have a contact of infinite order at the point xo : cf. Section 16.5, Problem 9). (d) Let X o E Bcand xo E M. Show that [Xo , (Yc] &o = if and only if XO(xo)# 0. (Use Problem 3 to show that the condition is necessary.) (e) Let M, N be two compact differential manifolds. Show that if there exists an isomorphism of the Lie algebra FA(M) onto the Lie algebra Y;(N), then M and N are diffeomorphic and every isomorphism of FA(M) onto YA(N) is of the form
+
XH T(u) . X, where u is a diffeomorphism of M onto N. (First observe that there is a canonical bijective correspondence between the closed subsets of M and the ideals of the Lie
15 EXTERIOR DIFFERENTIAL O F A DIFFERENTIAL p-FORM
313
algebra J t ( M ) : to each ideal a corresponds the set of all x E M such that X(x) = 0 for all X E a. Deduce that an isomorphism v of 9A(M) onto F t ( N ) defines a homeomorphism u of M onto N; then show with the help of (d) that if x,, E M is such that X(xd # 0, then we have (o(X))(u(xo))# 0, and deduce that if f~ b ( M ) , we have f o u-' E &N), by considering the vector fieldfXand its image under v . ) 15. T H E EXTERIOR DIFFERENTIAL O F A DIFFERENTIAL p-FORM
Let M be a dserential manifold. The mappingf t+ df (1 6.20.2) is a real differential operator of order 1 from the trivial line bundle M x R to the cotangent bundle T(M)*, as follows immediately from its local expression. We shall now define, for each p 2 1, a differential operator of order 1 from
(1 7.15.1)
T(M)* to KT(M)*. (1 7.15.2) Let M be a differential manifold. For each integer p 2 0, there exP
ists a unique real differential operator d of order 1 from A T(M)* to satisfying the following conditions: (i)
P+ 1
A T(M)*,
I f a is a C" p-form and p a C" q-form on M, then d(a A
(1 7.15.2.1)
p) = (dcr) A /3
+(-1)"~
A
dp
>= 0) (in other words (A.18.4) d is an antiderivation of the algebra of differential forms on M). (ii) When p = 0, d is the differential f Hdf (1 6.20.2). (iii) For each function f E b ( M ) , we have d(df) = 0.
(p 2 0 q
In the domain of definition V of a chart of M, every differential p-form may be written as the sum of a finite number of forms of the type f d g , A dg, A - - - A dg,, where f and the gk are real-valued functions of class C" on V. Condition (i), applied by induction on p, together with conditions (ii) and (iii), shows that we must have dcfdg,
h dgz A
dgp)= d f A dg,
dg2 A
...
dgp, which proves the uniqueness of d. By virtue of this uniqueness, we need only establish the existence of d in the domain of definition U of a chart (U, cp, n) of M. Then a p-form a is uniquely expressible as (1 7.15.2.2)
a= il
*.*
A
... < i p
ai,ir..ipdcp"
A
A
*.
A
dqiP
314
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
and du is defined unambiguously by
It remains to verify (i) and (iii). As to (iii), if F is the local expression off relative to the chart (U, q, n), we have
df=
n
C DiF
i=1
*
dq'
and therefore
d(df) =
n
n
n
C d ( D i F ) A dq' = C jC Dj(DiF) dq' i=1 =1
A
i=1
dq'.
Since Dj(DiF) = Di(DjF) and dq' A dqj = -dqj A d q i for i # j , and dq' A d q i = 0, (iii) is clear. As to (i), we may by linearity assume that u =f dqil A dq'p, jl = g dqjl A * A d&, so that
--a
--
A
jl =f g dqil
A
A
dqiP A d q j l
A
* - -
A
dqjq
and consequently
- + g - d f ) A d q i l A -..A dq'p A dq"
d(a A jl) = (f d g
= (dfA d q i l A
*
a
+ (-l)P(fdqil
*
d@)
A
... A dq'p) A (dg A d q j l A
h
.-*
A
dq'q
A
dq'q)
... h dq'q)
I\
A
(9 dq"
A
having regard to the anticommutation relations in the exterior algebra (A.13.2.9). Hence (i) is verified and the proof is complete. The differential operator d just defined is called the exterior diferential in the bundle /\T(M)*, the direct sum of the
P
A T(M)*.
Example (17.15.2.4) For any differential manifold M we have defined (16.20.6) the canonical differential 1-form I C on ~ the manifold T(M)*. The 2-form - d ~ is, called the canonical diflerentiul2-form on T(M)*. In the notation of (1 6.20.6),
its local expression is (17.15.3)
c dt' n
i=l
A
dvi.
(i) For each C" diferentialp-form u on M we have
1 5 EXTERIOR DIFFERENTIAL OF A DIFFERENTIAL p-FORM
31 5
d(da) = 0.
(17.15.3.1)
(ii) For each C"-mapping u : M'
+M
and each p-farm a on M , we have
d('u(a)) = 'u(da).
(1 7.15.3.2)
(jii) For each C" vectorfield X o n M and each C" diferentialp-form a on M , we have
8, * (da) = d(0, * a),
(17.15.3.3)
8, . a = ,i
(17.1 5.3.4)
- da + d(i, - a).
(iv) If a is a C" diferential p-form on M and i f Xo , X , , ... , X p arep + 1 C" vectorfields on M , then (1 7.15.3.5)
(da,Xo
A
X IA
* * *
A
x,>= f (-1)'ex;
(a,
x oA
* * *
A
j=O
+ O S i C j $ p ( - 1 y + j ( U , [xi,xj]A xo A
* * *
8, A
x,> A ttj A . - A xp),
A 8 j A
* - -
A
* * *
*
where as usual the circumflex over a symbol means that the symbol is to be omitted. In particular, for each C" diferential 1-form w on M and two vector fields X , Y , we have
(i)
It is enough to verify this locally, in the domain of definition of a chart
of M, and then it follows immediately from (17.1.2.3) and the fact that d(df) = 0. (ii) Again the question is local, so that we may assume that
whence and 'u(da) = 'u(df) A 'u(dg,)
A
A
'u(dg,)
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
316
by virtue of (16.20.9.5). The result therefore follows from the relation d ( f 0 U) = % ( d !(1 6.20.8.2). (iii) To prove (17.15.3.3) we may again assume that c1 is given locally by the expression in (ii). By virtue of (17.14.7.5), we have 8,
- (dfA dg, A + df (8,
* * *
A
*
A
dg,)
dg, A - A dg, + df A dg, A A dg, +
dg,) = (8, A
*
d)'
A
*
*
.*.
A
(8,
'
dg,)
and A - - - A dg,) = (ex .f>dg, A . -.A dg, +f(e, .dg,) A - - - A dg,, + ..- +f d g , A - . A (ex . dg,),
ex
(fdg,
so that the result follows from (17.14.10.1). To prove (17.15.3.4) we observe that, since i, and d are antiderivations, i, d + d ix is a derivation (A.18.7), and by virtue of (17.14.7.5) it is therefore enough to verify that 8, and this derivation take the same values for functions fof class C" and differential I-formsfdg. Since i, * f = 0, the first property is nothing but the definition of 8, f = (df, X ) = i, df. Also, by (17.14.10.1) and (1 7.14.7.5), we have 0
0
-
8,
*
c f d d = (0,
-
-
*f)dg + f @ x
d d = (0, .f)dg + f d @ , g)
and ix
*
dg) = (df, X ) dg - (dg, X ) df = (ex *f)dS ix * ( f d g ) =f (dg7 X > = f ( 4 .g). (&A
- (8,
. S ) df,
Consequently
d(i, * ( f d g ) ) = (0,
g ) df
*
+fd(&
*
9)
and (17.15.3.4) is verified for a =f dg. (iv) The proof is by induction on p. For p = 0, the formula (17.15.3.5) reduces to (df, X) = 8, * f for f E b ( M ) , which is just the definition of the operator 8,. If p > 0, by (1 7.1 4.7.7) we may write
<ex, a, X , A ..-A X J +
-
2 (a,X ,
A
=
ex, -
... A Xi-,A
[KO,X j ]
A
x,>
A xj+l A
.'*
A
x,).
j= 1
Also, by (17.15.3.4), (Ox,
- a,X , - - A X,) A
da, XI A * * - A X,) (d(i,, * a), X, A ... A X,,).
= (i,,
+
*
15 EXTERIOR DIFFERENTIAL OF A DIFFERENTIAL 9-FORM
317
Now by definition we have (1 6.18.4.5)
and on the other hand, by the inductive hypothesis,
Using once again in this last expression the definition of the operator ixo, we obtain immediately the formula (1 7.15.3.5). (17.15.4) For eachp 2 0 and each compact subset K of M, the mapping dis a continuous linear mapping of the Frtchet space 9 J M ; K) into the FrCchet space 9 p + l ( M ;K), and therefore by transposition (12.15.4) defines a linear operator of the space 9L+l(M; K) of (p 1)-currents into the space 9L(M; K) ofp-currents on M. We denote this operator by THCT; the current is called the boundary of the current T. Hence, for each compactly supported differential p-form a on M and each (p + 1)-current T on M, we have by definition
+
n
(17.15.4.1)
If n : M + M’ is a proper mapping of class C‘ ( r 2 l), then
For if a‘ is any differential p-form on M’ with compact support, we have
A similar argument shows that (17.15.4.3)
31 8
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
Examples (17.1 5.5) Let X be an oriented pure manifold of dimension n. If n : X + X' is = 0. To prove this it is a proper mapping of class Cr (r 2 I), then C(n(TFPx)) enough to show that, for each C" differential (n - ])-form a on X with compact support, we have
b
(17.15.5.1)
da = 0.
Let K be a compact neighborhood of Supp(a). Then K can be covered by a finite number of relatively compact open sets Uj (1 s j 5 r) such that each Uj is the domain of definition of a chart (Uj, 'pi,n) of X. There exists a family of C"-mappings of X into [0, 11 such that Supp(hj) c Uj and r
1hj(x) = 1 for all X E K(16.4.2).
We then have a =
r
h ja , so that it is
j= 1
j= 1
enough to show thatjx d(hja) = 0 for eachj; hence we may assume (by virtue of (17.15.3.2) and (16.24.5.1)) that X is an open subset of R", and then we can write A
fl
a=cfidt' A . - - A ~ ~ ~ A . . . A @ " so that
j= 1
Hence we are reduced to proving that iffis any C"-function on R" with compact support, then
D j f ( x )dt' dt2* * * dT" = 0, and this follows immediately from the Lebesgue-Fubini theorem. This result justifies the terminology " n-chain element without boundary " introduced in (17.5.2). (17.15.6) With the hypotheses of the elementary version of Stokes' theorem A dt", then the formula (16.24.11.1) (16.24.11), if we put a = dt2 A dt3 A may be written in the form (1 7.15.6.1)
15 EXTERIOR DIFFERENTIAL OF A DIFFERENTIAL p-FORM
319
for any function f E &(U). Now, given any two functions fi E 9(E,) and f, E g(E,), there exists a function f E 9(U) whose restrictions to E, and E, are, respectively,f, and f, . To see this, we may assume that U = I x V, where V is open in R”-’ and 1 is an open interval in R. If g, (resp. 9,) is a C“mapping of R into [0, 11 which is equal to 1 at the point a (resp. b) and vanishes outside a neighborhood W, (resp. W,) of a (resp. b) in I, where
a,
W, n W, = then we define f ( r , y ) = gl(t)f,(y) for t E W, ; f ( t , y ) = g2(t)fZ(y) for t E Wz ; and f(t, y ) = 0 otherwise. The function f so defined clearly satisfies the required conditions. Since by hypothesis the form a is nonzero almost everywhere in E and in E, , it follows therefore that if a,, az are compactly supported C“ dherential (n - l)-forms on E,, Eb, respectively, then there existsf E 9(U) such that a,,or, are induced by for on E,, E,, respectively. Hence from the formula (17.15.6.1) we deduce that (with the abuse of notation of (17.5.2)) (17.15.6.2)
Cua, b
= Eb
- Ea.
In Chapter XXlV we shall generalize this result by defining a large class of “open n-chain elements ” whose boundaries can be explicitly determined. (17.15.7) Now let M be a complex-analytic manifold. Starting from the differential operator f ~ d from f the trivial bundle M x C to the cotangent bundle T(M)*, we can repeat the construction of the exterior differential in the bundle A T(M)*, by replacing C” real-valued functions by holomorphic functions throughout; but, furthermore, if M, = MI, is the differential manifold underlying M, then the complex-analytic structure on M enables us to define canonically new differential operators on the bundle A (T(Mo)*)(c), the exterior algebra of the complexzjication (T(Mo)*)(c) of the cotangent bundle of M, . For there is a canonically defined endomorphism ‘J (16.20.16) o i the complex vector bundle (T(Mo)*)tc), with square - I, such that
p’ = + ( I - i . ‘ J ) ,
p ” = $ ( I + i - ‘1)
are projections on E = (T(Mo)*)(c) whose sum is I and whose images may be canonically identified with T(M)* and the “conjugate ” bundle T(M)* (whose sections are, respectively, generated locally by the differentials df of holomorphic functions, and the differentials d’of complex conjugates of holomorphic functions). For each complex-valued C“-function f on M, , we put (17.1 5.7.1)
d’f =p ’
0
df,
d ” f = p ” o’ d
320
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
so that d ' is a differential operator of order 1 from (T(Mo)*)(c)to T(M)*, and d" a differential operator of order 1 from (T(Mo)*)cc, to T(M)*, such that d = d' d" and such that the relation d"f= 0 characterizes the holomorphic functions on M, and the relation d'f= 0 characterizes their complex conjugates. From the decomposition of E = (T(Mo)*)cc, as the direct sum of
+
T(M)* and T(M)*, we have a canonical decomposition of sum of p 1 vector subbundles
+
P
A E as the direct
(17.1 5.7.2)
+
(0 5 r 5 p , r s = p). I f (U, rp, n) is a chart of M, and rp = the sections of k s E over U form a free B(U)-module with a basis consisting of the sections (1 7.15.7.3)
(dpj'A
9
*
A
d@)
A
(d@ k
l * ~ *.
A
dqkk,)
(jl c j , , k , < * * *
*
-
a
(17.15.7.4)
d' : R,'E + A'+'*SE,
d" : Ks SE
+
A',
S+ 1
E,
which for r = s = 0 are the operators already denoted by d' and d" and which have the following properties: (i) d ' o d ' = d " o d " = O , (ii) d'(a A d "(a A
d'od"+d"od'=O,
p) = (d'a) A /3 + (- I)"a A (d'p), p) = (d "a) A f l + (- l)pa A (d"P)
d=d'+d".
for a E k"E and j3 E K'."'E, with r + s = p . (iii) d"(d'f) = 0 and d ' (d " f) = 0 for each holomorphicfunctionfon M. The proof is analogous to that of (17.15.2), by reducing to the case where E is the product of a function and a form of type (17.15.7.3); the form a E Ks" the only point to be noticed is that iffis a complex-valued C"-function on an open subset of C", so that
we have
which follows from the fact that the operators a/azj and 8/aZj commute.
15 EXTERIOR DIFFERENTIAL OF A DIFFERENTIAL p-FORM
321
(17.15.8) The definition of the exterior differential du can be easily extended to the case of a vector-valued dixerentialp-form a (16.20.15), with values in a
finite-dimensional vector space F. If a = of F, then we define da =
4 i= 1
1u ie,, where (ei)l 4
i s qis
a basis
i=l
(dui)e,,and it is immediate that this definition
is independent of the choice of basis of F. Again we have d(da) = 0. If o is a vector-valued C" differential 2-form and X, Y are two C" vector fields on M, the formula which replaces (17.15.3.6) is
PROBLEMS 1. Let H be a vector subspace of F:(M) which is stable under all diffeomorphisms of M onto itself. Let L : H -+Y:(M) be a linear mapping such that L(u(s)) = u(L(s))for all diffeomorphisms u : M + M and all s E H. Show that, if I # s, then L is necessarily a differentia1 operator. (By virtue of Problem 1 of Section 17.13, the question reduces to verifying the condition (L)of (17.13.1). Observe that if SI U = 0 for some open neighborhood U of x , there exists a diffeomorphism u of M onto itself such that u is the identity on M - U, u(x) = x and TJu) is a hornothety whose ratio can be arbitrarily prescribed (Section 16.26, Problem 1 l).) Give an example in which r = s = 0 and L is not a differential operator (cf. (16.24.2)). 2. Let a be a C" differentialp-fom on M.At a point xo E M a system of local coordinates ((pl)lg~Snis said to be priuileged relative to a if, in some neighborhood of x o , the form a can be written either as c dv' Adq2 A . . * Ad@', where c is a constant, or as qP+' dq' A dy2A ' ' ' A d v . Show that for each point x o E M there exists a neighborhood U of xo in which a is the sum of a finite number ofp-forms, for each of which there exists a privileged local coordinate system in U (depending o n the p-form). (Observe that, for each C"-function g defined in a neighborhood of xo and each local coordinate system (I$) a t x o , there exists e > 0 such that (TI,.
. ., v p , q'+'
+ Eg, TP+*, ...,
(p")
is a local coordinate system.)
3. Let M be a connected differential manifold of dimension n, and let P be a differential P
< +
operator from A T(M)* to A T(M)* (0 < p , q n). If P . (u(a)) = u ( P . a) for all diffeomorphisms u of M onto M, show that (1) if q # p, p 1, then P = 0; (2) if q = p , t h e n P . a = ca,where c is a constant; (3) ifq = p 1 , then P . a = c . da, where c is a constant. (Use (17.1 3.3). Problem 11 of Section 16.26, and (16.26.8) to reduce to thecase where M = R" and to prove that for x = 0 and all p-forms a on R" we have (P. a)(O) = 0 q
+
322
XVll DISTRIBUTIONS A N D DIFFERENTIAL OPERATORS
+
ifq # p , p 1; ( P a a)(O)= ca(0) i f q = p ; and (P * a)(O)= c da(0) i f q = p + 1 . Then using Problem 2, reduce to the case where a is one of the two forms and apply the condition on P by taking u to be a translation or a linear mapping given by a diagonal matrix.) 4.
(a) Let a = f -d t l Adt2 A .. A d p be a p-form on R",with p 6 n - 1 and f a C"function with support contained in the ball B : llxll< 1, and such that f(x) = @'+ in the ball rB : I(x11 < r < 1. Let u be a diffeomorphism of R"onto itself which is equal to a homothety in a neighborhood of the support off, is the identity on Rn- B and is such that u(Supp0) C rB (Section 16.26, Problem 11). Let u(a) = g . d t l A d e A .. .Ad@, where Supp(g) C rB. Show that, for all sufficiently small 6) 0, the mapping u :( e l , .. . ,P)H(f'.
...,5", 5"" + &&),
...,4")
is a diffeomorphism of R" onto itself (Section 16.12, Problem 1 ) . We have then u(a) = a EU(GL), so that a = U-'U(LZ/E) - u - ~ ( G L / E ) . (b) Let M be a compact differential manifold of dimension n. Show that, for each p 5 n 1 , every C" differentialp-form a on M is the sum of a finite number ofp-forms p,, such that for each p, there exists a point x, E M and a local coordinate system (@) at x j relative to which is equal to V+ldq+ A * * . AdqP in a neighborhood of x,. (UseProblem 2 and remark that we may write 1 = A p + l + (1 - AP+').) (c) If M is a compact differential manifold of dimension n and if p < n, show that if a
+
-
linear form L on A T(M)* satisfies L(u(a))= @(a)) for all diffeomorphisrns u of M onto itself, then L = 0. (Use (a) and (b).)
5.
On a pure differential manifold M of dimension n, let X b e a vector field, a a differential n-form , and f a real-valued function (all of class C"). Show that
dfA (ix . OC) = (Ox . f ) a . 6.
Let M 1 , M I be two differential manifolds, f a real-valued function of class C" on M l x M I . For each point x = (xi,xz) E M 1 x M s , the tangent space Tx(M1x M I ) may be canonically identified with Tx1(Ml)x TA2(MZ).L.etd$"f(resp. di2)ndenote the covector (hl, h2)- (resp. (hl, hd- < h 2 ,d , , f ( x l , . )>I. Then d"'f: xHd$')f (resp. d @ ' f:x++d$')l) are two C" differential forms on MI x M,, called the partial differentials off. We have df = d(l'f+ dC2'f. With the notation of Section 16.25, Problem 14, show that one can define uniquely two differential operators d(l),d ( * )of order 1:
d"':
&r.t+&r+l.sy
d'":
gr.s+g,,s+l,
which for r = s = 0 are the operators defined above, and which have the following properties: d(1) d(1)= d(2), d(2)= 0, d(1) d(2)+ d(Z) d(1)= 0
d = d(1)+ &I),
+ + &,.$ and ?/ E a,,..,, , with p = r + s.
d")(aAR = (dcl)~)A\B (-l)paA(d")m, d("(a A @) = (d%) A p (- I)% A (d(')P)
for a G
15 EXTERIOR DIFFERENTIAL OF A DIFFERENTIAL p-FORM
7. For the Duac p-current
E,,
calculate g. E., ,'ix . E., and 'Ox
323
. E., .
Let ( e h sn be the canonical basis of R",and let V', V" be any neighborhoods of el and -el, respectively. Show that there exist two n-forms u', U" of class C" on Rn such that the supports of u', u" are contained in V', V respectively, and such that there exists an (n - 1)-form u of class C", with compact support, satisfying do = U' - u", and
8. (a)
finally such that
s s u' =
u"> 0. (Take u' to be
Me1 - 1)h((e2)'
+ .. +
df' A dtzA
*
... A&,
where h is a nonnegative C"-function on R with arbitrarily small support, and such that h(0)> 0. Define U" similarly.) (b) Let M, N be oriented connected differential manifolds of the same dimension n, and let f be a proper (17.3.7) C"-mapping of M into N, such that the inverse image ' f ( u o ) of an n-form uo belonging to the orientation of N is 2 0 at all points of M (relative to the orientation of M). If f i s not surjective, show that all points of M are critical points o f f (16.23). (The set f(M) is closed in N. Suppose that there exists a point y1 E N -f(M) and a point x, E M which is not critical for f, and let y z = f(.x2)E ~ ( M ) Then . there exists an open subset U C N, diffeomorphic to R" and containingyl and y, (16.26.9). Use (a) to show that there exist two n-forms u', u" on N and an (n - 1)-form u such that du = u' - u", the forms u', u" and u being of class C" and compactly supported, and such that
'f(u")> 0, contradicting
(17.15.5.1).)
M be a pure differential manifold of dimension n. Let = - d K M be the canonical 2-form (17.15.2.4) on the cotangent bundle N = T(M)*. For each z E N, the mapping h,wi(hz). n(z) is an N-isomorphism of the tangent bundle T(N) onto the cotangent bundle T(N)*. For each C" differential 1-form a on N, there exists therefore a unique C" vector field X, such that a = ix,. 0. (a) If a,,!? are C" differential 1-forms on N, their Poisson bracket {a,,!?}is defined to be the differential 1-form - icx,,xgl . 0.Show that
9. Let
{a,/?I = -Ox,
. /? + Ox, .a + d(ir, . (ix, . a)).
For any three 1-forms a,p, y on N, show that { a ,{P, YH
+ {P, {Y,4 )+ {Y,I@,PI} = 0.
IfJ g are real-valued C"-functions on M,their homogeneous Poisson bracket is defined to be the function
{ f , s= } - i X d / .( j X d g . - -'.%'d/
'
= exdB
'
f.
Show that d{f,g} = {dJ dg} and that
{f,{g,hH + 19, {h,fH + v, {J9 ) )= 0, { J gh}= Nf,d + df,h3 for any three functions f,9, h.
324
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
With the notation of (16.20.6), if the local expression of
+
n
i s c dt' A d v i , then I s 1
((ti),
(b) Replace n by n 1 and denote a point of N = T(M)* by (7,)) (0 $ i 5 n), where M is an open set in R"+'.Given any function F(z, x', ... ,xn,y , , ...,).y defined on M x R",construct the function
defined on the open subset U : vo # 0 in N. If G is another function on M x R"and g the corresponding function on U, and if F, G are both of class C", then the function TO{ f,g} corresponds to the function
which is denoted by {F, G} if there is no risk of ambiguity, and is called the nonhornogeneous Poisson bracker of F and G. If w is the differential I-form
then we have
16. C O N N E C T I O N S IN A VECTOR BUNDLE
(17.16.1) Let G be a Lie group. We have seen (16.15.6) that for each pair of points x,y of G there is a well-determined isomorphism hx-yx-' . h, of the vector space T,(G) onto T,,(G), which depends only on the points x, y ; but the existence of such an isomorphism depends essentially on the supplementary Liegroup structure on the differentiable manifold G. In the absence of such a structure, there is no canonicalisomorphism of the tangent space at a point x of a pure manifold M onto the tangent space at another point y, i.e., no isomorphism which is determined uniquely by the manifold structure of M and the two points x, y. The notion of a connection in a vector bundle E over M is the mathematical expression of the idea of defining, for each point x E M, a procedure for providing an isomorphism of Ex onto E,, for points y "infinitely near" to x . Since the question is local, we shall consider first the case of an open subset U of R" and a trivial bundle E = U x RP.Let x be a point of U, and
16 CONNECTIONS IN A VECTOR BUNDLE
suppose that for each vector h rs R" such that x isomorphism (17.1 6.1 .I)
(x, U) H ( x
+ h, F(h)-'
*
325
+ h E U we have a linear U)
+
of the fiber Ex= { x } x Rp onto the fiber Ex+* ={x h} x RP,so that h H F(h) is a mapping of a neighborhood V of 0 in R" such that x V c U, into the vector space Y ( R p )of all endomorphisms of RP (a space which is isomorphic to Rp2).Suppose that F(0) = I,, and that F is of class C" in V, so that the mapping (17.16.1.2)
(h, U ) H ( X
+
+ h, F(h)-' - U)
of V x RPinto U x RPis indefinitely differentiable. Its derivative at the point (0, u) is the value at the point (x, u) E Exof the linear isomorphism "infinitely near" to the identity which we wish to consider; this value is, by virtue of (8.1.5), (8.9.1), and (8.3.2), (k, v ) H ( ~ ,v - (DF(0) * k) * u),
(1 7.1 6.1.3)
a linear mapping of R" x RPinto itself; DF(0) belongs to S(R"; S(Rp)), and therefore (5.7.8) the mapping (k, u) H (DF(0) * k) . u is a bilinear mapping of R" x RP into RP,which we shall denote by
Conversely, if we prescribe arbitrarily such a bilinear mapping and put
+
F(h) := I,, T,(h, .), then F(h) is an automorphism of RPfor all sufficiently small h (8.3.2), such that (DF(0) * k) * u = Tx(k, u).
(17.1 6.2) Let us now interpret these remarks in terms which are independent of the trivialization of E chosen. Since E = U x Rp and since (x, u) is a point
of the fiber Ex,the tangent space T(x,u)(E)may be identified with
TAU) x T,(RP), and therefore with ({x} x R")x ({u} x Rp).To a pair of vectors ( x , k) E Tx(M) and ( x , u) E Ex we associate, by means of (17.16.1.4), the vector (17.1 6.2.1)
CAX, k), (x, u)) = ((x, u), (k, - u k , UN)
326
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
Moreover, T(E) is a vector bundle over E (16.15.4); if oE: T(E) + E is its projection, then we have oE((x,u), (k, v)) = (x, u). Finally, ((x, k), (x, u)) may be identified with the vector (x, (k, u)) of the fiber over x of the vector bundle T(U) @ E over U (16.16.1). Hence the mapping C, defined by (17.16.2.1) satisfies the following conditions:
the mapping
is a linear mapping of T,(U) into T(,, u)(E);and the mapping
is a linear mapping of Exinto T(n)-'(x, k). It is now easy to give the definition of a connection (or linear connection) in an arbitrary vector bundle E over a differential manifold M. We have merely to replace in the definitions and conditions of (1 7.1 6.2) the vector (x, u) by a vector u, E Ex,and the tangent vector (x, k) by a vector k, E Tx(M). We denote again by n : E + M, 0, : T(M) + M, and oE : T(E) E the canonical projections, and we observe that T(E) is a vector bundle over T(M) with projection T(n) (1 6.1 5.7). Finally, the composite mappings n oE and oY 0 T(n) of T(E) into M are equal, and if w denotes this mapping, then the triple (T(E), M, w ) is again aFbration, but not a vector bundle (16.15.7). Having said this, a connection (or linear connection)in the vector bundle E is defined to be an M-morphism (17.16.3)
0
(17.16.3.1)
C : T(M) @ E + T(E)
of fiber bundles over M (1 6.12.1), having the following properties:
the mapping (17.16.3.3)
k,
H C,(k
9
u3
16 CONNECTIONS IN A VECTOR BUNDLE
327
of T,(M) into Tux(E)is linear; the mapping
is a linear mapping of Ex into (T(E)), ,the fiber over k, of T(E) considered as a vector bundle over T(M) with projection T(n) (16.15.7). In particular, for each scalar c E R, we have
where m, is the mapping u, H c * u, of E into itself. We remark that the conditions (17.16.3.2) imply that the linear mappings k, H C,(k, , u,) and u, HC,( k, , u,) are injective. For each u, E Ex, the subspace of Tux(E)which is the image of T,(M) under the mapping k,
H C,(k
3
4)
is supplementary to the subspace TUx(n)-'(O,) formed by the vertical tangent vectors to E at the point u, (16.12.1). This supplementary subspace is sometimes called the space of horizontal tangent vectors to E at the point u,, relative to the connection C. For each vector field X on M, the horizontal lifting (relative to C) of X is a vector field on E, denoted by relc(X), and defined by (1 7.16.3.6)
reMX)(u,)
= CX(X(4, UX).
(17.16.4) Let (U, rp, n) be a chart of M such that E is trivializable over U, and let (n-'(U), 6, n + p ) be a corresponding3bered chart of E (16.15.1), with 6(n-'(U)) = q ( U ) x Rp. Then, for each point x E q(U)and each vector (k, u) E R" x RP,we have (1 7.16.4.1)
cq-l(x)(Tx(v-l). (x, k), O-'(x, u)) = q,, u)(e) . ((x, u), (k, - u k , u)),
where (k, u) H r,(k, u) is a bilinear mapping of R" x RPinto RP (1 7.16.1); the mapping (k, u ) H ( ~ ,-T,(k, u)) is called the local expression of C corresponding to the fibered chart. More explicitly, this mapping may be written as
(k',
..., k " , u l , ..., u P ) w ( k l ,..., k", - r i ( k , u ) , . . . ,
(x E rp(U)), where each ri is a bilinear form (1 7.16.4.2)
Tt(k,
U)
=
1l-L(x)khu', h, 1
the
being C"-functions on rp(U).
-rXp(k,u))
328
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
Now consider another fibered chart, corresponding to a chart (U, rp’, n) defined on the same open set U. The transition diffeomorphism $ : rp(U) x Rp +. rp’(U) x RP is of the form (1 7.16.4.3)
( X l , . . . , 2,U 1 ,. . . ,UP) H(@(x), A(x) * u),
where @(x) = (Q1(x), . .., @,,(x)) and A(x) = (uij(x)) is a square matrix of order p, so that the derivative D$ is the linear mapping
(k’, . ..,k”, d , . . . , v p ) ~ ( k ‘ ’ , . . . , k‘”, of‘, ..., dP) given by
2 Dh@,g kh, dY= uYld + 2 DhuYI khU’ I I
k” = (I7.16.4.4)
h
*
h,
((8.9.1) and (8.9.2)). Now let (k”,
. . ., k”’, u”, .. ., U ’ ~ ) H ( ~ ’.’.., , k‘”, -rk?(k’, u’), .. ., -T!$(k;
u’))
(x’ E rp’(U)) be the local expression of the connection C relative to the second chart, and let
(17.1 6.4.5)
T!Jk’,
u’) =
2 rG(X’)k’’U‘’.
8. Y
Also put @ - l ( x ’ ) = (ai1(X’), . . . , ai“(X’)),
Then we have from above, for x’
A-’(@-’(x’)) = (Hay(X’)).
-
= @(x) and u‘ = A(x) u,
also the first of the formulas (1 7.1 6.4.4) gives kh= 1 D, aih(x’) * k‘8 8
and we have UI =
c HIY(X’)U’Y, Y
so that finally we obtain the following expression for the T;(x‘):
16 CONNECTIONS IN A VECTOR BUNDLE
329
We see from this that, contrary to what one might have expected from (17.16.1), the presence of a connection on M does not enable us to define, for all x E M, a bilinear mapping rxof T,(M) x Ex into Ex; the mapping rxcor-
responding to a trivialization of E depends on this trivialization. In particular, to say that a connection is “zero” has no meaning, because all the ril can vanish without the l?g vanishing.
Since, for each vector (k,, u,) in T(M) 8 E, the value C,(k,, u,) of a connection belongs to the tangent space TUx(E),it follows that the sum of the values of two connections at the point (kx, u,) is defined, as is also the product of C,(k, , u,) by a scalar. We may therefore form linear combinations Cj of connections in the bundle E, the coefficients being real-valued (17.16.5)
cfi
fi
J
(2%’-functionson M; but in general the M-morphisms so obtained are not connections, by reason of the first condition (1 7.16.3.2). Nevertheless,there are two important particular cases. In the first place, if C and C’ are two connections in E, their difference C‘ - C is no longer a connection, but is an M-morphism (17.16.6)
(1 7.16.6.1)
A : T(M) @ E + T(E)
such that (1 7.1 6.6.2)
T(n) * A,@, , u,) = O x , o,(A,(k,,
u,)) = U,
.
The first of these relations shows that A,(k,, ux) is an element of Tux(E,), the tangent space to the3ber Ex at the point u, , identified with the subspace of *‘ vertical ” tangent vectors in Tux(E). Furthermore, k, I+ A,( k, , u,) is a linear mapping of Tx(M) into TUx(Ex),and u, H A,(kx, u,) a linear mapping of Ex into (T(E)),x, the fiber of T(E) over the point 0, E T,(M) relative to the fibration (T(E), T(M), T(n)), which is isomorphic to the tangent bundle T(E,), identified with Ex x Ex. Also we have a canonical isomorphism T,_ : Tux(E,) --* Ex (1 6.5.2) which, modulo the preceding identifications, is the restrictiop to Tux(E,) of the second projection. It follows that, if we put
330
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
B, is a bilinear mapping of T,(M) x Ex into Ex.In other words, we have defined a bilinear M-morphism B : T(M) 0 E + E, which, by abuse of language, is called the diference of the connections C' and C. Relative to a local trivialization of E in which C and C' correspond, respectively, to the bilinear functions r and r' (17.16.1), the morphism B corresponds to -r'+ r. Conversely, if B : T(M) 0 E + E is any bilinear M-morphism, then if we put A,(k,,
u 3 = Gx1(Bx(kx9UX)),
C f A is a connection, for every connection C in E.
(17.1 6.7) Let (U, ,cp, , n) be afamily of charts of M such that E is trivializable over each U, , and such that the family (U,) is locallyfinite. For each a, let C, be a connection in the vector bundle n-'(U,), and let vh> be a Cw-partitionof unity subordinate to (U,) (16.4.1). Then cf.C, is a connection in E (with the understanding that we replace &(x)(C,),(k, space TUx(E)for each point x 4 U,).
, u,) by the origin of the tangent
Since for each point x E M there is a neighborhood of x which meets only a finite number of the U,, and since c & ( x ) = 1, the only point that needs to be checked is that the functionsfa C, are Cw-mappingsof the whole of T(M) 8 E into E. If x is a frontier point of some U, ,there is a neighborhood V of x in M which does not meet the support of fa, and therefore, for each point (k,,, u,) of T(M) 0 E lying over a point y E V, by definition f.( y)(C,),,(k, , u,) is the origin of the vector space TUy(E).The assertion now follows by taking a trivialization of T(E) (considered as a bundle over M) in a neighborhood of x. In particular : (1 7.16.8)
There exists a connection in any vector bundle.
Choose a family of charts of M having the properties stated in (17.6.7), and define each connection C, by taking a particular trivialization of n-'(U,) and taking the corresponding mapping (17.16.1.4) to be, e.g., zero for all x E (PU(U3.
If (f,g) is an isomorphism of a vector bundle E over M onto a vector bundle E' over M', then every connection C in E is transported by (A g) to a connection C' in E', such that
(17.1 6.9)
17 DIFFERENTIAL OPERATORS ASSOCIATED WITH A CONNECTION
331
17. DIFFERENTIAL OPERATORS ASSOCIATED WITH A C O N N E C T I O N
(17.17.1) We take up again the situation considered in (17.16.1). Let Z be an open subset of some Rq,and let G be a C"-mapping of Z into the bundle E, so that we may write G(z) = (f(z), g(z)), wheref(resp. g) is a C"-mapping of Z into U (resp. into Rp). Let W be a neighborhood of 0 in R4 such that z W c Z, and for each w E W put h(w) =f(z + w) -f(z). Since the point G(z + w) lies in the fiber Ef(,)+,,(+, we may consider the point
+
(f(z), F(h(w)) . g(z + w)), which lies in the fiber EfCz,,and we are thus led to take as "derivative" of G the derivative of the mapping wHF(h(w))
*
g(z
+
W)
at the point w = 0; bearing in mind that F(0) = 1R P , and using (8.1.4) and (8.2.1), this gives us the following linear mapping of Rqinto RP: (17.17.1 .I)
WH
Dg(z) * w
+ (DF(0)
*
(Df (z) * w)) * g(z).
Since DG(z) . w = (Df(z) * w, Dg(z) * w), the right-hand side of (17.1 7.1 .I) can also be written as
-
DG(4 . w - (DfW w, -rf(z)(Df(z) * w, g(z)>) in which appears the value Cf(z)(Df(z)* w, g(z)) of the connection C . (17.17.2) We shall now give an intrinsic definition of the covariant derivative relative to a connection C in a vector bundle E over M. Let N be a differential manifold and let G be a Cr-mapping (r 1 1) of N into E. It follows from (17.16.3.2) that, for each z E N and each tangent vector h, E Tz(N),the vector Tz(G)
. hz - Cn(G(z))(Tz(B G, . hz
9
G(z))
belongs to the tangent space TG(Z)(En(G(z))) to the fiber through the point G(z:)of the bundle E. Since this fiber is a vector space, we may apply the canonical translation zG(z)(16.5.2) to the above vector: the vector so obtained in tliefiber Es(G(=))
is called the covariant derivative of G at the point z (relative to the connection C ) in the direction of the tangent vector h, .
332
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
When M and N are open sets in R" and R" (a situation to which we can always reduce by a suitable choice of charts), G is a function of the form ZH(~(Z), g(z)) and the connection C is given by (17.16.2.1). We have then, with h, = (2, h),
or, expressing everything in terms of coordinates as in (1 7.16.4),
for the ith component of V h z G in Rp. It is clear that h, HV h z * G is a linear mapping of T,(N) into If GI, G, are two mappings of class C' (r 2 1) of N into&, such that A Gl = II G2 (in other words, two liftings to E of the same C'-mapping of N into M), then we have 0
0
For every scalar function CJ of class C' on N, the mapping z Ho(z)G(z) is a C'-mapping of N into E which we denote by aG; clearly 7t 0 (aG) = A G. Bearing in mind the definition of (17.14.9), it is easily seen that 0
by reducing to the case where the covariant derivative is given by (17.17.2.2). Finally, if u : N1 H N is a mapping of class C', then for each point z, E N, and each tangent vector h,, E TzI(Nl),it follows immediately from the definitions that (1 7.17.2.6)
where z
= u(zl)
VhZl
*
(G 0 U ) = Vhz . G,
and h, = T,,(u) . h,, .
Remark By virtue of (17.17.2.1), the relation v h z * G = 0 signifies that the tangent vector T,(G) . h, is horizontal (17.16.3) at the point G(z).
(17.17.2.7)
(17.17.3) Having defined the covariant derivative of the mapping G : N + E at a point z E N in the direction of a tangent vector h, E T,(N), it is now easy
18 CONNECTIONS ON A DIFFERENTIAL MANIFOLD
333
to define the covariant derivative of G (relative to C ) in the direction of a vectorjield 2 E T;(N);this is the mapping V, * G of N into E which at each point z E N has the value VZ(=). G. The results of (17.17.2) give rise to the formulas
for two vector fields Z,, 2, on N;
V,
(17.1 7.3.2)
*
G = c(VZ * G )
for every scalar function cr of class C' on N;
for two liftings G,, G, to E of the same mappingf: N-, M of class C'; (17.1 7.3.4)
v, - ( c r ~=) (e, - c r ) +~ cr(~, - G)
for every scalar function cr of class C' on N. In the particular case where N = M and n G = 1, (that is, where G is a section of E), we see that for each C" vector field Xon M, the operator V x is a real differential operator of order 6 1 from E to E. 0
18. CONNECTIONS ON A DIFFERENTIAL MANIFOLD
(17.18.1) Given a differential manifold M, a connection (or linear connection) on M is (by abuse of language) a connection in the tangent vector bundleT(M), that is to say an M-morphism of T(M) @T(M) into T(T(M)) satisfying the conditions of (17.1 6.3), with E replaced by T(M). Given such a connection C we define, for each vector h,ET,(M) and each vector field Y on M, the covariant derivative (relative to C ) VhX* Y of Y at the point x in the direction of h, (17.1 7.2.1). Hence, for each vector field X on M, we have a differential operator YHV, . Y from T(M) to T(M) (17.17.3), with the following properties: (17.1 8.1 .I)
v,,, v,
(17.1 8.1.2) (17.1 8.1.3)
V,.(Y,
*
Y = v,,
Y + v, . Y ,
*
Y = O(V,
*
+ Y,)=Vx-
Y),
Y , +VX' Y , ,
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
334
vx - (or)= (ex
(17.18.1.4)
Y
+ o(vx- Y ) ,
where Q is any C" scalar function on M. In particular, if U is an open subset of M over which T(M) is trivializable, so that the d(U)-module FA(U) of C" vector fields on U admits a basis Y,, .. . , Y,,,then the above formulas show that a knowledge of the vector fields VYi. Y j on U for 1 S i,j 6 n determines Vx Y completely for any two vector fields ,'A Y E FA(U). In particular, if (U, cp, n) is a chart of M, and if (Xi)lsi6nis the basis of FA(U) over B(U) associated with this chart (16.15.4.2), then it follows from (17.1 7.2.3) that
-
(17.18.1.5)
VX,
*
XI, =
r;kXi, i
which shows that if two connections give rise to the same covariant derivative, they are identical (cf. Problem 2). (17.18.2) Let C be a connection on a differential manifold M , and let f : N + M be a mapping of class C" (m 2 1). For each pair of integers r 2 0, s 2 0, let W : ( f ) denote the vector space of all C" liftings o f f to the bundle T:(M). Then, for each Z E N and each tangent vector h,ET,(N), there exists a unique linear mapping G w V h Z G of W',(f) into the fiber (T:(M)),.(,, which: (1) for r = s = 0 coincides with Oh, (17.14.1); (2)for r = 1 and s = 0 coincides with the covariant derivative defined in (17.17.2.1); (3) satisjies the following conditions :
-
for G'
E
Wi:( f ) and G" E W$(f);
(17.18.2.2)
ohz
(G, G*) = (vhz
for G E W:(f) and G*
E
. G, G*(Z)) + (G(Z), v h ,
*
G*)
Bs(f).
The proof follows that of (17.14.6) step by step, replacing the vector field X by the vector h, , and tensor fields on M by liftings off. (It is also possible to obtain (17.14.6) and (17.18.2) simultaneously as corollaries of the same algebraic lemma: see Problem 1.) (17.18.3) If now 2 is a C" vector field on N, we define V, * G as in (17.17.3) for a lifting G E a:(f ) , by putting (V, * G)(z) = VZ(,) . G for all z E N; it is a lifting off to T:(M), of class C'"-'. The formulas of (17.17.3) remain valid without any change.
18 CONNECTIONS ON A DIFFERENTIAL MANIFOLD
335
(17.18.4) Consider the particular case where M = N and f is the identity mapping. The liftings off are then tensorfields on M . If U E F : ( M ) is a tensor field, the mapping (V*, X)H (V, * u,V * )
of F:(M) x FA(M) into I ( M ) is b(M)-bilinear by virtue of (17.17.3.2). Consequently this morphism defines a tensorfieldon M, belonging to F I +l(M), which is called the covariant derivative ofU (relative to the connection C) and is denoted by VU or VcU.Thus we have (17.1 8.4.1)
(VU,v* 0 X )
= (VX
- u,V * )
and it is clear that U H VU is a diferential operator of order II from T:(M)to T:+l(M).If o E d ( M ) is any scalar function, then
V ~= T do
(17.1 8.4.2)
by virtue of (17.14.1.1) and (17.18.2). Furthermore, we have (17.1 8.4.3)
V(&)
= .(VU)
+ U 0 do.
For with the notation introduced above, we have
(v, . (q, v * ) = o(v,. u,v*> + (ex . by use of (17.17.3.4); but 8,
*
v*)
o = (do, X), and
(do, X)(U, V * )
= (U 0 do,
V* 0 X )
by the definition of duality in tensor products (A.II.I.4); (17.18.4.3) now follows.
the formula
Let M be a diferential manifold and N a closed submanijold of M . Then every connection on N is the restriction of a connection on M. (17.1 8.5)
There exists a locally finite denumerable covering of M by open sets which are domains of definition of charts of M for which the conditions of (16.8.1) are satisfied for the submanifold N. Using a partition of unity subordinate to this covering and (17.16.7), we reduce to the situation in which N is an open subset of R"'and M = N x P, where P is an open subset of R"-",so that T(N) may be identified with N x R", T(P) with P x R"-", and T(M) with T(N) x T(P). Then if y E N and h', k' E R", the connection C' given on N may be written (17.1 6.2.1) in the form C:(h', k') = (h', -l-i(h',
k')),
336
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
where r6 is a bilinear mapping of R"'x R" into R"',and y H r; is of class C" on N. We then take for C the connection on M given by Cdh, k) = (h, -r;,ix(Prl
where x
E
h, pr, k)),
M and h, k E R" (considered as the product R"' x R"-"').
(17.18.6) Let f:M' + M be a local difleomorphism (16.5.6) and let C be a connection on M. Then there exists a unique connection C' on M' such that, for each open subset U' of M' for whichf 1 U' is a diffeomorphism of U' onto the open subsetf(U') of M, the restriction of C tof(U') is the image underf of the restriction of C' to U' (17.16.9). For this requirement determines C;, at every point x' of M, because if U', V' are two open neighborhoods of x' such that f 1 U' and f 1 V' are diffeomorphisms onto f(U') and f(V'), respectively, then f l (U' n V') is a diffeomorphism onto f(U' n V'). The connection C' is said to be the inverse image of C underf. This remark may be applied in particular to the situation in which (M', M,f) is a covering of M. If x i , x; are two distinct points of M' such that f ( x ; ) = f ( x ; ) = x E M, then there exist disjoint open neighborhoods U; of x i and U; of x; in M' such that f(Ui) =f(U;), and a diffeomorphism g : U; --f U; such that f I U; = g (fl U;). With the notation used above, if C; and C; are the restrictions of C' to U; and U;, respectively, then C; is the image of C; by g . Conversely, if a connection C' on the covering space M' has this property, it is immediately seen that we may define a connection C on M by taking (with the notation of the beginning of this subsection) the restriction of C tof(U') to be the image underfl U' of the restriction of C' to U'. The above condition on C' then ensures that C is unambiguously defined. 0
PROBLEMS
1. Let A, A be two commutative rings; p : A + A a surjective ring homomorphism; E, F two free A-modules with bases (eJl r j n and (A)] 1 6 n ; Q a bilinear form on E x F such that cD(ei,fi) = S1,. Also let E , F be two free A-modules with bases (e;)lsisn and (f;)]blJn, and W a bilinear form on E X F such that W(e;J;)= Si,. Let pb : E -+ E' be the A-module homomorphism defined by pb(el) = e; (1 6 i 5 n), and py : F + F the A-module homomorphism defined by p?(h) =f,'(l 5 i 5 n). Suppose also that we are given a mapping L : A + A such that L(ab) = L(a)p(b) p(a)L(b), and a linear mapping LA : E + E such that Lb(ax) = L(a)pb(x) + p(a)Lb(x) for all a, b E A and x E E.
+
(a) Show that there exists a unique mapping Lp : F -+ F such that
+
LB(ay) = L ( ~ ) P ? ( Y )p(a)L?(y)
19 THE COVARIANT EXTERIOR DIFFERENTIAL 337
for all a E A and y
E
F, and
L(@(X,Y)) = (P'(Lb(x), pP69
for all x E E and y
E
+ @'(pb(x), LPQ)
F.
(b) Show that for each pair (p, q) of integers 2 0 there exists a unique mapping L; : E@P@F@q+E'@P@F'@q
such that L%a4= L ( a ) p ! m
+ p(a)Y(z)
for a E A and z E E@P0Feq,where p; is the canonical extension of pb and p? to the tensor product); L:,+:(u
0v ) = L W 0pXv) + p%u) 0LXv)
for u E E m P@ Faqand v E E@'@ F*s; and L ( W , v*))
= WLZ(U),
p%*))
+ @'(p:(u), Y ( v * ) )
and u* E Eeq 0Fa', where (9 and (9' are the canonical extensions of for u E E e P @ F@* the bilinear forms to the tensor product. (Follow the proof of (17.14.6).) 2.
Let E be a vector bundle over M, and let Diff,(E) denote the B(M)-module of differential operators of order sl from E to E. Show that every B(M)-linear mapping X - P X of Y;(M) into Diff,(E) such that P x .( a V ) = (Ox . u)V u(Px . V ) for all u E B(M) and all Cm-sections Y of E, is of the form XHV, relative to a unique connection in E. (Show first that if X vanishes in an open set U, then Px I U = 0.)
+
3. Generalize the result of (17.18.1) to linear connections in an arbitrary vector bundle E over M. Consider in particular a Cm-sectionG* of the dual E* of E, and associate with it. the scalar function on E given by u,~S2(u,) = (u,, G*(x)). Show that
4.
.
With the notation of Section 16.19, Problem 1Is, how that a linear connection in E is a mapping C : E x.T(B)+T(E) such that po C = lexsT(.), and such that C is a bundle morphism of E x T(B) into T(E) both as bundles over E and as bundles over T(W.
5. With the notation of (17.18.4), show that V(c$U) = c;(VU) for any contraction c$.
19. T H E COVARIANT EXTERIOR DIFFERENTIAL
(17.19.1) The formula (17.15.3.6) enables us to calculate at each point x E M the value (d(w(x),h , k,) ~ of the exterior differential of a I-form w by considering two C" vector fields X , Y on M, such that X ( x ) = h, and Y(x)= k,, and calculating the value of the right-hand side of (17.15.3.6) at
338
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
the point x . It is remarkable that although each of the terms on the righthand side depends on the values of the fields X, Y in a neighborhood of x and not merely at the point x , yet the left-hand side does not. We shall see that an analogous phenomenon presents itself when the Lie derivative is replaced by the covariant derivative relative to a connection. (17.19.2)
In detail, let E be a vector bundle over M, and let f be a C"P
mapping of a differential manifold N into M. If o :/\T(N)+E is a C"mapping such that (f, o)is a oector bundle morphism (1 6.15.2), then 0 is said to be a C" differential p-form on N with values in E (relative to the mapping f)(cf. Section 20.6, Problem 2). If Z1, Zz , ... , 2, are C" vector fields on N, then for each point z E N the element o(Zl(z) A Z,(z) A * A Zp(z)) belongs by definition to E,,,,; moreover, it is immediate that the mapping Z H O(z,(Z) A
zz(Z) A
''* A
zp(Z))
is a C"-lifting off to E, which we denote by
o (2,A &
A
~ 2 ~ ) .
(1 7.1 9.3). Now suppose that we are given a connection C on E, and consider first the case p = 1. Let 0 be a differential 1-form of class C" on N, with values in E, and let X , Y be two CcO-vector fields on N. By analogy with (17.15.8.1) we form the C"-lifting off to E (17.19.3.1)
v, - (0
*
Y )- v
y
*
(0* X
) - 0 * [X, Y].
We shall show that the value of this mapping at a point z E N depends only on the values X(z) and Y(z) of the3elds X , Y at the point z. For this it is enough to show that if we replace X (resp. Y ) by O X (resp. aY), where a is a scalar function of class C" on N, the value of (1 7.1 9.3.1) is obtained by multiplying by a(z) the value for X and Y. For, reducing to the case where M, N are open subsets of RP, Rq and E = M x R", we have X(z) = ( z , g(z)) and Y(z) = (2,h(z)), and the formula (17.17.2.2) shows that (17.19.3.1) is a bilinear function of the vectors (g(z), Dg(z)) and (h(z), Dh(z)); hence, by virtue of (8.1 .4), the condition stated above is necessary and sufficient for this function not to depend on Dg(z) (resp. Dh(z)). Now we have vex.(0 * Y) = aV* (a* Y ) by (17.17.3.2), and
-
0
-
*
(ax) = a(o * X),
vy(o (ax)) = (e,
- a)(O - x)+ o(v, - . x)) (0
19 THE COVARIANT EXTERIOR DIFFERENTIAL
339
by (17.17.3.4), and finally
by (17.14.4.2). Hence the result. Since (17.19.3.1) is an alternating bilinear function of ( X , Y ) , there exists a unique C" differential 2-form on N with values in E, called the covariant exterior differential of o (relative to C ) and denoted by d o , such that for any two C" vector fields X , Yon N we have (17.19.3.2)
d o * ( X A Y ) = V , * (o Y ) - V , * (o* X ) - o * [ X , Y l .
(1 7.19.4) This result generalizes easily to differentialp-forms on N with values in E, where p > 1. We take the analog of the formula (17.15.3.5) by proving that if o is a C" differential p-form on N with values in E and if Xo , XI, .. ., X p are p + 1 C" vector fields on N, then the C"-lifting off to E (1 7.19.4.1) A
8, A
has at each point z E N a value which depends only on the values X j ( z ) of the vectorJields X j at z (0 5 j 5 p). The method of proof is exactly the same: we replace each X j successively by axj. In this way we establish the existence and uniqueness of a C" differential (p + ])-form d o on N, with values in E, such that d o ( X , A ... A X,) has as its value (17.19.4.1); d o is said to be the covariant exterior diRerentia1 of o (relative to C). 0
Finally, since by convention A T(N) = N x R (1 6.16.2), a C" differential 0-form on N with values in E is identified with a C" lifting G off to E. For each C" vector field X o n N, the value of V , * G at a point z E N depends only on X(z) by definition (17.17.3). Hence there exists a unique differential I-form of class C" on N, with values in E, which is called the covariant exterior differential of G and is denoted by dG, such that (1 7.19.4.2)
dG-X=V,.G.
340
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
(17.19.5) With the notation and hypotheses of (17.19.2), let u : N, -+ N be a C"-mapping, and consider the composite mappingf, = f o u : N, + M. It is
clear that the pair (A,a,),where o1= o 0 A T(u), is a morphism of vector bundles. The differential p-form o1on N, with values in E is called the inverse image of o by u. Now suppose that we are given a connection C on E. Then the covariant exterior differentials relative to f and tof, = f o u satisfy the relation P
d(o
(17.19.5.1)
P
A
P+ 1
A T(u)) = (do) o A T(u)
for any C" exterior differentialp-form o on N with values in E. We shall give the proof only for p = 0 and p = 1 (cf. Problem 1). For p = 0, in view of the definition (17.19.4.2), the formula (17.19.5.1) reduces to (17.17.2.6). For p = 1, since the question is local with respect to N, N,, and M, we may assume that M yN, N, are open subsets of R"', R",R"', respectively, and that E = M x Rq. Then o may be written as (z, h)w(f(z), A(z) h), where ZHA(Z) is a C"-mapping of N into 9 ( R " ; Rq); this implies that DA(z) is an element of 2Z2(R"; R4) ((5.7.8) and (8.12)). It now follows from the formula (17.17.2.2), the definition (17.1 9.3.2), and the rules for calculating derivatives in vector spaces (Chapter VIII) that d o is the mapping
-
(17.19.5.2) (2,
h A k)H(f(Z), DA(z) * ((h, k) - (k, h)) + r,(z)(Df(z) * h, A@) * k) - I-/(&?f(4 * k Y 4 4 * h)).
(Incidentally, this * calculation provides another proof of the fact that (17.19.3.1) depends only on the values X(z) and Y(z).) Likewise, o1may be written as (z,,hl)i+(fl(zl), A,(z,) . hl), wheref, = f o u and
A,(z,) = A(u(z1)) O Du(z1). We have then Dfl(z,) h, = Df(u(z,)) * (Du(z,) * h,); also the mapping h, H DA,(z,) (h,, k,) is the derivative of z1wA,(z,) k, ; consequently, in view of the definition of Al, we have
-
-
-
DA,(z,) (hl, k,) = DA(u(z1)) * (Du(z1) * h,, Du(z1) * k,) + A(u(z1)) * (DZU(Zl)* (hl9 kl)). Inserting these values of Df,(z,) and DA,(z,) into the expression for do, analogous to (17.19.5.2), we obtain, using the symmetry of D2u(z,) (8.1 2.2),
(do,)(z,, h, A k,)
= (do)(z,
h A k),
where z = u(z,), h = Du(z,) * h,, k = Du(z,) k,. This proves (17.19.5.1) in the casep = 1.
19 THE COVARIANT EXTERIOR DIFFERENTIAL
341
PROBLEMS
1. Prove the formula (17.19.5.1) for arbitraryp, as follows. To calculate the value of the
-
+
P+ 1
left-hand side at a ( p 1)-vector ko A kl A * * A k, E A Tsl(Ni), consider separately the cases where the vectors T.,(u) * k, (0 sj s p ) are linearly independent or linearly dependent. In the first case, reduce to the situation where Nl is a submanifold of N of dimension p 1 by use of a chart, and use the fact that in calculating the value of (17.19.4.1) we may assume that the X , are fields of tangent vectors to N1. In the second case, we may assume that T,,(u) ko= 0 and that the fields X, such that X,(z,) = k, for 0 s j s p are such that [X,,X,]= 0; use the formula (17.17.2.6).
+
2. Let M be a differential manifold. Given two integers p 2 1, q 2 0, and M-morphisms P
4
P : A T(M) + T(M), Q : A T(M) +T(M), we define an M-morphism
by the formula (P
Q) * (Xi
A Xz A
s . 0
A XP+,-,)
(When q = 0, Q is identified with a vector field X , and Q * (Xu(i)A A Xu(,))has to be replaced by X.) Likewise, for each scalar-valued differentialp-form a on M, we define a (p q - 1)form a K Q by the formula -1.
+
A X Z A * * * A Xp+,-l>
(with the same convention for q = 0). Extend this definition to the casep = 0 by putting f ilQ = 0 for all functions f~ b(M). Show that: (a) If p 2 1 , q 2 1 and if a is a scalar-valued r-form, we have (a K P ) K Q - a 7 1 ( P 7 T Q ) = ( - l ) c p - ' ) t q - 1((a ) 71 Q) 7 i P - a
K (Q FPN.
(b) ~ T ( M ) K Q = Q ; P K ~ T ( M ) = P P (c) If (I,v are endomorphisms of T(M) (1-forms with values in T(M)), or tensor fields of type (1, l), then (I
~ v = u o v ,
for any vector field X. 3. The notation is as in Problem 2.
.71X=u*X
XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
342
(a) The mapping a-a iT Q of the exterior algebra at of scalar-valued differential forms on M into itself is an untiderivation of degree q - 1 which is zero on Qo This antiderivation is denoted by i Q . Conversely, every antiderivation of at of degree q - 1 which vanishes on do is of this form. (Observe that such an antiderivation is a
.
9
differential operator from T(M) to A T(M).) (b) The mapping D : a*(da) i T Q + ( - l ) q d ( a T Q ) of at into itself is an antiderivation of degree q such that
D d = (-1)gdo D.
(1)
0
(We have D = iQ 0 d - (- 1)q-ld iQ .) This antiderivation is denoted by dQ . Conversely, every antiderivation D of d of degree q which satisfies condition (1) is of the 0
form da . (Observe that D is a differential operator from M x R into A T(M).) (c) Every antiderivation D of d of degree r can be written uniquely as D = ip Q
r+l
r
+ dQ,
where P (resp. Q) is an M-morphism of A T(M) (resp. A T(M)) into T(M). (Determine Q by the condition that dQ coincides with D on 80.)(Frohlicher-Nijenhuis theorem.) (d) If Q is a vector field X (i.e., if q = 0), then iQ coincides with the interior product ix If Q is an endomorphism u of T(M) (i.e., if q = l), then
.
and, in particular, ilTcM) * a =pa. If Q is a vector field X,then dQ coincides with the Lie derivative O x . We have d l T ( M ) - d (exterior differentiation in 4. (el With themme notation, there exists a unique M-morphism [P,Q ] of T(M) such that [dP
(2)
We have
9
P+4
A T(M) into
~ Q= I d[p,Q l .
[Q,pI = (-l)pq+’[p, Q1, / ~ T ( M )9
Ql
= 0,
+
(-1)“[P, [Q, Rl1+ (-1)‘”[Q, [R, PI] ( - 1 ) ” W
[ P , Q11= 0.
(f) If P = X and Q = Yare vector fields, then [P,Q ] is the usual Lie bracket [ X , Y ] . For each Q , [ X , Q ] is the Lie derivative Ox . Q (where Q is identified with a tensor field of type (1,q)). If P = u and Q = u are endomorphisms of T(M), then [U, U ] *
-Id
*
+
+
+
Y ] [u-X,U.Y ] . - ( a . [ X , Y ] ) u * ( u - [ X , Y ] ) [u * x,Y ] - D * [U . x,Y ] - U * [X, 0 . Y ] - D . [X, U .Y ]
( X A Y )= [ u . X ,
U
S
and, in particular, &[Id, U ]
*
( X A Y ) = [ u . X , U ‘ Y ]+ U ’ ( u . [ X , Y ] ) -
(the Nijenhuis torsion of u). (9) Show that IiQ, dpl= d
U .
[ u . X, Y ] - U
K Q + (- I)%P. Q I .
*
[X,U * Y ]
20 CURVATURE AND TORSION OF A CONNECTION 4.
343
Let M be a pure differential manifold of dimension n. Consider the canonical exact sequence (Section 16.19, Problem 11)
o
--f
T(M) x T(M) LT(T(M))1:T(M) x T(M) +0.
p is called the certical endomorphism of T(T(M)). Its local The endomorphism J = expression relative to a chart of M at a point x is 0
(X,
h,, Ux,khx)-(X,
hx,0, Ux).
It is a T(M)-morphism for the vector bundle structure on T(T(M)) with projection not for the bundle structure with projection T(oM).It is of rank n at every point, and we have J J = 0.
o ? ( ~ )but ,
0
(a) With the notation of Problems 2 and 3, show that J TJ=O, [ i j ,d j ] = 0,
[J,J]=O, d j d j = 0.
[d,d j ] = 0,
0
(b) Show that, for all vector fields Z, Z'on T(M), we have
+
.
[ J . z,J Z ]= J . [ J . z,2 1 J * [Z,J . 2 1 , [ i ~ , i ~ -]i J= . z .
20. CURVATURE A N D TORSION OF A C O N N E C T I O N
Let E be a vector bundle over M, let C be a connection in E, let E a C"-lifting off. Since dG is a C" differential 1-form on N with values in E, we may consider the differential 2-form d(dG) on N with values in E. By contrast with the case of the exterior differential (17.1 5.3.1), however, d(dG) is not identically zero in general; but, for all h, , k, in T,(N), the vector d(dG) * (h, A k,) E Eft,, depends only on the value G(z) E E,,,, of G at the point z (and not on its values in a neighborhood of z). To see this, let X , Y be two C" vector fields on N such that X ( z ) = h, and Y(z) = k,. Then the vector d(dG) . (h, A k,) is by definition ((17.19.3) and (17.9.4)) the value at z of the following lifting o f f t o E: (17.20.1)
f:N -+ M be a C"-mapping and G : N
(17.20.1 .I)
V,.
(Vy
*
G) - V y . (V, . G) - Vr,,
yl
*
G.
By the same argument as in (17.19.3), it is enough to show that if o is any C" scalar-valued function on N, the value of (17.20.1.1) for oG is obtained by multiplying the corresponding value for G by o(z). Now, by virtue of (17.17.3.4), we have
vx . (vy. ( 0 ~ )=) v, . ((e, . o ) +~o(vy. GI) = (ex . (e, . O ) ) G + (e, o)(v,
+ (0,
+
*
a)(Vy * G)
+ o(VX
*
G)
(Vy
. G)).
344
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
Interchanging X and Y in this formula, and remembering that V [ x ,Y] * (4= ( e [ x , Y] * 4 G + d v f x , rim G),
it follows that our assertion is a consequence of the definition of [ (1 7.14.3). The expression (17.20.1.1) is clearly a linear function of G. Hence there is an endomorphism R, (h, A k,) of Eft,, such that
(17.20.2)
d(dG) (h,
A
k,) = R,(h,
A
-
k,) G(z).
Furthermore, it is immediately seen that the mapping
r f : h, A k, H R,(h,
(17.20.2.1)
A
k,) 2
is such that (f, r,) is a vector bundle morphism of A T(N) into Hom(E, E) = E * @ E (in other words, r, is a diferential 2-form on N with values in E* 8 E). (17.20.3) Now let u : N, + N be a C"-mapping. By applying the formula (17.19.5.1) for p = 0 and p = 1 we obtain 2
(17.20.3.1)
d(d(G u)) = (d(dG)) o 0
A T(u).
In the notation introduced in (17.20.2), this takes the form (17.20.3.2)
for if G, : N,
+E
is any lifting of f o u, and if z1E N,, then there exists a
lifting G o f f to E such that G(u(z,)) = G,(z,) ((1 6.15.1 -2) and (1 6.19.1)).
Consider in particular the case where N = M and f M-morphism r l Mis denoted simply by
(17.20.4)
=
1,.
The
2
r : AT(M)+E*@E
and is called the curvature M-morphism (or simply the curvature) of the connection C in E. Knowledge of this morphism determines all the differentials d(dG), by virtue of (17.20.3.2); in other words, with the notation of (17.20.1), if G is any (?-lifting off, we have
20 CURVATURE AND TORSION OF A CONNECTION
345
When M is an open set in R" and E = M x R4,then we have h, = (x, h) and k, = (x, k) with h, k E R" for all x E M ; if u, = (x, u) is any element of Ex,an easy calculation using (17.1 7.2.2) and (17.1 9.5.2) gives
(Since x H r , is a mapping of M into B2(R", R4;Rg), it follows that Dr, belongs to S?(R"; Y,(R", R4; R4)), identified with the space of trilinear mappings Y,(R", R", R4; R4).) (17.20.5) Consider in particular the case where E = T(M), so that C is a linear connection on M. Then the curvature morphism r of C defines a bilinear M-morphism (hx, k,)wr * (h, A k,) of T(M) 0 T(M) into T:(M) and hence may be identified with a tensorfield r of type (1, 3) (16.1 8.3), called the curuature tensorfield (or, by abuse of language, the curvature tensor) of the connection C. If (U, cp, m) is a chart of M, and ( X i ) t j i J mthe basis of FA(U) over b ( U ) associated with this chart (16.1 5.4.2), then from (17.20.4.2) we have (17.20.5.1)
from which we obtain the corresponding components of the curvature tensor r: (17.20.5.2)
ar:, art..
r!. = -- 2 + C (rLi lJk
ax)
axk
h
-
ri,).
(17.20.6) Again assume that E = T(M). The identity mapping 1,(,, of T(M) can be considered as a drflerential 1-form on M with values in T(M). Its covariant exterior differential r = d( 1T(MJ is therefore an M-morphism of
A T(M) into T(M), which is called the torsion M-morphism (or simply the 2
torsion) of the linear connection C on M. So, by definition (17.19.3), we have
(17.20.6.1)
t.(X/\Y)=V,- Y - V y '
x- [ X , Y ]
for any two C" vector fields X , Y on M. This morphism defines a bilinear M-morphism (h,, k3-r * ( h , ~k,) of T(M) @T(M) into T(M), and hence may be identified with a tensor$eld t of type (I, 2), called the torsion tensor
346
XVll
DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
field (or, by abuse of language, the torsion tensor) of the connection C. If (U, cp, m)is a chart of M and ( X i ) l 6 i 6 mthe basis of 9-:(U) over B(U) associated with this chart (16.1 5.4.2), then from (17.1 8.1.5) we have
from which we obtain the corresponding components of the torsion tensor
t:
Iff: N + M is any C"-mapping, Tcf) is a lifting off to T(M),and we may write it as T ( f ) IT(M). The formula (17.19.5.1) consequently shows that, for any two vector fields X , Y of class C" on N, we have 0
Let C', C" be two linear connections on M , t' and t" their respective torsions. If B is the bilinear M-morphism of T(M)@ T(M) into T(M) which is the difference of C' and C" (17.1 6.6), then we have
(17.20.7)
t" * ( X A Y)
(17.20.7.1)
- t'
*
(XA Y ) = B(X, Y ) - B(Y, X ) .
If we denote byVLx and VLx the covariant derivatives relative to C' and C", respectively, then it follows immediately from (17.1 7.2.1) that
VkX* Y - VLX * Y
= ~y(,)(
- C:(h,,
= -B,(h,,
Y(x))
+ C:( h, , Y ( x ) ) )
Y(XN
from the definition of B. Hence we have (17.20.7.2)
V i * Y -V'*
Y
=
-B(X, Y )
and the formula (17.20.7.1) follows immediately from this and the definition (17.20.6.1) of the torsion.
APPEND1 X
MULTILINEAR ALGEBRA
(The numbering of the sections in this Appendix continues that of the Appendix to Volume I.)
8. MODULES. FREE MODULES
(A.8.1) None of the results of (A.l .I)-(A.3.5) inclusive involves the field structure of K, and therefore all these results remain valid without modification when K is replaced by an arbitrary commutative ring A (with identity element). In place of K-vector spaces we speak of A-modules (in (A.2.3), h, is bijective if and only if 1 is invertible in A). By abuse of language, the elements of an A-module are sometimes called vectors and the elements of A are called scalars. (A.8.2) The definitions of a free family and of a basis of a vector space, given in (A.4.1) and (A.4.4), require no modification for an A-module, and the same is true of (A.4.3). On the other hand, the condition given in (A.4.2) for a family to be free is no longer valid in general (in the Z-module Z, for example, the number 1 does not belong to the Z-module 2 2 generated by 2, but (1, 2) is not a free family). An A-module possessing a basis is said to be free.
Everything in (A.5.1)-(A.6.6) inclusive, on determinants and matrices, also remains valid when the field K is replaced by an arbitrary commutative ring A. In (A.6.4) it is merely necessary to choose the formf, so that it takes the valuef,(b,, .. ., b,) = 1 for a basis (bi) of E. We remark that these results prove that any two bases of the same A-module E (assumed to have a (A.8.3)
347
348
APPENDIX:
MULTILINEAR ALGEBRA
finite basis) necessarily have the same number of elements (which one might call the “dimension” of E, but it should be realized that most of the results concerning dimensions of vector spaces do not generalize to free modules). The determinant calculations in (A.6.8) and (A.7.4) are also valid for an arbitrary commutative ring A.
9. DUALITY FOR FREE MODULES
(A.9.1) If E is an A-module, the A-module Hom(E, A) of linear forms on E (A.2.4) is called the dual of the module E and is often written E*. If x E E and x* E E*, we shall often write (x, x*) or (x*, x) in place of x*(x). The mapping (x, x*)H(x, x*) is a bilinear form, called the canonicaZ bilinear form, on
E x E*. For each x E E, the mapping X*H (x, x*) of E* into A is a linear form on E*, in other words an element c ~ ( xof ) the bidual E** = (E*)* of E, and the mapping cE(called the canonical mapping) of E into E** is linear.
(A.9.2) Suppose that E is a free module having a j n i t e basis (ei)l,i6n (also called a ,finitely-generatedfree module). For each index i, let e: be the linear form on E (called the ith coordinatefunction) such that (ei ,eT ) = dij (Kron-
ecker delta). Then for each x =
n
tiei E E, where ti E A (1
S i 5 n), we have
i=1
(x, e?) = ti. Hence (A.5.1) (e:)lsisn is a basis of the dual A-module E*, called the basis dualto the basis (ei). From this definition it follows immediately that if (e?*)l is the basis of E** dual to ($1, then we have cE(ei)= e:* for all i, and hence cEis an isomorphism, by means of which we shall identifv E** with E, so that each element x E E is regarded as a linear form on E*, namely the form x* H(x, x*).
(A.9.3) Let E and F be two A-modules and u a linear mapping of E into F. Then for each linear form y* E F*, the function y* o u is a linear form on E, hence an element of E*, and it is immediately verified that the mapping y* H y* 0 u of F* into E* is linear. This mapping is called the transpose of u and is denoted by ‘u. Clearly we have (A.9.3.1)
‘(U1
+
u2)
= ‘U1
+ ‘U2,
‘(nu) = 1 . ‘24
for all u l , u 2 , u ~ H o m ( E ,F) and AEA. If G is another A-module and u : F -P G a linear mapping, then (A.9.3.2)
‘(0 0 2.4) = ‘u 0 ‘27.
9
DUALITY FOR FREE MODULES
349
The definition of the transpose is contained in the fundamental duality formula:
for all x E E and y* E F*. If u is an isomorphism of E onto F, then ‘u is an isomorphism of F* onto E*. Its inverse ‘u-’ (which is also the transpose of u-’) is called the isomorphism contragredient to u. It satisfies the relation (u(x), ‘u-’(x*)) = ( x , x * )
(A.9.3.4)
for all x E E and x*
E
E*.
Suppose now that E and F are free modules with finite bases (aJlsisn and (bJlslSm, respectively. If U = (orji) is the matrix of u with respect to these bases (A.5.2), then orji = (u(ai),b;), and the formula (A.9.3.3) therefore shows that ( a i , ‘u(b;)) = aji . Since (ai) is the basis dual to (a:), the matrix of ‘U with respect to the bases (b;) and (a:) is obtained by interchanging the rows and columns of the matrix U of u (it is called the transpose of U and is denoted by ‘ U ) .Further, it is immediately clear that (A.9.4)
(A.9.4.1)
‘(54)
= u.
(A.9.5) Suppose that the finitely-generatedfree A-module E is the direct sum M @ N of two finitely-generated free A-modules (A.3.1). To the canonical projections p : E + M, q : E + N (A.2.3) there correspond by transposition canonical injections * p : M* + E*, ‘q : N* E*, such that E* is the direct sum
of the submodules No = ‘p(M*) and Mo = ‘q(N*).To see this, we take a basis of E consisting of the elements of a basis (ai)’ ism of M and a basis (bj), of N. It is then immediate that if (a:) and (b;) are the bases dual to (ai) and (bj),respectively (A.9.2), the elements ‘p(a:) and ‘q(br)form the basis of E* dual to the chosen basis of E. The submodule Mo (resp. No)can also be defined as the set of linear forms x* E E* such that (x, x*> = 0 for all x E M (resp. all x E N), and is called the annihilator of M (resp. N) in E*. By reason of the identification of a finitely generated free module with its bidual, it is clear that (M’)’ = M and (No)’ = N. Finally, note that if i : M -+ E and j : N --t E are the canonical injections (A.2.3), their transposes ‘i : E* + M* and 7 : E* + N* are the canonical projections, when we identify E* with the direct sum M* @ N+. (A.9.6) If E is a finite-dimensional vector space over a (commutative) field K, every vector subspace M of E admits a supplement N in E, and the results of
350
APPENDIX: MULTILINEAR ALGEBRA
(A.9.5) can be applied to the direct sum decomposition M 0 N of E. The annihilator Mo of M in E* does not depend on the choice of the supplementary subspace N. It is clear that (A.9.6.1)
codim Mo = dim M,
(A.9.6.2)
(M0)O = M.
Also, if M,,M, are two subspaces of E, (A.9.6.3)
(Ml
+ M,)'
( M , n M,)'
= MY n Mg,
= MY
+Mi;
this is easily seen by taking a basis of E as in (A.4.12). Let F be another finite-dimensionalvector space over K, and let u : E + F be a linear mapping. Then, with the same notation, we have (A.9.6.4)
(Ker u)'
= Im('u),
(Im u)' = Ker('u)
as is easily shown by decomposing E into the direct sum of Ker(u) and a supplementary subspace, and F into the direct sum of Im(u) and a supplementary subspace. It follows that rk('u) = rk(u).
(A.9.6.5)
10. T E N S O R P R O D U C T S O F FREE M O D U L E S
(A.IO.1) Let El, ..., E, be A-modules, and for each index j let xi* E E? be a linear form on Ej . Then the mapping
of El x E, x . . * x E, into A is an r-linear form (A.6.1), which is called the tensor product of the forms x:, . .. ,x,* and is denoted by x:: @ x; @ *
(A.10.1 .I)
* *
0 x:.
The set 2',(El, . .., E, ;A) of all r-linear forms is an A-module, and it follows directly from the definition that the mapping (A.10.1.2)
of Er x Ef x
. ..,X,*)HX:: 0 x: 0 x: into Y,(E1,..., E, ;A) is r-linear.
(x?, x:,
- .-x Ef
*.-
10 TENSOR PRODUCTS OF FREE MODULES
351
(A.10.2) Suppose now that each Ej is a free A-module with a finite basis ( e j k ) l s k j n jThen . it follows from (A.6.2) that the mapping (A.10.1.2) is bijective and that the elements
form a basis consisting of nlnZ * * . n, elements o f the A-module 9 , ( E 1 , . . . , E, ;A). This A-module is called the tensor product of E:, ... , E,*, ET 0 * * * @ E,* . and is denoted by E: @ A E: @ A * * @ A E,*, or simply E: (A.10.3) Since, under the hypotheses of (A.10.2), Ej is identified with Ej** (A.9.2), it follows that we may define in the same way the tensor product El @A E, @ A .* . oAE, (or El @ E, @ * . . @ E,) which has a basis consisting
of the nln,
*
n, elements
(A.10.3.1)
el, kl
8 eZ,
k2
@
'.
'
@ er,
k,
(1 5 k j S n j , 1 5 j 6 r). The fundamental property of this A-module is the following:
x E , into an arbitrary For every r-linear mapping u of El x E, x A-module F , there exists a unique A-linear mapping v of El @ E, @ * * * @ E , into F such that
for all xi E Ej (1 5 j 5 r). For if u(el,k , , e,, tions
k2,
..., e,, k,)
= C k l k 2 ...k , E
v(el,kl Be,,,, 8 * * * 8 er,k,)
F , we can define u by the condi= Cklk2...kr)
and the mapping u so defined clearly has the required properties. If E and F are two finitely-generated free A-modules, and if (ei)l i s m and are bases of E and F, respectively, then the ei @fi form a basis of E 63 F, and hence every element of E @ F is therefore uniquely expressible in the form z = C t i j e i@fi. This expression can also be written as
(fi), s j g ,
iJ
(A.10.3.3).
z=
n
m
j=1
i= 1
cxj@fj= 1eioyi,
where the xi (resp. y i ) are elements of E (resp. F) uniquely determined by z.
352
APPENDIX:
MULTILINEAR ALGEBRA
(A.10.4) The above definition implies the existence of canonical isomorphisms between tensor products of finitely-generated free A-modules. If El, E, , E, are three finitely-generated free A-modules, there is a unique isomorphism (the associativity isomorphism)
(El @Ez)8E3+El 8 Ez B E 3
(A.10.4.1)
which maps (x, @ x,) 8 x, to x, @ x, 8 x 3 .It can be dejned by this property for the basis elements (el,k,@ e2,k2)8 e,,k3 of the left-hand side. Likewise, there is a unique isomorphism (the distributivity isomorphism) (A.10.4.2)
(E,
eE,) 6E,
-+
(E, 8 E,)
e(E, 8 E,)
which maps (xi @ x,) @ x3 to (x, @ x,) 8 (x, @ x,); it is defined in the same way as before. (A.10.5) We have already seen (A.10.3) that there is a canonical isomorphism of Hom(El @ E, , F) onto the A-module Y,(E,, E2; F) of bilinear mappings of El x E, into F. Moreover, there is also a canonical isomorphism (A.10.5.1)
Hom(E,, Hom(E,, F)) -+ Hom(E, 8 E, , F).
For if x, H V , , is a linear mapping of El into Hom(E,, F), then (xl, x,)Hu,,(x,) is a bilinear mapping of El x E, into F, and we apply (A.10.3).
Now let El, E, , F,, F, be four finitely-generated free A-modules. To each pair of A-linear mappings u1 : El + F,, u, : E, + F, we associate the bilinear mapping (x,, x2)wu1(x1) @ uz(xz)of El x E, into F, 8 F, ;to this bilinear mapping there corresponds (by (A.10.3)) a linear mapping ui
Ei
+Fi OF2
such that (A.10.5.2)
(4 0 UZ)(Xl@
X2)
= Ul(X1) @ uz(xz).
Furthermore, the mapping (u,, U,)HU, there corresponds to it a linear mapping (A.10.5.3)
Hom(E,, F,) 0 Hom(E,, F,)
@ u, is bilinear; hencz (A.10.3)
-+
Hom(E, 63 E, , F, @ F,),
which is in fact an isomorphism (and therefore the fact that the symbol u, @ u, has different meanings in the two sides of (A.10.5.3) is unimportant). If (ai), (bj),(c,,), (dk)are bases of El, E, , F,, F, ,respectively, if uih E Hom(E,, F,) is defined by the conditions U i h ( a i ) = ch, and Ui,,(am)= 0 for m # i, and if likewise
10 TENSOR PRODUCTS OF FREE MODULES
353
wjk E Hom(E, , F2) is defined by wjk(bj) = dk, Wjk(b,) = 0 for n # j , then it is immediate that the linear mappings uih@wjk form a basis of Hom(E, €3 E2 , F, 0 F2). In particular, taking F, = F, = A, since A 0 A is canonically isomorphic to A (considered as the free A-module with basis consisting of the element l), we obtain a canonical isomorphism
If on the other hand we take F,
= E, = A, we
obtain a canonical isomorphism
E* @ F + Hom(E, F)
(A.10.5.5)
under which x* @ y (where x* E E* and y mapping XH ( x , x * ) y of E into F.
E
F) corresponds to the linear
(A.10.6) Let B be a commutative ring containing A which is a finitelygenerated free A-module and has the same identity element as A. If E is any finitely-generated free A-module, there is a unique B-module structure on E OAB such that (x @/?)A = x @ (/?A) for all x E E and /?,A E B. For if (ej)lsi4n is a basis of E, every element of E @AB is uniquely of the form e, @ pi with pi E B, and the B-module structure required may be defined by i
e, €3 /?. A C ei 8 (Bin.This B-module is said to be obtained from E by (T extension of the ring of scalars to B and is denoted by E(B). The elements =
1)
i
g j 5 n) form a basis of E(B), and E may be identified with the sub-A-module of E(B)generated by these basis elements, by means of the canonical injection XH x 8 1. Every A-linear mappingf of E into a B-module G extends uniquely to a B-linear mappingfof E(B)into G, such thatf(x @ fl) = f ( x ) / ? .We may definef by the conditionsf(ei 0 1) =f(ei) for 1 5 i 5 n. In particular, if F is another finitely generated free A-module and if j : F -+ F(B)is thecanonicalinjection, thentoeachA-homomorphismu : E + F there corresponds the extension o f j 0 u to E(B), which is a B-linear mapping u ( ~ :, E(B)+ F ( B ) such that u ( ~ ) (0 x p) = U(X) @ p. In this way we define a canonical isomorphism ei €3 I (1
(A.10.6.1)
F))(B)
+
HomB(E(B)
9
which, in particular, gives an isomorphism (A.10.6.2)
(E*)(B)
+
(E(B))*.
F(B)),
354
APPENDIX:
MULTILINEAR ALGEBRA
Finally, there is also a canonical isomorphism (A.10.6.3)
E(B) @B
F(B)
--*
(E @ A F)(B)3
where E, F are finitely generated free A-modules; the element (x @ B) @ (Y 0 B'>
(x e E, Y E F, B, B'
E B)
is mapped to (x 0 Y ) 0 (BP'). 11. TENSORS
If E is a finitely generated free A-module, we denote by E@"or T"(E) or T:(E) the tensor product of n copies of E, for n 2 2. This A-module is called the nth tensor power of E. We also define TA(E) to be E itself and T:(E) to be the ring A, considered as an A-module. Likewise we denote by T;(E) the tensor product (E*)@", with the convention that Ty(E) = E*. Finally, if p and q are two integers >O, we denote by T:(E) the tensor product (E*)@4' @ (EBP). The elements of T",E) (resp. T;(E)) are called n-foid contruvariant (resp. n-fold covariant) tensors; the elements of T:(E), for p > 0 and q > 0, are called mixed tensors of type ( p , q), and p (resp. q) is the contravariant (resp. covariant) index. It follows from (A.10.5.4) and (A.10.5.5) that T:(E) may be canonically identified with Hom(EB4, EBP). Hence by (A.10.5.3) we have a canonical isomorphism (A.ll.l)
(A.ll .I.I)
T:(E) 0 TXE) -,T::S(E)
-@ in which the product u @ u of a tensor u = x ~ @ * * * @ x ~ @ x l @ - -xp and a tensor v = y : 0 * * . 0 y*, @ y1 0 .* * 0 y, corresponds to the tensor XT
0. * * 0 x:0
y r 0
..* 0 ys*@ X I 0 . * .0 xpOy , 0 ... 0 y , .
When p = q = 0 or r = s = 0, the isomorphisms (A.ll .I.I)are the linear mappings corresponding to the bilinear mappings (A, Z)HAZ and (2,A)- Az, respectively. With these definitions it is immediate that for any three tensors u, u, w we have (A.ll .I.2)
(u 0v) 0 w = u 0 (v 0 w).
Again, by reason of the identification of a finitely generated free module with its bidual, and the canonical isomorphism (A.10.5.4), we have a canonical isomorphism (A.11 .I .3)
(T,P(E))*
+
T:W
11 TENSORS
355
such that, if the dual of T:(E) is identified with T:(E) by means of this isomorphism, we have
(A.11.2) If (ej)l elements
ism
is a basis of E and (e?) the dual basis of E* (A.9.2), the
of T:(E), where the indices ih and jk run independently through the set {1,2, ... , m}, form a basis of T,P(E)which is called the basis associated to (ei). A tensor belonging to T:(E) then has a unique expression of the form
C cx{,$:i2e71
.
* *
ei*,
ej,
* * *
0 eip
. .
the sum being over all mp+qfamilies of indices ( j l , .. ., j q , zI, . . ., i,). (A.11.3) Given two indices i, j such that 1 5 i 5 p and 1 5 j 5 q, there exists a unique linear mapping c j : T,P(E) 4T:I i(E),
called the contraction of the contravariant index i and the covariant index j such that, for xlr . . . , x p E E and x:, .. ., x: E E* ,
wherein the circumflex accent signifies that the term underneath it is to be omitted from the tensor product. The mapping ci may indeed be defined by this formula for the basis elements (A.11.2.1). In particular, taking p = q = I , we have c:(x* O x) = (x, x*) E A. The elements of E* @ E correspond canonically to the endomorphisms of E (A.10.5.5); the value of the contraction c : for the tensor corresponding to an endomorphism u is called the trace of u and is denoted by Tr(u). If (with the above notation) u corresponds to the tensor eT @ e i , i.e., if u is the endomorphism XH (x, e 7 ) e i , then its trace is d i j (Kronecker delta). It follows
356
APPENDIX:
MULTILINEAR ALGEBRA
easily that if U = ( a i j )is the matrix of u with respect to the basis (e,), then Tr(u)
(A.11.3.2)
= i
aii,
the sum of the diagonal elements of the matrix U ; this is also called the trace of the matrix U and is denoted by Tr(U). It is immediately verified that, for any two endomorphisms u, u of E, we have Tr(u 0 u ) = Tr(u 0 u)
(A.11.3.3)
(it is enough to consider the case where u, u correspond to "decomposed" tensors a* 8 a and b* 8 b).
12. SYMMETRIC A N D ANTISYMMETRIC TENSORS
From now on we shall assume that the ring A contains the field Q of rational numbers, so that for each a E A and each integer m # 0 the element m-'a belongs to A and is the only element E A such that m5 = a. (A.12.1) Consider the A-module T"(E) of n-fold contravariant tensors over a finitely-generated free A-module E. We define an action of the symmetric group 6, on Tn(E)as follows. For each permutation u E 6, , the mapping ( X I > x2
9
* * *
9
xn) Hxu- '(1)
8 x u - '(2) 8 . ..8 x u - '(n)
of E" into T"(E) is n-linear, hence factorizes as (XI, x2,
.
-,X,)HXI 0 x2 0 . * . 0 X,HO.
(xi 0 x 2 0
Ox,),
where u is an endomorphism of the A-module T"(E), defined by
From this definition it follows immediately that, if 6,T are any two permutations in G,, we have (A.12.1.2)
T
. ( 0 . Z) = (TO) . Z
for all z E T"(E). A tensor z E T"(E) is said to be symmetric (resp. antisymmetric) if o * z = z (resp. CT * z = E,,z, where E, is the signature of the permutation a) for all
12 SYMMETRIC AND ANTISYMMETRIC TENSORS
357
6,. If we take the basis of T"(E)associated with a basis (ei) of E (A.11.2), a tensor is symmetric if and only if its components satisfy the conditions 0E
(A.12.1.3)
and antisymmetric if and only if (A.12.1.4)
for all indices il, i2, .. ., inand all CT E G,,. It is sufficient that these relations should be satisfied for all transpositions T E G,,. (A.12.2) If z is any tensor belonging to T"(E), we obtain from z a symmetric tensor (called the symmetrization of z) s*z=
(A.12.2.1)
c
OE
b ' Z
Gn
and an antisymmetric tensor (called the antisymmetrization of z)
c
a .z =
(A.12.2.2)
&&.
z).
O€G,
It is evident that s . z is symmetric; to show that a * z is antisymmetric, we observe that, for any p E 6, , p
* (@
*
2)
=
c CAP
oeQ,
O
(a . z ) ) = & p
*
c
ae6.
&pO((Pc) . 2)
= Ep . ( a * z ) .
If z is already symmetric, then (A.12.2.3)
s .z = n ! z ,
and if z is already antisymmetric, (A.12.2.4)
a * z = n !z.
The n-linear mapping (x,, .. . , x,,)++s(x~Q Q x,,) of E" into T"(E) is symmetric, and the n-linear mapping (x1, . ,x,,)++u(x, Q 0 x,,) of Eninto T"(E) is alternating (or antisymmetric). In particular, if xi = xi for some pair of distinct indices i, j , then a(xl Q * . . Q x,,) = 0.
..
APPENDIX: MULTILINEAR ALGEBRA
358
If z E T"(E) and z* E T"(E*),then for each permutation a E 6,we
(A.12.3)
have
( 5 . z,
(A.12.3.1)
z*) = (z, a-l * z*)
by virtue of the formula (A.ll .I.2), because
If we identify covariant tensors with multilinear forms on E (A.10.3), we have therefore (a
-
z*)(x1,
. ..,x,)
= Z * k ( l )7
a .
-
Y
xu,,,);
consequently, symmetric (resp. antisymmetric) covariant tensors may be identified with symmetric (resp. antisymmetric or alternating) multilinear forms (A.6.3).
13. T H E EXTERIOR ALGEBRA
All the tensors considered in this section are contravariant. (A.13.1) Let E be a finitely generated free A-module, (eJlsiSma basis of E. The antisymmetrization a(ei, @ ei2€3. * 69 ein)is zero whenever two of the indices ik are equal (A.12.2). On the other hand, if the indices ik are all distinct (which requires that n 5 m), there is a unique permutation a E G, such that < iu(z)< * * . < iU(,,).For each subset H = { i l , i z , . .., in} of n elements of the set {1,2, .. .,m} such that i, < i2 < * . . < in, the elements eH =
-
a (ei, @ ei2@ .*
Bein)therefore form a basis (consisting of
(3
elements)
of the A-module A,,(E) of antisymmetric tensors of order n over E. (A.13.2)
Given two antisymmetric tensors zp E. AJE), zq E A,(E), their exdenoted by zp A zq ,is defined to be the antisymmetric tensor of q given by the formula
terior product,
order p
+
(A.13.2.1)
1
Zp A Zq = -a ( Z P @ Zq).
p !q !
We shall prove the following two fundamental properties:
13 T H E EXTERIOR ALGEBRA
(A.13.2.2)
359
(Anticommutativity) I f z , E A,(E) and z,, E A,(E), then
.
Z,,A 2, = (- l)”ZP A Z,,
(A.13.2.3)
(Associativity) Ifz, E A,(E), z,, E A,(E), z, E AJE), then Zp A (Zq A 2,)
= (2, A 2,) A 2,.
To begin with, we shall establish two preliminary results: (A.13.2.4)
If t, (resp. t,,) is a tensor of order p (resp. q), then aMr,) 6 t,,) = P!ao, 6 tq), a@, 6 4t,N = q !4,6 t$.
We have a(a(tp) 6 t q ) =
C
C
~u Erg usSP+, roep
*
((7
tp)
6 tq);
but we can identify G, with the subgroup of G,+, which fixes the integers > p in {1,2, . . . , p + q}. We have then
=p !
c
-
E p P ( t p 6 t,,) = P ! U ( t , @ t,,)‘
PESP+4
The other formula is proved in the same way. (A.13.2.5)
Zft, (resp. t,,) is a tensor of order p (resp. q ) then
360
APPENDIX:
MULTILINEAR ALGEBRA
On the other hand, if we put y i = x ~ ( ~then ) , the permutation c’ such that ye,(i)= xer(i)is equal to z-’az, so that E,. = E,, . Consequently
c
&e~e(,,+l)@...@~e(,,+q)@~e(~)@...~~e(,,)
UEQp+*
=
c
E d Ye,(,)
e’ E B p + q
= a(r,
€3* * * @ Y a y q )@ Y e y q + 1) @ * * * @ Y o ’ ( p + q )
@ t,).
The general case now follows by linearity. (A.13.2.6) that
To prove (A.13.2.2) and (A.13.2.3) it is now enough to observe
The relation (A.13.2.2) is then an immediate consequence of (A.13.2.5). As to (A.13.2.3), we have
by the second formula of (A.13.2.4). Similarly the first formula of (A.13.2.4) gives us (Z,,A Zq) A Z,.=
1
p! q ! r !4
z p
@ Zq @ z r )
and (A.13.2.3) is therefore proved. This proof shows, by induction, that for any family of h antisymmetric tensors zpkE A,,(E) (1 5 k 5 h),
In particular, for n vectors xi E E (1
s j 5 n), we have
13 THE EXTERIOR ALGEBRA
(A.13.2.8)
XI A X 2 A
”.
AX,,
= a(Xl @X2 @ *
361
@Xn).
* *
Consequently
for all permutations u E 6,; and if xi = xi for two distinct indices i, j , then XiAX2A
*.’
AX,,=O.
(A.13.3) By reason of the last formula, the module A,,(E) is called the nth exterior power of the finitely-generated free A-module E, and is denoted by n
A E. The basis of A E associated with the basis (ei)of E consists of the fl
elements eH= ei,
(A.13.3.1)
A
ei,
A
* *
. A ein,
where H runs through the set of subsets {il,i,, .. ., in} of (1,2, i, < i2 < *..
. .. ,m}, and
For each alternating n-linear mapping u (A.6.3) of En into an arbitrary n
A-module F , there exists a unique A-linear mapping v : A E + F such that
for all xi E E.
.
.
< i,, in (1, 2, .. , m}, For if u(e,, , ei2, . ., ei,) = cH E F for i, < i, < we dejine u by the conditions u(eH)= cH;clearly u has the required properties, in view of the hypothesis on u. (A.13.4) Consider two finitely generated free A-modules E, F, and let u : E 4F be an A-linear mapping. Then the mapping
( x , , x2, . . . ,X,)H
u(xJ A u(x,)
A
..-
A
u(x,,)
n
A F is clearly alternating and n-linear. By virtue of (A.13.3), there exists therefore a unique linear mapping u : A E + A F such that of Eninto
n
v(xl
for all x j E E (1
A
x2 A * .*
AX,,)
= u(xl) A u(x2)A
n
*
. A u(xJ
s j 5 n). The mapping u is called the nth exterior power of u
and is denoted by
A u. n
APPENDIX:
362
MULTILINEAR ALGEBRA
be a basis of E, and let cf)15i 6 p be a basis of F. Suppose Let (ej), that n 5 inf(m, p) (otherwise A u = 0) and let X = (uij) be the matrix of u (with p rows and m columns) relative to these two bases. We shall calculate n
n
the matrix of A u relative to the associatedbases (eK)of A E and cfH)of A F, where K (resp. H) runs through the set of all subsets of n elements of {1,2,
...,m} (resp. {1,2, ..., p}), these
n
c) (3) (resp.
sets being arranged
in an arbitrary order. We have by definition
so that, if K = {jl,j2, ...,j,} with j , <j 2 < *
= ( i i , iz,.
Ui,j, UiZj2*
.. ,in)
* *
- <jn,
a. W .n f. 11
Ah2 A
* * *
A
fin
,
the summation being over all sequences (il, i, , ..., i,,) of n distinct elements of the index set {1,2, .. .,p}. Now group together all the terms in the sum for which the set H = {il, i, , . .. , in}is the same; among the n! sequences having H as underlying set there will be one for which il < i, < < in, and the others will all be of the form (iu(l),in(,),.. . , i,(,)), where IT runs through 6, ; hence, by virtue of (A.13.2.9), we have
where H runs through the set of subsets of n elements of {1,2, ..., p}. The Scalar factor multiplying fH iS the determinant of the n x n matrix XHK= (&)1 s h , R B n , where Phk = uihjk(A.6.8.1). Hence, with this notation, we may write (A.13.4.1) n
and therefore the matrix of A u relative to the bases (eK) and matrix (det(XHK))formed by the n x n minors of X.
In particular, if x i =
(fH)
is the
1 tijej are n elements of the A-module E, these m
j= 1
considerations can be applied to the mapping u :A" + E which maps the elements a,, a, , . ..,a, of the canonical basis of Anrespectively to xl, x2 , .. . , xn ;we obtain
13 THE EXTERIOR ALGEBRA
363
(A.13.4.2)
where H runs through the set of all subsets of n elements j , <j , < * <j , of {1,2,. ..,m}, and xHis the n x n matrix (I]),&, where qhk= &,, j k for 1S h , k S n .
In particular, if n (A.13.4.3)
= m,
AX,=det(X)elAe2A
X1AX2A
* . - he,,
where X is the square matrix whose j t h column consists of the components tij (1 6 i 6 m) of x i . (A.13.5) The definition of the exterior product (A.13.2.1) enables us to define a structure of an (associative) A-algebra on the A-module which is the direct sum of the exterior powers of E,
0
(where conventionally defined by the formula
A E =A
1
and
A E = E),
the multiplication being
(A.13.5.2)
A E. (When j = 0, z j is an element 1 of A, and the product z j A Z ; is taken by definition to be Azi in the module A E.) Associativity follows directly from (A.13.2.3), and the identity element of A is also the identity element of A E. This algebra is called the exterior algebra of E; as for all z j , zj in
j
k
A-module it admits a basis consisting of the 2" elements eH,where H runs through the set of all 2" subsets of (1, 2, .. . , m} (we define eD to be 1 E A). The multiplication table for this basis is given by (A.13.5.3)
eH/\eK=O if H n K f 0 , eHAeK= pH,KeHUK if H n K = 0 ,
with pH,K= (- l)", where v is the number of pairs (i, j) E H x K such that i > j . This follows immediately from (A.13.2.9), by considering the permutation which is the product of the transpositions T~~ (where T i j ( i ) =j,z i j ( j ) = i, and .rij(h)= h for h # i, j in H u K) for i E H,jE K, and i > j .
364
APPENDIX:
MULTILINEAR ALGEBRA
It follows immediately from above that if E is the direct sum of two submodules M, N, then submodules of
A E.
A M and A N may be identified canonically with n
n
n
(A.13.6) Let B be a commutative ring which contains A, has the same identity element as A, and is a finitely-generated free A-module. Then the isomorphism (A.10.6.3) generalizes to an arbitrary finite number of factors, and in particular gives rise to a canonical A-module isomorphism
T"(E(B))
(A.13.6.1)
(Tn(E))(B)*
It is immediately clear that this isomorphism transforms antisymmetric tensors into antisymmetric tensors, and therefore induces a canonical isomorphism
which is a B-algebra isomorphism, as is easily verified. If E is a vector space of finite dimension m over a field K, the notion of exterior product enables us to express in a simple form the linear independence of n vectors xl, .., x,, in E: a necessary and sufficientcondition for this is that (A.13.7)
.
(A.13.7.1)
X i A X 2 A " ' AX,,
# 0.
-
For it is clear that the exterior product x1A x2 A * AX,, will vanish if one of the xi is a linear combination of the others. Conversely, if the x j are linearly independent, then there exists a basis of E in which xl, .. ., x,, are the first n vectors (A.4.8); hence x1 AX^ A * . * AX,, is an element of the basis of
A E associated with this basis of E, and therefore is #O. n
The elements of over a ring A).
A E are often called p-vectors (even when E is a module P
14. DUALITY IN THE EXTERIOR ALGEBRA
(A.14.1) Let E be a free A-module with a finite basis ( e J l j i 6 , , , .It follows from the definitions (A.13.1) that the A-module T"(E) of n-fold contravariant
tensors over E splits up into the direct sum of the submodule
n
A E = A,,(E)
14 DUALITY
IN
THE EXTERIOR ALGEBRA
365
-
and the submodule spanned by the basis elements e i , @I eil @I . @ ein for which the sequence ( i l , i 2 , ..., in) either has two terms equal, or all terms distinct but not in increasing order. Hence (A.9.5) every linear form on the A-module A,(E) = A (E) is the restriction to this submodule of a linear form ZI+(Z, t * ) on T"(E), where t* E T"(E*) (A.11.1.4). Now, by virtue of (A.12.3.1), we have (2, t*) = E,(O z, t * ) = (z, E, cr-' * t*) for all z E An(E) and cr E 6, ; summing over all permutations u, we obtain n
-
1
(z, t * ) = - (z, a(t*)).
(A.14.1 .l)
n!
In other words, if for each z* E A,(E*) we denote by 6(z*) the restriction of the linear form Z H ( l / n ! ) < z ,z*) to A,(E), then 6 is a surjectiue linear mapping of An(E*)onto the dual (A,(E))* of A,(E); but 6 is also injectiue, because if z* E An(E*) is such that (a(t),z * ) = 0 for all t E T"(E), the same argument together with the fact that a(z*) = n ! z* gives ( t , z*) = 0 for all t E Tn(E), that is to say, z* = 0. n
The mapping 6 therefore identifies the exterior power /\(E*) with the
( /I
dual E)*. If xi (1 5 j 5 n) are elements of E and xi* (1 5 j of E*, then by virtue of (A.14.1 .I)we have
5 n) elements
that is to say (A.6.8.1), (A.14.1.2)
(xl
A
x2 A . . . A x,, x r
A
x;
A
..
*
A
x:> = det((x,, xT>). n
If ( e r ) is the basis of E* dual to (e,), and if ( e H and ) (e:) are the bases of and
n
A E* associated with (e,) and (et),
(A.14.1.3)
( e H ,e:)
= 6,,
AE
respectively, then it follows that (Kronecker delta)
for any two subsets H, K of n elements of {I, 2, . .. , m}. In other words, ( e i ) is the basis dual to ( e H )when
n
A E* is identified with the dual of A E. n
366
APPENDIX:
MULTILINEAR ALGEBRA
The elements of A E* may be canonically identified with the alternating n-linear forms on E* (A.13.3); these are also called exterior forms of degree n on E, or n-forms (or n-couectors when E is a vector space). (A.14.2)
Suppose that E is the direct sum M 8 N of two finitely generated n
n
AnM and A N may be n identified with submodules of A E. For each z* E A E*, considered as a n n linear form on A E, we may therefore speak of the restriction of z* to A M free modules M, N. We have seen (A.13.5) that
(or, as is also said, the restriction of z* to M, considered as an alternating n-linear form on M). P
4
Let p , q be two integers >O, let u E A M,u E A N, and let u* be an eleP
AX,* an element of ment of A E* and u* = x: A xf A linear forms xi* (1 S j q) are zero on M. Then we have
(A.14.2.1)
( U A V , U* A U * ) = ( U , U * ) ( U ,
A E* such that the 4
U*).
It is enough to verify this relation when u = a, ~a~ A * * * A U ~and u = b, A b2 A * * . A b, and u* = c: A cf A * * A c,* , where the ai belong to M, the bj to N and the cr are arbitrary elements of E*. The formula then follows from (A.14.1.2) and the rule for calculating the determinant of a matrix by blocks (A.7.4.1), since ( a i , x,*) = 0 for 1 5 i 5 p and 1 2 k 2 q.
-
15. INTERIOR PRODUCTS
(A.15.1) We retain the hypotheses and notation of Section 14. Let p , q be two integers 20 and let z4 E T4(E) be a contravariant tensor; then the mapping v p w up @ z4 of Tp(E) into Tp'4(E) is linear, and its transpose (A.9.3) may therefore be identified with a linear mapping of TP+4(E*)into Tp(E*), which is denoted by
*
Up+qHZq
4+,
. this definition we have and is called the interior product of z4 and u , * + ~From (A.15.1 .I)
= ( u p @ zq , .;+,>
15 INTERIOR PRODUCTS
367
it follows immediately that (ej, @ * *
(A.15.1.2)
8 ej,,) _J (etl @ * * @ ek*,+,) = 0
*
unless& = kP+,,for 1 S h 6 q ; and that (A.15.1.3)
(ej, @ * - * @ ej,) J (ez1 8 * . * @ eK*,+,) = ef, @
*
a
*
€31ek*,
if j h= kP+, for 1 5 h 5 q. (A.15.2) It follows directly from the definition (A.15.1) and the associativity of the tensor product (A.ll .I .2) that, for any three integers p , q, r 2 0,
@ .a1 Uf+,+,
z; 1 (z; -I u,*+4+,) = (.I,
(A.15.2.1)
In particular, the interior product u,*+~ H X J u,*+~by an element x E E is sometimes denoted by i(x). The interior product by a tensor x1 @ x2@ * * * @xpE Tp(E)
may therefore be written as i(xl) 0 i(xJ (A.15.3)
0
-
*
. i(xp). 0
Now consider the linear mapping up++ up
where z4 is an element of
A
zq of
P
A E into A E, P+4
A E; its transpose, which is a linear mapping of 4
A E* into A E*, is denoted by
P+4
P
and is called the interior product of z4 and u,*+, . However, it should be remarked that this interior product is not the same as the restriction to antisymmetric tensors of the interior product defined in (A.15.1) (this double use of the same notation does not in practice cause any confusion). Hence we have now (A.15.3.1) P
for all up E A E and u,*+, E
A E*. With the notation of (A.13.5), we have
P+4
<eH, eK J e t > = <eHheK, e*,>
and therefore, by virtue of (A.13.5.3), (A.15.3.2)
eK A e t = O {eKJ
e *, -pL-K,Ke*,-,
if K g L , if K c L .
368
APPENDIX:
MULTILINEAR ALGEBRA
(A.15.4) From the associativity of the exterior product (A.13.2.3) we deduce immediately (A.15.4.1)
J (2: J u,*+,+,) = ( Z J A Z t ) -I U,*+,+,.
2;
The interior product u;+~H X J u;+, by an element x E E is denoted by i(x), so that the interior product by x1 A X , A AX^ may be written as i(xl) 0 i(xz) 0 * * . 0 i(xp).In particular, i(x) 0 i(x) = 0.
(A.15.4.2)
Explicitly, i(x) is given by the formula
with the usual convention that the symbol below the circumflex is to be omitted. By linearity, it is enough to consider the case where x = e j and x: = ei*,,where i, i, < * * * < i p + l ,and then the above formula follows from
-=
(A.15.3.2).
16. NONDEGENERATE ALTERNATING BILINEAR FORMS. SYMPLECTIC GROUPS
(A.16.1) Let E be a uector space of dimension movera(commutative) field K, and let B be an alternating bilinearform on E x E, so that B may be identified
with an element of A E*. We shall show, by induction on m, that there exists a basis (ei)lsis, of E and an integer r 2 0 such that, if (e:) is the dual basis of E*, then we have 2
(A.16.1 .I)
B
= e: A
e:
+ e:
A
ez
+
*.*
+ e:l-l
A
e2*,.
We may assume that B # 0, or there is nothing to prove. Then there exists a bivector el A e , such that B(el, e z ) # 0 . The vectors e,, e, are therefore linearly independent, and by multiplying one of them by a nonzero scalar we may assume that B(e,, e,) = 1. Let F be the subspace of E consisting of the vectors x such that B(e,, x ) = B(e, ,x) = 0. Then F is of codimension 5 2 in E (A.9.6); and if P is the plane spanned by el and e , , then P n F = {0} (for if B(e,, clel + De,) = B(e2, mel + Bez) = 0 we obtain p = 0 and -a = 0).
16 NONDEGENERATE ALTERNATING BILINEAR FORMS
369
Hence the subspaces F and P are supplementary. If B , is the restriction of B to F x F, then by the inductive hypothesis there exists a basis (ej)35 of F such that
B, = e:
A
ez
+ * - . + e,*,-,
A
e2*,.
In view of the definition of F, we conclude that
which is equivalent to (A.16.1 .I). The number r is independent of the basis (e,) satisfying (A.16.1 .I),for it follows from this relation that if B A Sdenotes the product of s bivectors equal to B in A E*, then Bh(r+l)= 0, because this product is a sum of products of 2r 2 factors taken from the set {e:, . .., e:r}, and each such product must contain a repeated factor and therefore vanishes. On the other hand, since the bivectors .efi and e;j-l ~ e commute, : ~ we have
+
(A.16.1.3)
B A ' = r ! e : A e , * A e ~ A e , * A . . . A e , * , -A,e , * , # 0 .
The integer 2r is called the rank of B, and if 2r = m, then B is said to be nondegenerate. Equivalently, B is nondegenerate if and only if there exists no vector x # 0 in E such that B(x, y ) = 0 for all y E E. When m = 2r and B is nondegenerate, a basis (ei)l i s m of E for which (A.16.1 .I)holds is called a symplectic basis of E (relative to B). (A.16.2) For each x E E, i(x)(B)= @(x) is a vector belonging to the dual space E*. If B is givenby(A.16.1.l),itisimmediatelyseenthat@(ezj) = - e f i - l and @ ( e z j - l )= e z j for 1 S j r, and that Q,(e,) = 0 for k > 2r (A.15.4.3), so that CP is a linear mapping of rank 2r (A.4.16) of E into E*. Consequently Q, is bijective if and only if B is nondegenerate.
Suppose for the rest of this section that B is nondegenerate (so that m = dim(E) is euen). Two vectors x , y E E are said to be orthogonal (relative to B ) if B(x, y) = 0 ; by (A.15.3.1), this is equivalent to (y, aS(x)) = 0. If V is any vector subspace of E, the set of vectors y E E orthogonal to all x E E is a vector subspace V' of E, which is equal to Q,-'(Vo) in the notation of (A.9.6), and is called the orthogonal supplement of V (relative to B). Since Q, is bijective, it follows from (A.9.6) that: (A.16.3)
(A.16.3.1)
codim V' = dim V,
370
APPENDIX: MULTILINEAR ALGEBRA
=v,
(V')'
(A.16.3.2)
(V,
(A.16.3.3)
+ V2)* = V:
n V:
,
(V, n V,)' = Vi f V i
.
A subspace V of E is said to be isotropic if V n V' # {0}, and totally isotropic if V c V'. Every one-dimensional subspace of E is totally isotropic. If V is isotropic, then V n V' is totally isotropic by virtue of (A.16.3.2) and (A.16.3.3). It follows from (A.16.3.1) that if V is totally isotropic, then dim V g +m, and this upper bound is attained by the subspace generated by the vectors el, ...,e, (where 2r = m) of a symplectic basis of E. (A.16.4)
The bijective linear mappings u : E
E such that B(u(x), u(y)) = 2
B(x, y ) for all x, y E E (or, equivalently, such that A ('u)(B) = B) are called symplectic automorphism of E (relative to B) and form a group called the symplectic group of E (relative to B), which is denoted by Sp(E, B). The relation A ('u)(B) = B implies immediately (A.13.3.2) that 2
m
A('u)(B"') = BAr,
where r = t m ; hence, from (A.16.1.3) and (A.13.4.3), we have det(u) = 1 for all u E Sp(E, B). In other words, Sp(E, B) is a subgroup of the special linear group SL(E) c GL(E) (the group of automorphisms of the vector space E which have determinant equal to 1). Relative to a symplectic basis of E, the matrices of symplectic automorphisms are the matrices U which satisfy the relation
'UsJ . U = J,
(A.16.4.1)
where J is the square matrix of order 2r = m: 0 0 1 0 0 0 0 0 0 0 -1
-1
J=
0 *** 0 *.. 1
*..
0
*.'
............., .a .
0 0 0 0
0 0 0 0
-.. .**
0 0 0 0
0 0 0 0
0 1 -1 0
All the symplectic groups Sp(E, B) relative to different nondegenerate alternating forms B on E x E are therefore isomorphic. The group of matrices U satisfying (A.16.4.1) is denoted by Sp(m, K) (it is therefore defined only for even m).
17 THE SYMMETRIC ALGEBRA
371
17. T H E SYMMETRIC ALGEBRA
The developments of (A.13) can be repeated by replacing antisymmetric tensors throughout by symmetric tensors, and the antisymmetrizationoperator u by the symmetrizationoperator s. The A-module S,,(E) of symmetric tensors of order n has a basis obtained as follows: For each integer p >= 1, let e?P denote the tensor product e , @ * * @ e , with p factors, and for each multiindex u = (a1, . .,ad E N", put
-
.
e" = s ( e f ' " 1 eFa2 ~ @ - * - o e:"m). Then the e" such that Iu I = n form a basis of S,,(E). Next we define the symmetric product zpzqof an element z p E S,(E) and an element zq E Sq(E)by the formula 1 zpzq= -s(zp€3zqh
(A.17.1)
p!q!
and just as in (A.13) one proves that (A.17.2)
z pzq = zqz p
(A.17.3)
(z, zq)z, = zp(zqz,)
(commutativity), (associativity).
In particular, for n elements xl, ..., x, of E we have (A.17.4)
x,x,
..* x,
= s(x1 Ox2 @ * . - Ox.).
For this reason the module S,(E) is called the nth symmetric power of E. The (infinite) direct sum of the symmetric powers of E:
where So(E) = A and S,(E) = E, becomes a commutative and associative A-algebra with the multiplication defined by (A.17.6)
(when p = 0, so that zp = 1 E A, the product z p z i is taken to be the product Azi in the A-module Sq(E)). This algebra S(E) is called the symmetric algebra
372
APPENDIX: MULTILINEAR ALGEBRA
of the A-module E; it has a basis consisting of the em,where a E N" (and e0 is taken to be the identity element 1 of A), and the multiplication of basis elements is given by = em+@.
(A.17.7)
This proves that the symmetric algebra S(E) is isomorphic to the polynomial algebra in m indeterminates A[X,, .. .,X,]. Finally, as in (A.14), we can put Sn(E)and S,(E*) in duality, in such a way that
but it should be remarked that S(E*) is only a submodule of the dual of S(E).
18. DERIVATIONS A N D ANTIDERIVATIONS O F GRADED ALGEBRAS
(A.18.1) In this and the following section, the algebras under consideration will not be assumed necessarily to be associative. An algebra E over a commutative ring A with identity element is therefore an A-module E together with an A-bilinear mapping E x E --f E, denoted by (x, y) H xy. The algebra E is said to be graded if E is the direct sum of a sequence (En)n2o of submodules, such that
for all pairs of integers m, n 2 0. If E admits an identity element e (so that ex = xe = x for all x E E), we assume also that e E E, . The elements of En are said to be homogeneous of degree n. The zero element 0 is therefore homogeneous of all degrees, but a homogeneous element x # 0 belongs to only one En, and the integer n is called its degree. The exterior algebra (A.13.5) and the symmetric algebra (A.17) of a finitely-generatedfree A-module M are graded algebras, graded, respectively, by the submodules A M and SJM). n
Let E be an A-algebra. A mapping d : E + E is called a derivation of E if it is A-linear and satisfies (A.18.2)
18 DERIVATIONS AND ANTIDERIVATIONS
373
for all x, y E E. For example, if E is associative and a E E, the mapping ad@) :
(A.18.2.2)
XHUX
- xu = [a, x ]
is a derivation (called an inner derivation). If E is a graded algebra, define a linear mapping d : E + E by putting d(xp)= px, for all p 2 0 and all x, E Ep. Then d is a derivation, for if x, E E, and x4 E E, , we have
4x,x4) = (P + dxpx4 = (dxp)x,
+ x p (dx,).
If E is associative, then by induction on n we obtain from (A.18.2.1) (A.18.2.3)
d(x,x2
x,,) =
i= 1
x1
xi-, (dxi)xi+,
x,,
for all x,, ..., x,, E E and any derivation d : E + E. If E has an identity element e # 0, we have d(e) = 0
(A.18.2.4)
because from e2 = e we obtain d(e) = d(e2) = e * d(e) + d(e) * e = d(e) + d(e). (A.18.3)
Let d,, d2 be two derivations of an algebra E. Then the linear
mapping
is a derivation, since we have
4 (d2(XY)) = dl ((d2X ) Y + x (d2 Y ) ) =
(4 (d2 4
Y
+ (d2 ( 4 Y ) + (d,x) (d2 Y ) + x (dl(d2Y))
and our assertion follows by interchangingthe indices 1 and 2 and subtracting. (A.18.4) When E is a graded A-algebra, a derivation d of E is said to be of degree r (where r is an integer, positive, negative, or zero) if (A.18.4.1)
d(E,,) = Em+,
for all n 2 0 (with the convention that Em = (0)for m < 0).
374
APPENDIX:
MULTILINEAR ALGEBRA
With the same convention, an antiderivation of E of degree r is defined to be an A-linear mapping d : E --+ E which satisfies (A.18.4.1) and the relations d(x, x,,) = (dx,)
(A.18.4.2)
X,
+ (- 1 ) " ' ~(dx,,) ~
for all m, n 2 0 and x, E Em, xn E En. An antiderivation of even degree r is therefore a derivation of degree r. (A.18.5) Let M be a finitely generated free A-module and let E = A (M*) be the exterior algebra of the dual module M*. For each element x E M, the
mapping i(x) is defined on A M*for all integers n 2 1 (A.15.4), and we extend it to the whole of E by linearity and by taking it to be the zero mapping on n
0
A = /"\ M*. This mapping is an antiderivation of E of degree - 1. To prove this assertion we must show that
for all u,* E
A M* and u: E A M*. By linearity, we may assume that u*, = P
4
xf A X ; A * - . A X * , and u,* = x,*+, A AX,*+,, where the xi* are arbitrary elements of M*, and then the result follows immediately from (A.15.4.3).
If the graded algebra E has an identity element e (necessarily of degree O), then we have d(e) = 0 for every antiderivation d of E, for the same reason as in (A.18.2.4). Suppose that E is associative and let xi E Em,for 1 SJ 5 n ; then if d is any antiderivation of E of degree r, we have
(A.18.6)
(A.18.6.1)
by induction on n. From this formula and from (A.18.2.3) it follows that if two derivations (resp. antiderivations)of E of the same degree coincide on a set of homogeneous generators of E, then they are identical. (A.18.7) (i) r f d is an antiderivation of odd degree r, its square d 0 d is a derivation of degree 2r. (ii) Ifdl (resp. d,) is an antiderivation of degree r (resp. s), then the linear mapping
19 LIE ALGEBRAS
375
XMdi(d2X) - (-l)'"d,(dIx)
(A.18.7.1)
is an antiderivation of degree r + s. Let x be a homogeneous element of E of degree n. Then, for all y E E, we have dl(d,(XY)) = (dl(d2X ) ) Y + (- l P + " Y 4X)(dlY) (- l)"(d,x)(dZ y ) (- 1)('+s)"~(d~(d2 y)).
+
+
If dl = d2 = d and r = s is odd, this shows that d d is a derivation. Next, if we interchange dl and d2 in this equation, we obtain 0
dl(d,(XY)) - (- l)"d,(dl(XY)) = (4 (d2 X ) ) Y - ( - 1 )'"(d,(dlX))Y + (- l)'""'"~(d,(d2 y ) ) - (- l)rs+(r+s)n x(d2(dl Y)),
which proves (ii).
19. LIE ALGEBRAS
A Lie algebra over a commutative ring A (with identity element) is an A-module g endowed with an A-bilinear mapping of g x g into 9, usually denoted by (x, y ) [x,~y ] , which satisfies the identities
[x,X I
(A.19.1) (A.19.2)
tx, [ Y , zll
= 0,
+ [ Y , tz, X I ] + [z, [ x , yl1 = 0
for all x , y , z E g. The identity (A.19.2) is called the Jucobi identity. From (A.19.1) we deduce immediately that (A.19.3)
b,Y l = - [ Y ,
XI.
If 1: is an associative A-algebra, then the A-module E endowed with the A-bilinear mapping (x, y ) ~ x -y y x is a Lie algebra. A subalgebra fj of a Lie algebra g (i.e., a submodule fj of g such that [x,y ] E b for all x, y E b) is clearly a Lie algebra. If E is an arbitrary Aalgebra, the subalgebra of the algebra End,(E) of A-module endomorphisms of E, formed by the derivations of E, is a Lie algebra with respect to the bracket (A.18.3.1).
376
APPENDIX:
MULTILINEAR ALGEBRA
A submodule a of a Lie algebra g is an ideal of g if the relations x E g and y E a imply [x, y] E a (or equivalently [y, x] E a). If 2, 3 are two elements of the quotient A-module g/a (i.e,, cosets of a in g), then the values of [x, y] for all x E iand y E j belong to the same coset of a in g, because if x’ - x E a and y’ - y E a, we have [x’, y’] - [x, y] = [x’, y’ - y] [x’ - x, y]. This coset is denoted by [i31, , and it is immediately verified that the mapping (i j , ) [i ~ 31,defines a Lie algebra structure on g/a. The A-module g/a, endowed with this structure, is called the quotient Lie algebra (of g by a). If g, g’ are two Lie algebras, a homomorphismf of g into g’ is an A-linear mapping such that f([x, y]) = Lf(x),f(y)] for all x, y E g. The kernel off is an ideal a of g and the image off a Lie subalgebra of g’, canonically isomorphic to g/a. If gl, g2 are two Lie algebras over A, the product A-module g1 x g2 is a Lie algebra with respect to the law of composition
+
((x19 x21, (Yl, Y 2 ) ) H ([Xi, Yll, [x2 9 Y2l). The verification of the axioms (A.19.1) and (A.19.2) is immediate. This Lie algebra is called the product of g1 and g2. The mapping x1H (xl, 0) (resp. x2w(0, xz)) is an isomorphism of g1 (resp. g2) onto an ideal of g1 x g2 ; usually we shall identify g1 and g2 with these ideals of g x g2. The quotient algebra (gl x g2)/g1 (resp. (gl x g2)/g2) is then identified with g2 (resp. gl). Iff is a homomorphism of g1 into g2, its graph in g1 x g2 is a Lie subalgebra of g1 x g2, and the mapping x1 H (xl, f (xl)) is an isomorphism of g1 onto this subalgebra. Let g be a Lie algebra. For each x E g, the linear mapping y~ [x, y ] is a derivation of g, denoted by ad,(x) or ad(x). The mapping x ~ a d ( x is ) a homomorphism of g into the Lie algebra Der(g) of derivations of g. For each derivation D E Der(g) we have [D, ad(x)] = ad(Dx). (A.19.4)
In view of (A.19.3), the Jacobi identity may be written in the form
-
ad(x) * [Y, 4 = tad(x) * y, 21 + [Y, a W ) 21, which proves the first assertion. It can also be written in the form
-
ad([x, y]) z = ad(x) . (ad(y) * z ) - a d Q (ad(x) * z), which proves the second. Finally, we have from the definitions tD, ad(x)l * y
= D([x,
Yl) - 1x9 Dyl
-
= [Dx, y] = ad(Dx) y,
which proves the last assertion.
19 LIE ALGEBRAS
377
A derivation of g of the form ad(x) is called an inner derivation. A submodule a of g is an ideal if and only if it is stable under all inner derivations. If a and b are ideals in a Lie algebra 9, then a + b and a n b are ideals in g. If 6 is a Lie subalgebra of g and a is an ideal of 9, then $ + a is a Lie subalgebra of g, and ($ a)/a is the image of lj under the canonical homomorphism of g onto g/a, which is canonically isomorphic to $/($ n a). If a, b are two submodules of a Lie algebra, we denote by [a, b] (by abuse of notation) the submodule of g generated by all [x, y] with x E a and y E b. Clearly [a, b] = [b,a]. If a, b are ideals in 9,so is [a, b]; this follows immediately from the Jacobi identity. A Lie algebra g is said to be commutative if [x, y ] = 0 for all x , y E g. In that case every submodule of g is an ideal, and every quotient algebra of g is commutative. The ideal [g, g] is the smallest ideal a such that g/a is commutative. It is called the derived ideal of g and is denoted by 9 g . The derived series of g is the decreasing sequence of ideals defined by induction in the following way :
+
909 = g,
Wf'g = [ W g , Wg].
A Lie algebra g is called solvable if there is an integer p 2 0 such that W g = (O}.
REFERENCES
VOLUME I
[I] Ahlfors, L., “Complex Analysis,” McGraw-Hill, New York, 1953. [2] Bachmann, H., “Transfinite Zahbn” (Ergebnisse der Math., Neue Folge, Heft 1). Springer, Berlin, 1955. [3] Bourbaki, N., “ Elements de Mathkmatique,” Livre I, “Thkorie des ensembles” (Actual. Scient. Ind., Chaps. I, 11, No. 1212; Chap 111, No. 1243). Hermann, Paris, 1954-1956. [4] Bourbaki, N., “ElCments de MathCmatique,” Livre 11, “Alg&bre”(Actual Scient. Ind., Chap. 11, Nos. 1032, 1236, 3rd ed.). Hermann, Paris, 1962. [5] Bourbaki, N., “ ElCments de MathCmatique,” Livre 111, “ Topologie gknbale” (Actual. Scient. Ind., Chaps. I, 11, Nos. 858, 1142,4th ed.; Chap. IX,No. 1045, 2nd ed.;Chap. X,No. 1084,2nd ed.). Hermann, Paris, 1958-1961. [6] Bourbaki, N., “ ElCments de MathCmatique,” Livre V, “ Espaces vectoriels topologiques” (Actual. Scient. Ind., Chap. I, 11, No. 1189, 2nd ed.; Chaps. 111-V, No. 1229). Hermann, Paris, 1953-1955. [7]Cartan, H., Sminaire de I’Ecole Normale Supixieure, 1951-1952: “ Fonctions analytiques et faisceaux analytiques.” [8] Cartan, H., “ T h b r i e GlCmentaire des Fonctions Analytiques.” Hermann, Paris, 1961. [9] Coddington, E., and Levinson, N., “Theory of Ordinary Differential Equations.” McGraw-Hill, New York, 1955. [lo] Courant, R., and Hilbert, D., “Methoden der mathematischen Physik,” Vol. I, 2nd ed. Springer, Berlin, 1931. [I I] Halmos, P., “Finite Dimensional Vector Spaces,’’ 2nd ed. Van Nostrand-Reinhold, Princeton, New Jersey, 1958. [I21 Ince, E., “Ordinary Differential Equations,” Dover, New York, 1949. [I31 Jacobson, N., “Lectures in Abstract Algebra,” Vol. 11, “Linear algebra.” Van Nostrand-Reinhold, Princeton, New Jersey, 1953. [I41 Kamke, E., “ Differentialgleichungen reeller Funktionen.” Akad. Verlag, Leipzig, 1930. [I 51 Kelley, J., “ General Topology.” Van Nostrand-Reinhold, Princeton, New Jersey, 1955. [I61 Landau, E., “Foundations of Analysis.” Chelsea, New York, 1951. 378
REFERENCES
379
[ 171 Springer, G.,
“ Introduction to Riemann Surfaces.” Addison-Wesley, Reading, Massachusetts, 1957. [IS] Weil, A., “Introduction A 1’8tude des Varittks Kahleriennes” (Actual. Scient. Ind., No. 1267). Hermann, Paris, 1958. [19] Weyl, H., “Die Idee der Riemannschen Flache,” 3rd ed. Teubner, Stuttgart, 1955.
VOLUME II
[20] Akhiezer, N., “The Classical Moment Problem.” Oliver and Boyd, EdinburghLondon, 1965. [21] Arnold, V. and Avez, A., “Theorie Ergodique des Systemes Dynamiques.” GauthierVillars, Paris, 1967. [22] Bourbaki, N., “ Elements de Mathematique,” Livre VI, “Int~gration”(Actual. Scient. Ind., Chap. I-IV, No. 1175, 2nd ed., Chap. V, No. 1244, 2nd ed., Chap VII-VIII, No. 1306). Hermann, Paris, 1963-67. [23] Bourbaki, N., “ Elements de Mathematique: Theories Spectrales” (Actual. Scient. Ind., Chap I, 11, No. 1332). Hermann, Paris, 1967. [24] Dixmier, J., “ Les Algkbres d‘Opbateurs dans I’Espace Hilbertien.” Gauthier-Villars, Paris, 1957. [25] Dixmier, J., “ Les C*-Algkbreset leurs Reprksentations.” Gauthier-Villars, Paris, 1964. [26] Dunford, N. and Schwartz, J., “Linear Operators. Part 11: Spectral Theory.” Wiley (Interscience), New York, 1963. Vorlesungen uber Inhalt, Obefiache und Isoperimetrie.” Springer, (271 Hadwiger, H., ‘‘ Berlin, 1957. [28] Halmos, P., “ Lectures on Ergodic Theory. ” Math. SOC.of Japan, 1956. [29] Hoffman, K., “ Banach Spaces of Analytic Functions.” New York, 1962. [30] Jacobs, K., “ Neuere Methoden und Ergebnisse der Ergodentheorie ” (Ergebnisse der Math., Neue Folge, Heft 29). Springer, Berlin, 1960. [31] Kaczmarz, S. and Steinhaus, H., “Theorie der Orthogonalreihen.” New York, 1951. [32] Kato, T., “Perturbation Theory for Linear Operators.” Springer, Berlin, 1966. 1331 Montgomery, D. and Zippin, L., “Topological Transformation Groups.” Wiley (Interscience), New York, 1955. [34] Naimark, M., “Normal Rings.” P. Nordhoff, Groningen, 1959. 1351 Rickart, C., “General Theory of Banach Algebras.” Van Nostrand-Reinhold, New York, 1960. [36] Weil, A., “Adeles and Algebraic Groups.” The Institute for Advanced Study, Princeton, New Jersey, 1961.
VOLUME Ill
[37] Abraham, R., “ Foundations of Mechanics.” Benjamin, New York, 1967. [38] Cartan, H., Seminaire de 1’Ecole Normale Supkrieure, 1949-50: “Homotopie: espaces fibrks.” [39] Chern, S. S., “Complex Manifolds” (Textos de maternatica, No. 5). Univ. do Recife Brazil, 1959. [40] Gelfand, I. M. and Shilov, G. E., “Les Distributions,” Vols. 1 and 2. Dunod, Paris, 1962.
380
REFERENCES
[41] Gunning, R., “Lectures on Riemann Surfaces.” Princeton Univ. Press, Princeton, New Jersey, 1966. [42] Gunning, R., “Lectures on Vector Bundles over Riemann Surfaces.” Princeton Univ. Press, Princeton, New Jersey, 1967. [43] Hu, S. T., “Homotopy Theory.” Academic Press, New York, 1969. [44]Husemoller, D., “ Fiber Bundles.” McGraw-Hill, New York, 1966. [45] Kobayashi, S., and Nomizu, K.,“ Foundations of Differential Geometry,” Vols. 1 and 2. Wiley (Interscience), New York, 1963 and 1969. [46] Lang,S., “Introduction to Differentiable Manifolds.” Wiley (Interscience), New York, 1962. [47] Porteous, I. R., “Topological Geometry.” Van Nostrand-Reinhold, Princeton, New Jersey, 1969. [48]Schwartz, L., “Thhrie des Distributions,” New ed. Hermann, Paris, 1966. [49] Steenrod, N., “The Topology of Fiber Bundles.” Princeton Univ. Press, Princeton, New Jersey, 1951. 1501 Stemberg, S., “ Lectures on Differential Geometry.” F’rentice-Hall, Englewood Cliffs, New Jersey, 1964.
INDEX
In the following index the first reference number refers to the chapter in which the subject may be found and the second to the section within the chapter. A
Analytic manifold defined by a holomorphic function : 16.8, prob. 12 Analytic mapping : 16.3 Analytically compatible charts : 16.1 Antiderivation of degree r : A.18.4 Antisymmetric tensor : A.12.1 Antisymmetrization : A.12.2 Apsidal transformation : 16.20, prob. 4 Arcwiseconnected : 16.27, prob. 1 Arcwiseconnected component : 16.27, prob. 1 Area of a face of a polyhedron : 16.24, prob. 3 Area of the frontier of a convex body : 16.24, prob. 4 Atlas : 16.1 Automorphism of a Lie group : 16.9 B
Base of a fibration : 16.12 Basis of a module : A.8.2 Basis of a tangent (cotangent) space associated with a chart : 16.5 Bernstein's theorem : 17.8, prob. 3 Bessel function : 17.1I , prob. 2 Bidual of a module : A.9. I Bieberbach's theorem on injective functions: 16.22, prob. 5
Bilinear morphism of vector bundles : 16.16 Blowing up a manifold at a point : 16.11, prob. 3 Boundary of a current : 17.15 Bundle : 16.12 Bundle associated with a principal bundle: 16.14 Bundle of antisymmetric (symmetric) tensors of order m : 16.17
C
Co-fibration : 16.25, prob. 8 Co-vector bundle : 16.25, prob. 8 Canonical basis of the module of differential p-forms on an open set in R": 16.20 Canonical bijection of T,(E) onto E (E a vector space) : 16.5 Canonical bilinear form on E x E* : A.9.1 Canonical chart on an open set in R" : 16.1 Canonical Lie group structure on a vector space : 16.9 Canonical manifold structure on a vector space : 16.2 Canonical mapping of a module into its bidual : A.9.1 Canonical morphisms : 16.18 E @E"+E' 0 E" Ei 00%0E d +(El 0E J 0 E3 381
382
INDEX
EiO(E,@Ed+ (El 0E d 0 (El 0 E d Hom(E@F, G)+ Horn (E, Horn (F, G)) Hom (E' @ E", F) + Horn (E', F) @Horn (E", F) Hom (E, F @ F") + Hom (E, F ) @Horn (E, F ) Hom (E, F ) 0Hom ( E , F ) + Hom (E' 0E , F 0F") E + E** Hom (E, F) + Hom (F*, E*) (E 0F)* +E* 0F E* 0F + Horn (E, F) E* @ E + B x R W E ) 0TXE) +T;++:(E) W E ) 0TXE) +Horn (T:(E), TXEN Cr%E))*+Ti@) m Earn+ A E
(A E)O(; (A El*+:
E ) + ~ ~ E (E*) I- 1
E O ( A E*)+A E* E ( o 0E:c) * (E 0E)cc, (Horn (E, E))w Horn (EcC,,EiCd (E*)(C, (E(C,)* +
+
7
E)(c, + h C J Canonical orientation of a complex manifold : 16.21 Canonical orientation of R" : 16.21 Canonical trivialization of T(M) (hl open in R"): 16.15 Canonical 2-form on T(M)* : 17.15 Canonical vector bundle over a Grassmannian : 16.16 Cauchy principal value of an integral : 17.9, prob. 1 Cauchy's formula for convex bodies : 16.24, prob. 4 Cauchy's formula for convex polyhedra: 16.24, prob. 3 Chart, chart at a point : 16.1 Class C', C" (functions) : 16.3 Coarse C'-topology : 17.1, prob. I Compatible atlases, charts : 16. I Compatible (group structure and manifold structure) : 16.9 Completely monotone function : 17.8, prob. 3
Complex-analytic atlas : 16.1 Complex-analytic fibration : 16.12 Complex-analytic Lie group : 16.9 Complex-analytic manifold : 16.1 Complex vector bundle : 16.15 Composition of two jets : 16.9, prob. 1 Cone of revolution : 16.1, prob. 1 Conjugate of a complex current : 17.8 Connected s u m of two manifolds : 16.26, prob. 15 Connection in a vector bundle : 17.16 Connectiononadifferential manifold : 17.18 Contact of order Z k (of mappings) : 16.5, prob. 9 Contact transformation : 16.20 Contingent : 16.8, prob. 5 Contractible manifold : 16.27 Contraction of a tensor: 16.18 and A.11.3 Contragredient of an isomorphism :A.9.3 Convergent integral : 17.5, prob. 4 Convex body : 16.5, prob. 6 Convex polyhedron : 16.5, prob. 6 Convolution of a finite sequence of distributions : 17.1 1 Coordinate functions : A.9.2 Coordinates relative to a chart : 16.1 Cotangent bundle : 16.20 Covariant derivative : 17.17 Covariant differential of a tensorfield : 17.18 Covariant exterior differential of a differential p-form with values in a vector bundle : 17.19 Covector (tangent) : 16.5 Covering homology theorem : 16.28, prob. 10
Covering space : 16.12 Critical point : 16.5 Critical value of a function : 16.5 Current : 17.3 Curvature of a connection, curvature morphism, curvature tensor field, curvature tensor : 17.20 Curve : 16.1 D
D'Alembertian : 17.9 Degree of a homogeneous element of a graded algebra : A.18.1 Derivation in an algebra : A.18.2
INDEX
Derivation of degree r in a graded algebra : A.18.4
Derivative of a distribution : 17.5 Derivative of a function in the direction of a tangent vector: 17.14 Derived ideal of a Lie algebra, derived series : A. 19 Diffeomorphic manifolds : 16.2 Diffeomorphism : 16.2 Difference of two linear connections : 17.16 Differentiable action (of a Lie group on a manifold) : 16.10 Differentiable mapping : 16.3 Differential fibration : 16.12 Differential manifold : 16.1 Differential manifold underlying an analytic manifold : 16.1 Differential of a function : 16.20 Differential of a function at a point : 16.5 Differential operator : 17.13 Differential p-form : 16.20 Differential p-form with values in a vector bundle : 17.19 Dilatation : 16.20, prob. 4 Dimension of a chart : 16.1 Dimension of a convex body : 16.5, prob. 6 Dimension of a manifold at a point : 16.1 Dimension of a pure manifold : 16.1 Dirac p-current : 17.3 Direct sequence of vector fields : 16.21 Direct sum of vector bundles : 16.16 Distribution : 17.3 Divisor, divisor of a function :16.14, prob. 3 Domain of definition of a chart : 16.1 Dual basis : A.9.2 Dual of a module :A.9.1 E Embedding of a manifold : 16.8 Equivariant actions : 16.10 . Etale mapping : 16.5 Euclidean sphere : 16.2 Exact homotopy sequence : 16.30, prob. 5 Exact homotopy sequence of fiber bundles : 16.30, prob. 6 Exact sequence of vector bundles : 16.17 Extensions of sections : 16.25, prob. 9 Exterior algebra : A.13.5 Exterior differential of a differentialp-form : 17.15
383
Exterior powers of a finitely-generated free module: A.13.3 Exterior powers of a linear mapping : A.13.3 Exterior powers of a vector bundle : 16.16 Exterior product of antisymmetric tensors, p-vectors : A.13.2 F
Face of a convex polyhedron : 16.5, prob. 6 Fiber, fiber at a point : 16.12 Fiber bundle : 16.12 Fiber product : 16.12 Fiber-type : 16.12 Fibered chart: 16.15 Fibered chart of T(M)* associated with a chart of M : 16.20 Fibered manifold : 16.12 Fibration : 16.12 Fibration underlying an analytic fibration : 16.12
Field of point-distributions : 17.13 Fine C'-topology : 17.1, prob. 2 Finite part of an integral : 17.9 Frame, frame at a point, framing : 16.15 Frame dual to a frame of E : 16.16 Frame of T(M) associated with a chart: 16.15
Frame of T(M)* associated with a chart : 16.20
Frame of T;(E) induced by a frame of E : 16.16
Free family of elements of a module :A.8.2. Frohlicher-Nijenhuis theorem : 17.19, prob. 3
Function of class Cr, C" : 16.3 Function of support : 16.5, prob. 7 Fundamental divisor on Pl(C) : 16.14,prob. 3
Fundamental group of a connected manifold : 16.27 Fundamental groups of a manifold at a point : 16.27 Fundamental 1-form on T(M)* : 16.20 G
Galois covering : 16.28, prob. 2 Gaussian mapping : 16.19, prob. 8 Global section : 16.12
384
INDEX
Gradedalgebra : A.18.1 Grassmannian (real, complex, quaternionic) : 16.11 H
Heaviside function : 17.5 Hessian matrix : 16.5 Hessian of a function at a point : 16.5 Holomorphic mapping : 16.3 Holomorphic vector bundle : 16.15 Homogeneous contact transformation : 16.20 Homogeneous element : A.18.1 Homomorphism of Lie algebras : A.19 Homomorphism of Lie groups : 16.9 Homotopic mappings : 16.26 Homotopy, Cp-homotopy : 16.26 Homotopy equivalence : 16.26, prob. 2 Homotopy groups : 16.30, prob. 3 Homotopy type : 16.26, prob. 2 Hopf fibration : 16.14 Horizontal lifting of a vector field : 17.16 Hypergeometric function : 17.11, prob. 2 Hypersurface : 16.8
I Ideal of a Lie algebra : A.19 Image of a current : 17.3 and 17.7 Imaginary part of a current : 17.6 Immersion : 16.7 Implicit function theorem : 16.6 Induced covariant tensor field on a submanifold : 16.20 Induced current on an open set : 17.4 Induced differential form on a submanifold : 16.20 Induced fibration on a submanifold : 16.12 Induced orientation on an open set : 16.21 Induced principal bundle on a submanifold : 16.14 Induced structure of differential manifold on an open set : 16.2 Induced vector bundle on a submanifold : 16.19 Inner derivation in a Lie algebra : A.19.4 Inner derivation in an associative algebra : A.18.2 Integral of a differential form along the fibers : 16.24, prob. 1 1
Integral of an n-form : 16.24 Interior product of ap-vector and a (p 9)form : A.15.3 Invariant distribution : 17.8, prob. 4 Inverse image of a covariant tensor field : 16.20 Inverse image of a current under a local diffeomorphism : 17.4 Inverse image of a differential form : 16.20 Inverse image of a differential form with values in a vector bundle : 17.19 Inverse image of a distribution : 17.5, probs. 8 and 9 Inverse image of a fibration : 16.12 Inverse image of a section : 16.12 Inverse image of a vector bundle : 16.19 Invertible jet : 16.9, prob. 1 Isomorphism of differential manifolds : 16.2 Isomorphism of fibrations, B-isomorphism of fibrations : 16.12 Isomorphism of Lie groups : 16.9 Isomorphism of principal bundles, Bisomorphism of principal bundles : 16.14 Isomorphism of vector bundles, B-isomorphism of vector bundles :16.15 Isotopy : 16.26 Isotropic subspace :A.16.3
+
J
Jacobi identity :A.19 Jacobi polynomial : 17.11, prob. 2 James embedding: 16.25, prob. 16 Jet of order k from X to Y : 16.5, prob. 9 Jet of order k of a mapping at a point : 16.5, prob. 9 K
Klein bottle : 16.14 Kneser-Glaeser theorem : 16.23, prob. 1 Kronecker tensor field : 16.17 1
Lagrange’s method of undetermined multipliers : 16.20, prob. 5 Laplacian : 17.9
INDEX
Lebesgue measure : 16.22 Legendre transformation : 16.20 Lie algebra : A. 19 Lie bracket of two vector fields : 17.14 Lie group, Lie group homomorphism 16.9 Lie subgroup : 16.9 Lie’s transformation : 16.20, prob. 4 Lifting of a mapping to a fiber bundle : 6.12 Lifting of a path-homotopy : 16.28 Linear connection : 17.16 Linear differential operator : 17.13 Linear representation of a Lie group : 16.9 Local coordinates at a point : 16.1 Local diffeomorphism : 16.5 Local expression of a B-morphism of fiber bundles : 16.13 Local expression of a Hessian Hess,(f) : 16.5
Local expression of a mapping : 16.3 Local expression of a morphism of vector bundles : 16.15 Local expression of a p-form : 16.5 Local expression of a section of a vector bundle : 16.15 Local expression of a tangent linear mapping T,(f) : 16.5 Local expression of a tangent vector h, :16.5 Local expression of a vector field : 16.15 Local homomorphism : 16.9 Local triviality : 16.12 Locally arcwise-connected : 16.27, prob. 1 Locally integrable n-form : 16.24 Locally isomorphic Lie groups : 16.9 Lorentz group : 16.11 M
Manifold : 16.1 Manifold obtained by patching : 16.2 Manifold of class C‘ : 16.1, prob. 2 Measurable n-form : 16.24 Meromorphic function : 16.14, prob. 2 Mobius strip : 16.14 Module : A.8.1 Module obtained by extending the ring of scalars : A.10.6 Monodromy principle : 16.28 Morphism of differential manifolds : 16.3 Morphism of fibrations, B-morphism of fibrations : 16.12
385
Morphism of principal bundles : 16.I 4 Morphism of vector bundles, B-morphism of vector bundles : 16. I5 Morse index of a function at a point : 16.5, prob. 3 Morton Brown’s theorem : 16.2, prob. 5 Multiple of a vector bundle : 16.16 Multiplet layer: 17.10
N Natural continuation of a holomorphic function : 16.8, prob. 12 n-chain element without boundary : 17.5 Negative differential form (relative to an orientation) : 16.21 Negative sequence of vector fields : 16.21 Negligible n-form : 16.24 Nijenhuis torsion : 17.19, prob. 3 Nondegenerate alternating bilinear form : A.16.1
Nondegenerate critical point : 16.4 Nonhomogeneous contact transformation: 16.20, prob. 3 Normal bundle of a submanifold : 16.19 n-sheeted covering : 16.12
0 Open n-chain, open n-chain element : 17.5 Operator of local character : 17.13 Opposite (of a Lie group) : 16.9 Orbit manifold : 16.10 Order of a current : 17.3 Order of a differential operator : 17.13 Orientable manifold: 16.21 Orientation of a manifold : 16.21 Orientation off-’(y) induced from those of X and Y (f:X --f Y a submersion) : 16.21
Orientation of X induced from an orientation of Y by an &ale morphism f:X + Y : 16.21 Orientation preserving, reversing : 16.21 Oriented manifold : 16.21 Orthogonal supplement (of a vector subspace relative to an alternating bilinear form) : A.16.3
386
INDEX
Orthogonal vectors (relative to an alternating bilinear form) :A.16.3 P
Parallelizable manifold : 16.15, prob. I Paratingent : 16.8, prob. 5 Patching condition for fiber bundles : 16.13 p-covector :A.14.2 Periodic current : 17.9 p-form : A.14.2 Poincar&Volterra theorem : 16.8, prob. 11 Point-distribution : 17.7 Poisson bracket of differential I-forms : 17.15, prob. 9 Positive differential form (relative to an orientation) : 16.21 Positive sequence of vector fields : 16.21 Predivisor: 16.14, prob. 3 Principal bundle : 16.14 Principal Co-bundle : 16.25, prob. 8 Principal divisor : 16.14, prob. 3 Privileged local coordinate system (relative to a p-form) : 17.15, prob. 2 Product manifold : 16.6 Product of fibrations : 16.12 Product of Lie algebras : A.19 Product of manifolds : 16.6 Product of orientations : 16.21 Product of two manifolds over a manifold : 16.8, prob. 10 Projective bundle : 16.19, prob. 9 Projective space (real, complex, quaternionic) : 16.1I Proper mapping : 17.3 Pure differential manifold : 16.1 p-vector : A.13.6
Q Quotient bundle : 16.17 Quotient Lie algebra :A.19 R
Rank of a C'-mapping at a point : 16.5 Rank of a vector bundle at a point : 16.15 Rank of an alternating bilinear form : A.16.1
Real-analytic atlas : 16.1 Real-analytic fibration : 16.12 Real-analytic group : 16.9 Real-analytic manifold : 16.1 Real-analytic manifold underlying a complex manifold : 16.1 Real part of a current : 17.6 Real p-current : 17.6 Real vector bundle underlying a complex vector bundle: 16.15 Reciprocal polars : 16.20, prob. 4 Regular value : 16.23 Regularizing sequence : 17.1 Relative homotopy groups : 16.30, prob. 4 Restriction of a chart to an open set : 16.1 Restriction of a differential operator to an open set : 17.13 Restriction of an atlas to an open set : 16.2 Retrograde sequence of vector fields : 16.21 p-extension of a principal bundle : 16.14, prob. 17 Riemann sphere : 16. I 1 Riemann surface : 16.1 Riemann surface defined by a function of two variables : 16.8 Riernann surface defined by a holomorphic function : 16.8, prob. 12 Riemann surface of the logarithmic function : 16.8 Rotation group : 16.11 5
Sard's theorem : 16.23 Saturated atlas : 16.1 Section of a fibration, fiber bundle : 16.12 Simple layer : 17.10 Simply-connected manifold : 16.27 Singular support of a current : 17.5 Solid angle : 16.24 Sonine's formula : 17.1I , prob. 2 Source of a jet : 16.5, prob. 9 Space of a fibration : 16.12 Special linear group : 16.9 Special orthogonal group : 16.9 Special unitary group : 16.11 Sphere oriented toward the outside (inside) : 16.2 I Stationary at a point : 16.5 Steiner-Minkowski formula : 16.24, prob. 7
INDEX
Stereographic projection : 16.2 Stiefel manifolds : 16.11 Stokes’ theorem : 16.24 Strictly convolvable distributions : 17.11 Subalgebra of a Lie algebra :A.19 Subbundle : 16.17 Subimmersion, subimmersion at a point : 16.7
Submanifold : 16.8 Submersion, submersion at a point : 16.7 Summable distribution : 17.1 1, prob. 1 Supplement (of a vector subbundle) : 16.17 Support of a current : 17.4 Surface : 16.1 Surface measure on S. : 16.24 Symmetric algebra :A.17 Symmetric powers of a finitely-generated free module : A.17 Symmetric product of symmetric tensors : A.17
Symmetric tensor : A.12.1 Symmetrization of a tensor : A.12.2 Symplectic automorphism :A.16.4 Symplectic basis : A.16.4 Symplectic group : A.16.4 System of local coordinates : 16.1
T
Tangent affine-linear variety : 16.8 Tangent bundle : 16.15 Tangent covector : 16.5 Tangent (functions) : 16.5 Tangent hyperplane : 16.8 Tangent linear mapping : 16.5 Tangent plane : 16.8 Tangent to a curve : 16.8 Tangent vector at a point of a differential manifold : 16.5 Tangent vector field : 16.15 Tangent vector space at a point of a differential manifold : 16.5 Target of a jet : 16.5, prob. 9 Tautological vector bundle over a Grassmannian : 16.16 Tensor: A.ll.1 Tensor bundle of type (p, q) : 16.16 Tensor (by abuse of language) : 16.20
387
Tensor field of type (p, q) : 16.18 and 16.20 Tensor multiplication : 16.18 Tensor powers of a finitely-generated free module: A.1I.I Tensor powers of a vector bundle : 16.16 Tensor product of distributions : 17.10 Tensor product of elements of free A-modules : A.10.1 Tensor product of finitely-generated free Amodules : A.10.3 Tensor product of linear forms : A. 10.1 Tensor product of linear mappings : A.10.5 Tensor product of vector bundles : 16.16 Thorn’s transversality theorem : 16.25, prob. 17
Topological manifold : 16.1 Topological space underlying a manifold : 16.1
Torsion, torsion morphism, torsion tensor, torsion tensor field : 17.20 Totally isotropic subspace : A.16.2 Trace measure on G x G : 17.5, prob. 10 Trace of a matrix :A. 1 1.3 Trace of an endomorphism of a finitelygenerated free module : A.11.3 Trace of an endomorphism of a vector bundle : 16.18 Transition diffeomorphisrn : 16.13 Transition homeomorphism : 16.1 Transpose of a homomorphism of finitelygenerated free modules :A.9.4 Transpose of a matrix : A.9.4 Transpose of a morphism of vector bundles : 16.16
Transversal at a point, transversal over a submanifold (mappings) : 16.8, prob. 9 Transversal mappings : 16.8, prob. 10 Transversal submanifolds : 16.8, prob. 9 Trivial fiber bundle : 16.12 Trivial principal bundle : 16.14 Trivial vector bundle : 16.15 Trivializable, trivializable over an open set (bundle or fibration) : 16.I 2 Trivializable principal bundle : 16.14 Trivializable vector bundle : 16.15 Trivialization of a fiber bundle, fibration : 16.12
Trivialization of a principal bundle : 16.14 Trivialization of a vector bundle : 16.15 Twisted torus : 16.14
388
INDEX U
Unending path : 16.27 Unimodular group : 16.9 Unitary group : 16.1 1 Universal covering : 16.29 Universal covering group of a Lie group : 16.30 V
Vector bundle (real, complex) : 16.15 Vector field : 16.15 Vector part of a section : 16.15 Vector-valued differential p-form : 16.20
Vertical tangent vector : 16.12 Volume, volume form : 16.24 W
Weak integral of a variable current :17.8 Whitney sum of vector bundles : 16.16 Whitney’s embedding theorem :16.25, probs. 2 and 13 Whitney’s extension theorem : 16.4, prob. 6 Z
Zero section of a vector bundle : 16.15
Pure and Applied Mathematics A Series of Monographs and Textbooks Editors
Samuel Eilenberg and Hyman Bass Columbia University, N e w York
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