Gauge theory for fiber bundles

GAUGE THEORY FOR FIBER BUNDLES Peter W. Michor

Mailing address: Peter W. Michor, Institut fur Mathematik der Universitat Wien, Strudlhofgasse 4, A-1090 Wien, Austria. E-mail [email protected] 1980 Mathematics subject classi
cation 53-02, 53C05, 53C10, 58-02, 58D05, 58A35

Extended version of a series of lectures held at the Institute of Physics of the University of Napoli, March 28 | April 1, 1988

1

Table of contents

Introduction . . . . . . . . . . . . . . . . . . 1. Notations and conventions . . . . . . . . . . 2. Calculus of smooth mappings . . . . . . . . . 3. Calculus of holomorphic mappings . . . . . . . 4. Calculus of real analytic mappings . . . . . . . 5. Innite dimensional manifolds . . . . . . . . . 6. Manifolds of mappings . . . . . . . . . . . . 7. Dieomorphism groups . . . . . . . . . . . . 8. The Frolicher-Nijenhuis Bracket . . . . . . . . 9. Fiber Bundles and Connections . . . . . . . . 10. Principal Fiber Bundles and G-Bundles . . . . 11. Principal and Induced Connections . . . . . . 12. Holonomy . . . . . . . . . . . . . . . . . 13. The nonlinear frame bundle of a ber bundle . . 14. Gauge theory for ber bundles . . . . . . . . 15. A classifying space for the dieomorphism group 16. A characteristic class . . . . . . . . . . . . 17. Self duality . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . .

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2 5 6 10 12 14 17 22 27 36 44 60 76 81 83 86 90 98 101 102 103

2

Introduction Gauge theory usually investigates the space of principal connections on a principal ber bundle (P
p
M
G) and its orbit space under the action of the gauge group (called the moduli space), which is the group of all principal bundle automorphisms of P which cover the identity on the base space M. It is the arena for the Yang-Mills-Higgs equations which allows (with structure group U(1)
SU(2)) a satisfactory unied description of electromagnetic and weak interactions, which was developed by Glashow, Salam, and Weinberg. This electro-weak theory predicted the existence of massive vector particles (the intermediate bosons W + , W ; , and Z), whose experimental verication renewed the interest of physicists in gauge theories. On the mathematical side the investigation of self dual and anti self SU(2)-connections on 4-manifolds and of their image in the orbit space led to stunning topological results in the topology of 4-manifolds by Donaldson. The moduli space of anti self dual connections can be completed and reworked into a 5 dimensional manifold which is a bordism between the base manifold of the bundle and a simple manifold which depends only on the Poincare duality form in the second dimensional homology space. Combined with results of Freedman this led to the discovery of exotic dierential structures on 4 and on (supposedly all but S 4 ) compact algebraic surfaces. In his codication of a principal connection Ehresmann, 1951] began with a more general notion of connection on a general ber bundle (E
p
M
S) with standard ber S. This was called an Ehresmann connection by some authors, we will call it just a connection. It consists of the specication of a complement to the vertical bundle in a dierentiable way, called the horizontal distribution. One can conveniently describe such a connection as a one form on the total space E with values in the vertical bundle, whose kernel is the horizontal distribution. When combined with another venerable notion, the Frolicher-Nijenhuis bracket for vector valued dierential forms (see Fr
olicher-Nijenhuis, 1956]) one obtains a very convenient way to describe curvature and Bianchi identity for such connections. Parallel transport along curves in the base space is dened only locally, but one may show that each bundle admits complete connections, whose parallel transport is globally dened (theorem 9.10). For such connections one can dene holonomy groups and holonomy Lie algebras and one may prove a far-reaching generalization of the Ambrose Singer theorem: A complete connection tells us whether it is induced from a principal connection on a principal ber bundle (theorem 12.4).

Introduction

3

A bundle (E
p
M
S) without structure group can also be viewed as having the whole dieomorphism group Di(S) of the standard ber S as structure group, as least when S is compact. We dene the non linear frame bundle for E which is a principle ber bundle over M with structure group Di(S) and we show that the theory of connections on E corresponds exactly to the theory of principal connections on this non linear frame bundle (see section 13). The gauge group of the non linear frame bundle turns out to be just the group of all ber respecting dieomorphisms of E which cover the identity on M, and one may consider the moduli space Conn(E)= Gau(E). There is hope that it can be stratied into smooth manifolds corresponding to conjugacy classes of holonomy groups in Di(S), but this will be treated in another paper. The dieomorphism group Di(S) for a compact manifold S admits a smooth classifying space and a classifying connection: the space of all embeddings of S into a Hilbert space `2 , say, is the total space of a principal bundle with structure group Di(S), whose base manifold is the non linear Grassmanian of all submanifolds of `2 of type (dieomorphic to) S. The action of Di(S) on S leads to the (universal or classifying) associated bundle, which admits a classifying connection: Every S-bundle over M can be realized as the pull back of the classifying bundle by a classifying smooth mapping from M into the non linear Grassmanian, which can be arranged in such a way that it pulls back the classifying connection to a given one on E | this is the Narasimhan-Ramadas procedure for Di(S). Characteristic classes owe their existence to invariant polynomials on the Lie algebra of the structure group like characteristic coecients of matrices. The Lie algebra of Di(S) for compact S is the Lie algebra X(S) of vector elds on S. Unfortunately it does not admit any invariants. But there are equivariants like X 7! LX ! for some closed form ! on S which can be used to give a sort of Chern-Weil construction of characteristic classes in the cohomology of the base M with local coecients (see sections 16 and 17). Finally we discuss some self duality and anti self duality conditions which depend on some berwise structure on the bundle like a berwise symplectic structure. This booklet starts with a short description of analysis in innite dimensions along the lines of Frolicher and Kriegl which makes innite dimensional dierential geometry much simpler than it used to be. We treat smoothness, real analyticity and holomorphy. This part is based on Kriegl-Michor, 1990b] and the forthcoming book Kriegl-Michor]. Then we give a more detailed exposition of the theory of manifolds of mappings and the dieomorphism group. This part is adapted from

4

Introduction

Michor, 1980] to the use of the Frolicher-Kriegl calculus. Then we present a careful introduction to the Frolicher-Nijenhuis bracket, to ber bundles and connections, G-structures and principal connection which emphasizes the construction and recognition of induced connections. The material in the rest of the book, from section 12 onwards, has been published in Michor, 1988]. The version here is much more detailed and contains more results. The material in this booklet is a much extended version of a series of lectures held at the Institute of Physics of the University of Napoli, March 28 { April 1, 1988. I want to thank G. Marmo for his hospitality, his interest in this subject, and for the suggestion to publish this booklet. Wien, August 6, 1990 P. Michor

5

1. Notations and conventions

1.1. Denition. A Lie group G is a smooth manifold and a group such that the multiplication  : G
G ! G is smooth. Then also the inversion  : G ! G turns out to be smooth. We shall use the following notation:  : G
G ! G, multiplication, (x
y) = x:y. a : G ! G, left translation, a (x) = a:x. a : G ! G, right translation, a (x) = x:a.  : G ! G, inversion, (x) = x;1 . e 2 G, the unit element. If g = Te G is the Lie algebra of G, we use the following notation: AdG : G ! AutLie (g) and so on. 1.2. Let ` : G
M ! M be a left action, so ` : G ! Di(M) is a group homomorphism. Then we have partial mappings `a : M ! M and `x : G ! M, given by `a (x) = `x (a) = `(a
x) = a:x. For any X 2 g we dene the fundamental vector
eld X = XM 2

X(M) by X (x) = Te (`x ):X = T(e
x)`:(X
0x ). Lemma. In this situation the following assertions hold: (1) : g ! X(M) is a linear mapping. (2) Tx (`a ): X (x) = Ad(a)X (a:x). (3) RX
0M 2 X(G
M) is `-related to X 2 X(M). (4)  X
Y ] = ; X
Y ] . 1.3. Let r : M
G ! M be a right action, so r : G ! Diff(M) is a group anti homomorphism. We will use the following notation: ra : M ! M and rx : G ! M, given by rx (a) = ra (x) = r(x
a) = x:a. For any X 2 g we dene the fundamental vector
eld X = XM 2 X(M) by X (x) = Te (rx):X = T(x
e) r:(0x
X). Lemma. In this situation the following assertions hold: (1) : g ! X(M) is a linear mapping. (2) Tx (ra ): X (x) = Ad(a;1 )X (x:a). (3) 0M
LX 2 X(M
G) is r-related to X 2 X(M). (4)  X
Y ] = X
Y ] .

6

2. Calculus of smooth mappings 2.1. The traditional dierential calculus works well for nite dimen-

sional vector spaces and for Banach spaces. For more general locally convex spaces a whole ock of dierent theories were developed, each of them rather complicated and none really convincing. The main diculty is that the composition of linear mappings stops to be jointly continuous at the level of Banach spaces, for any compatible topology. This was the original motivation for the development of a whole new eld within general topology, convergence spaces. Then in 1982, Alfred Frolicher and Andreas Kriegl presented independently the solution to the question for the right dierential calculus in innite dimensions. They joined forces in the further development of the theory and the (up to now) nal outcome is the book Fr
olicherKriegl, 1988], which is the general reference for this section. See also the forthcoming book Kriegl-Michor]. In this section I will sketch the basic denitions and the most important results of the Frolicher-Kriegl calculus. 2.2. The c1 -topology. Let E be a locally convex vector space. A curve c : ! E is called smooth or C 1 if all derivatives exist and are continuous - this is a concept without problems. Let C 1(
E) be the space of smooth functions. It can be shown that C 1 (
E) does not depend on the locally convex topology of E, only on its associated bornology (system of bounded sets). The nal topologies with respect to the following sets of mappings into E coincide: (1) C 1(
E). (2) Lipschitz curves (so that f c(tt);;sc(s) : t 6= sg is bounded in E). (3) fEB ! E : B bounded absolutely convex in E g, where EB is the linear span of B equipped with the Minkowski functional pB (x) := inf f > 0 : x 2 B g. (4) Mackey-convergent sequences xn ! x (there exists a sequence 0 < n % 1 with n (xn ; x) bounded). This topology is called the c1 -topology on E and we write c1 E for the resulting topological space. In general (on the space D of test functions for example) it is ner than the given locally convex topology, it is not a vector space topology, since scalar multiplication is no longer jointly continuous. The nest among all locally convex topologies on E which are coarser than c1 E is the bornologication of the given locally convex topology. If E is a Frechet space, then c1 E = E.

2. Calculus of smooth mappings

7

2.3. Convenient vector spaces. Let E be a locally convex vector

space. E is said to be a convenient vector space if one of the following equivalent (completeness) conditions is satised: (1) Any Mackey-Cauchy-sequence (so that (xn ; xm ) is Mackey convergent to 0) converges. This is also called c1 -complete. (2) If B is bounded closed absolutely convex, then EB is a Banach space. (3) Any Lipschitz curve in E is locally Riemann integrable. (4) For any c1 2 C 1 (
E) there is c2 2 C 1 (
E) with c01 = c2 (existence of antiderivative). 2.4. Lemma. Let E be a locally convex space. Then the following properties are equivalent: (1) E is c1 -complete. (2) If f : k ! E is scalarwise Lipk , then f is Lipk , for k > 1. (3) If f : ! E is scalarwise C 1 then f is dierentiable at 0. (4) If f : ! E is scalarwise C 1 then f is C 1 . Here a mapping f : k ! E is called Lipk if all partial derivatives up to order k exist and are Lipschitz, locally on n. f scalarwise C 1 means that  f is C 1 for all continuous linear functionals on E. This lemma says that a convenient vector space one can recognize smooth curves by investigating compositions with continuous linear functionals. 2.5. Smooth mappings. Let E and F be locally convex vector spaces. A mapping f : E ! F is called smooth or C 1 , if f c 2 C 1 (
F) for all c 2 C 1(
E) so f : C 1 (
E) ! C 1 (
F) makes sense. Let C 1 (E
F) denote the space of all smooth mapping from E to F . For E and F nite dimensional this gives the usual notion of smooth mappings: this has been rst proved in Boman, 1967]. Constant mappings are smooth. Multilinear mappings are smooth if and only if they are bounded. Therefore we denote by L(E
F) the space of all bounded linear mappings from E to F . 2.6. Structure on C 1(E
F ). We equip the space C 1(
E) with the bornologication of the topology of uniform convergence on compact sets, in all derivatives separately. Then we equip the space C 1 (E
F ) with the bornologication of the initial topology with respect to all mappings c : C 1 (E
F ) ! C 1 (
F ), c (f) := f c, for all c 2 C 1 (
E). 2.7. Lemma. For locally convex spaces E and F we have: (1) If F is convenient, then also C 1(E
F) is convenient, for any E. The space L(E
F ) is a closed linear subspace of C 1 (E
F ), so it

8

2. Calculus of smooth mappings

also convenient. (2) If E is convenient, then a curve c : ! L(E
F) is smooth if and only if t 7! c(t)(x) is a smooth curve in F for all x 2 E. 2.8. Theorem. Cartesian closedness. The category of convenient vector spaces and smooth mappings is cartesian closed. So we have a natural bijection C 1 (E
F
G)

= C 1(E
C 1 (F
G))
which is even a dieomorphism. Of coarse this statement is also true for c1 -open subsets of convenient vector spaces. 2.9. Corollary. Let all spaces be convenient vector spaces. Then the following canonical mappings are smooth. ev : C 1 (E
F )
E ! F
ev(f
x) = f(x) ins : E ! C 1(F
E
F)
ins(x)(y) = (x
y) ( )^ : C 1 (E
C 1 (F
G)) ! C 1(E
F
G) ( )_ : C 1 (E
F
G) ! C 1 (E
C 1(F
G)) comp : C 1 (F
G)
C 1(E
F ) ! C 1 (E
G) C 1(
) : C 1(F
F 0)
C 1 (E 0
E) ! C 1 (C 1(E
F)
C 1(E 0
F 0)) (f
g) 7! (h 7! f h g) Y Y 1 Y Y : C (Ei
Fi) ! C 1 ( Ei
Fi)

2.10. Theorem. Let E and F be convenient vector spaces. Then the dierential operator

d : C 1(E
F) ! C 1 (E
L(E
F))
f(x + tv) ; f(x)
df(x)v := tlim !0 t exists and is linear and bounded (smooth). Also the chain rule holds: d(f g)(x)v = df(g(x))dg(x)v:

2.11. Remarks. Note that the conclusion of theorem 2.8 is the starting

point of the classical calculus of variations, where a smooth curve in a space of functions was assumed to be just a smooth function in one variable more.

2. Calculus of smooth mappings

9

If one wants theorem 2.8 to be true and assumes some other obvious properties, then the calculus of smooth functions is already uniquely determined. There are, however, smooth mappings which are not continuous. This is unavoidable and not so horrible as it might appear at rst sight. For example the evaluation E
E 0 ! is jointly continuous if and only if E is normable, but it is always smooth. Clearly smooth mappings are continuous for the c1 -topology. For Frechet spaces smoothness in the sense described here coincides with the notion Cc1 of Keller, 1974]. This is the dierential calculus used by Michor, 1980], Milnor, 1984], and Pressley-Segal, 1986]. A prevalent opinion in contemporary mathematics is, that for innite dimensional calculus each serious application needs its own foundation. By a serious application one obviously means some application of a hard inverse function theorem. These theorems can be proved, if by assuming enough a priori estimates one creates enough Banach space situation for some modied iteration procedure to converge. Many authors try to build their platonic idea of an a priori estimate into their dierential calculus. I think that this makes the calculus inapplicable and hides the origin of the a priori estimates. I believe, that the calculus itself should be as easy to use as possible, and that all further assumptions (which most often come from ellipticity of some nonlinear partial dierential equation of geometric origin) should be treated separately, in a setting depending on the specic problem. I am sure that in this sense the Frolicher-Kriegl calculus as presented here and its holomorphic and real analytic osprings in sections 2 and 3 below are universally usable for most applications.

10

3. Calculus of holomorphic mappings 3.1. Along the lines of thought of the Frolicher-Kriegl calculus of smooth mappings, in Kriegl-Nel, 1985] the cartesian closed setting for holomorphic mappings was developed. The right denition of this calculus was already given by Fantappie, 1930 and 1933]. I will now sketch

the basics and the main results. It can be shown that again convenient vector spaces are the right ones to consider. Here we will start with them for the sake of shortness. 3.2. Let E be a complex locally convex vector space whose underlying real space is convenient { this will be called convenient in the sequel. Let  be the open unit disk and let us denote by C ! (
E) the space of all mappings c : ! E such that  c : ! is holomorphic for each continuous complex-linear functional  on E. Its elements will be called the holomorphic curves. If E and F are convenient complex vector spaces (or c1 -open sets therein), a mapping f : E ! F is called holomorphic if f c is a holomorphic curve in F for each holomorphic curve c in E. Obviously f is holomorphic if and only if  f : E ! is holomorphic for each complex linear continuous functional  on F. Let C ! (E
F) denote the space of all holomorphic mappings from E to F . 3.3. Theorem of Hartogs. Let Ek for k = 1
2 and F be complex convenient vector spaces and let Uk  Ek be c1 -open. A mapping f : U1
U2 ! F is holomorphic if and only if it is separably holomorphic (i. e. f(
y) and f(x
) are holomorphic for all x 2 U1 and y 2 U2 ). This implies also that in nite dimensions we have recovered the usual denition. 3.4 Lemma. If f : E U ! F is holomorphic then df : U
E ! F exists, is holomorphic and -linear in the second variable. A multilinear mapping is holomorphic if and only if it is bounded. 3.5 Lemma. If E and F are Banach spaces and U is open in E, then for a mapping f : U ! F the following conditions are equivalent: (1) f is holomorphic. (2) f is locally a convergent series of homogeneous continuous polynomials. (3) f is -dierentiable in the sense of Fr echet. 3.6 Lemma. Let E and F be convenient vector spaces. A mapping f : E ! F is holomorphic if and only if it is smooth and its derivative is everywhere -linear.

3. Calculus of holomorphic mappings

11

An immediate consequence of this result is that C ! (E
F ) is a closed linear subspace of C 1 (E
F ) and so it is a convenient vector space if F is one, by 2.7. The chain rule follows from 2.10. The following theorem is an easy consequence of 2.8. 3.7 Theorem. Cartesian closedness. The category of convenient complex vector spaces and holomorphic mappings between them is cartesian closed, i. e. C ! (E
F
G)

= C ! (E
C ! (F
G)): An immediate consequence of this is again that all canonical structural mappings as in 2.9 are holomorphic.

12

4. Calculus of real analytic mappings 4.1. In this section I sketch the cartesian closed setting to real analytic mappings in innite dimension following the lines of the Frolicher-Kriegl calculus, as it is presented in Kriegl-Michor, 1990a]. Surprisingly

enough one has to deviate from the most obvious notion of real analytic curves in order to get a meaningful theory, but again convenient vector spaces turn out to be the right kind of spaces. 4.2. Real analytic curves. Let E be a real convenient vector space with dual E 0 . A curve c : ! E is called real analytic if  c : ! is real analytic for each  2 E 0. It turns out that the set of these curves depends only on the bornology of E. In contrast a curve is called topologically real analytic if it is locally given by power series which converge in the topology of E. They can be extended to germs of holomorphic curves along in the complexication E of E. If the dual E 0 of E admits a Baire topology which is compatible with the duality, then each real analytic curve in E is in fact topological real analytic for the bornological topology on E. 4.3. Real analytic mappings. Let E and F be convenient vector spaces. Let U be a c1 -open set in E. A mapping f : U ! F is called real analytic if and only if it is smooth (maps smooth curves to smooth curves) and maps real analytic curves to real analytic curves. Let C ! (U
F ) denote the space of all real analytic mappings. We equip the space C ! (U
) of all real analytic functions with the initial topology with respect to the families of mappings c

C ! (U
) ;! C ! (
)
for all c 2 C ! (
U) c C ! (U
) ;! C 1 (
)
for all c 2 C 1 (
U)
where C 1 (
) carries the topology of compact convergence in each derivative separately as in section 2, and where C ! (
) is equipped with the nal topology with respect to the embeddings (restriction mappings) of all spaces of holomorphic mappings from a neighborhood V of in mapping to , and each of these spaces carries the topology of compact convergence. Furthermore we equip the space C ! (U
F ) with the initial topology with respect to the family of mappings 

 ! C ! (U
F) ;! C (U
)
for all  2 F 0: It turns out that this is again a convenient space.

4. Calculus of real analytic mappings

13

4.4. Theorem. In the setting of 4.3 a mapping f : U ! F is real analytic if and only if it is smooth and is real analytic along each a ne line in E. 4.5. Theorem Cartesian closedness. The category of convenient spaces and real analytic mappings is cartesian closed. So the equation C ! (U
C ! (V )
F )

= C ! (U
V
F) is valid for all c1 -open sets U in E and V in F, where E, F , and G are convenient vector spaces. This implies again that all structure mappings as in 2.9 are real analytic. Furthermore the dierential operator d : C ! (U
F ) ! C ! (U
L(E
F)) exists, is unique and real analytic. Multilinear mappings are real analytic if and only if they are bounded. Powerful real analytic uniform boundedness principles are available.

14

5. Innite dimensional manifolds 5.1. In this section I will concentrate on two topics: Smooth partitions of unity, and several kinds of tangent vectors. The material is taken from Kriegl-Michor, 1990b]. 5.2. In the usual way we dene manifolds by gluing c1-open sets in

convenient vector spaces via smooth (holomorphic, real analytic) dieomorphisms. Then we equip them with the identication topology with respect to the c1 -topologies on all modeling spaces. We require some properties from this topology, like Hausdor and regular (which here is not a consequence of Hausdor). Mappings between manifolds are smooth (holomorphic, real analytic), if they have this property when composed which any chart mappings. 5.3. Lemma. A manifold M is metrizable if and only if it is paracompact and modeled on Fr echet spaces. 5.4. Lemma. For a convenient vector space E the set C 1(M
E) of smooth E-valued functions on a manifold M is again a convenient vector space. Likewise for the real analytic and holomorphic case. 5.5. Theorem. Smooth partitions of unity. If M is a smooth manifold modeled on convenient vector spaces admitting smooth bump functions and U is a locally nite open cover of M, then there exists a smooth partition of unity f'U : U 2 Ug with carr=supp('U )  U for all U 2 U . If M is in addition paracompact, then this is true for every open cover U of M. Convenient vector spaces which are nuclear admit smooth bump functions. 5.6. The tangent spaces of a convenient vector space E . Let a 2 E. A kinematic tangent vector with foot point a is simply a pair (a
X) with X 2 E. Let Ta E = E be the space of all kinematic tangent vectors with foot point a. It consists of all derivatives c0 (0) at 0 of smooth curves c : ! E with c(0) = a, which explains the choice of the name kinematic. For each open neighborhood U of a in E (a
X) induces a linear mapping Xa : C 1(U
) ! by Xa (f) := df(a)(Xa ), which is continuous for the convenient vector space topology on C 1 (U
), and satises Xa (f g) = Xa (f) g(a) + f(a) Xa (g), so it is a continuous derivation over eva . The value Xa (f) depends only on the germ of f at a. An operational tangent vector of E with foot point a is a bounded derivation @ : Ca1(E
) ! over eva . Let Da E be the vector space

5. Innite dimensional manifolds

15

of all these derivations. Any @ 2 Da E induces a bounded derivation C 1 (U
) ! over eva for each open neighborhood U of a in E. So the vectorQspace Da E is a closed linear subspace of the convenient vector space U L(C 1 (U
)
). We equip Da E with the induced convenient vector space structure. Note that the spaces Da E are isomorphic for all a 2 E. Example. Let Y 2 E 00 be an element in the bidual of E. Then for each a 2 E we have an operational tangent vector Ya 2 Da E, given by Ya (f) := Y (df(a)). So we have a canonical injection E 00 ! Da E. Example. Let ` : L2(E ) ! be a bounded linear functional which vanishes on the subset E 0  E 0 . Then for each a 2 E we have an operational tangent vector @`2 ja 2 Da E given by @`2 ja (f) := `(d2f(a)), since `(d2 (fg)(a)) = = `(d2f(a)g(a) + df(a)  dg(a) + dg(a)  df(a) + f(a)d2 g(a)) = `(d2f(a))g(a) + 0 + f(a)`(d2 g(a)):

5.7. Lemma. Let ` 2 Lksym (E )0 be a bounded linear functional which vanishes on the subspace kX ;1 i=1

;i (E ) Lisym (E ) _ Lksym

of decomposable elements of Lksym (E ). Then ` de nes an operational tangent vector @`k ja 2 Da E for each a 2 E by @`k ja (f) := `(dk f(a)): The linear mapping ` 7! @`k ja is an embedding onto a topological direct summand Da(k) E of Da E.PIts left inverse is given by @ 7! ( 7! @(( diag)(a+ ))). The sum k>0 Da(k) E in Da E is a direct one. 5.8. Lemma. If E is an in nite dimensional Hilbert space, all operational tangent space summands D0(k) E are not zero. 5.9. Denition. A convenient vector space is said to have the (bornological) approximation property if E 0  E is dense in L(E
E) in the bornological locally convex topology. The following spaces have the bornological approximation property: ( ), nuclear Frechet spaces, nuclear (LF)spaces.

16

5. Innite dimensional manifolds

5.10 Theorem. Tangent vectors as derivations. Let E be a convenient vector space which has the approximation property. Then we have Da E = Da(1) E

= E 00. So if E is in addition reexive, each operational

tangent vector is a kinematic one. 5.11. The kinematic tangent bundle TM of a manifold M is constructed by gluing all the kinematic tangent bundles of charts with the help of the kinematic tangent mappings (derivatives) of the chart changes. TM ! M is a vector bundle and T : C 1 (M
N) ! C 1(T M
T N) is well dened and has the usual properties. 5.12. The operational tangent bundle DM of a manifold M is constructed by gluing all operational tangent spaces of charts. Then M : DM ! M is again a vector bundle which contains the kinematic tangent bundle TM embeds as a splitting subbundle. Also for each k 2 the same gluing construction as above gives us tangent bundles D(k) M which are splitting subbundles of DM. The mappings D(k) : C 1 (M
N) ! C 1 (D(k) M
D(k) N) are well dened for all k including 1 and have the usual properties. Note that for manifolds modeled on reexive spaces having the bornological approximation property the operational and the kinematic tangent bundles coincide.

17

6. Manifolds of mappings 6.1. Theorem. Manifold structure of C 1(M
N). Let M and N be

smooth nite dimensional manifolds. Then the space C 1(M
N) of all smooth mappings from M to N is a smooth manifold, modeled on spaces C 1 (f  TN) of smooth sections of pullback bundles along f : M ! N over M. A careful description of this theorem (but without the help of the Frolicher-Kriegl calculus) can be found in Michor, 1980]. I include a proof of this result here because the result is important and the proof is much simpler now with the help of the Frolicher Kriegl-calculus. Proof: Choose a smooth Riemannian metric on N. Let exp : TN  U ! N be the smooth exponential mapping of this Riemannian metric, dened on a suitable open neighborhood of the zero section. We may assume that U is chosen in such a way that (N
exp) : U ! N
N is a smooth dieomorphism onto an open neighborhood V of the diagonal. For f 2 C 1(M
N) we consider the pullback vector bundle M
N TN

N f

f  TN ;;;;! TN ?

f  N ? y

f

? ? yN

M ;;;;! N: 1  Then C (f TN) is canonically isomorphic to the space Cf1 (M
TN) := fh 2 C 1 (M
T N) : N h = f g via s 7! (N f) s and (IdM
h)  h. We consider the space Cc1 (f  TN) of all smooth sections with compact support and equip it with the inductive limit topology Cc1 (f  TN) = inj lim CK1 (f  T N)
K

where K runs through all compact sets in M and each of the spaces CK1 (f  TN) is equipped with the topology of uniform convergence (on K) in all derivatives separately. For f, g 2 C 1 (M
N) we write f g if f and g agree o some compact subset in M. Now let Uf := fg 2 C 1 (M
N) : (f(x)
g(x)) 2 V for all x 2 M
g f g
uf : Uf ! Cc1 (f  TN)
uf (g)(x) = (x
exp;f (1x) (g(x))) = (x
((N
exp);1 (f
g))(x)):

Then uf is a bijective mapping from Uf onto the set fs 2 Cc1 (f  T N) : s(M)  f  U g, whose inverse is given by u;f 1(s) = exp (N f) s, where

18

6. Manifolds of mappings

we view U ! N as ber bundle. The set uf (Uf ) is open in Cc1 (f  TN) for the topology described above. Now we consider the atlas (Uf
uf )f 2C 1 (M
N ) for C 1(M
N). Its chart change mappings are given for s 2 ug (Uf \ Ug )  Cc1 (g TN) by (uf u;g 1 )(s) = (IdM
(N
exp);1 (f
exp (N g) s)) = (f;1 g ) (s)
where g (x
Yg(x) ) := (x
expg(x) (Yg(x) ))) is a smooth dieomorphism g : g TN  g U ! (g
IdN );1 (V )  M
N which is ber respecting over M. Smooth curves in Cc1 (f  TN) are just smooth sections of the bundle  pr2 f  TN !
M, which have compact support in M locally in . The chart change uf u;g 1 = (f;1 g ) is dened on an open subset and obviously maps smooth curves to smooth curves, therefore it is also smooth. Finally we put the identication topology from this atlas onto the space C 1 (M
N), which is obviously ner than the compact open topology and thus Hausdor. The equation uf u;g 1 = (f;1 g ) shows that the smooth structure does not depend on the choice of the smooth Riemannian metric on N. 6.2. Lemma. Smooth curves in C 1(M
N). Let M and N be smooth nite dimensional manifolds. Then the smooth curves c in C 1 (M
N) correspond exactly to the smooth mappings c^ 2 C 1 (
M
N) which satisfy the following property: (1) For each compact interval a
b]  there is a compact subset K  M such that the restriction c^j(a
b]
(M n K)) is constant. Proof: Since is locally compact property (1) is equivalent to (2) For each t 2 there is an open neighborhood U of t in and a compact K  M such that the restriction c^j(U
(M n K)) is constant. Since this is a local condition on and since smooth curves in C 1 (M
N) locally take values in charts as in the proof of theorem 6.1 it suces to describe the smooth curves in the space Cc1 (E) of sections with compact support of a vector bundle (E
p
M
V ). It is equipped with the inductive limit topology Cc1(E) = inj lim CK1 (E)
K

where K runs through all compact subsets of M, and where CK1 (E) denotes the space of sections of E with support in K, equipped with the

6. Manifolds of mappings

19

compact C 1 -topology (the topology of uniform convergence on compact subsets, in all derivatives separately). Since this injective limit is strict (each bounded subset is contained and bounded in some step) smooth curves in Cc1(E) locally factor to smooth maps in CK1 (E) for some K. Let (U
 : E ! U
V ) be a vector bundle atlas for E. Then Y CK1 (E) 3 s 7! (pr2  sjU ) 2 C 1 (U
V ) 

is a linear embedding onto a closed linear subspace of the product, where the factor spaces on the right hand side carry the topology used in 2.6. So a curve in CK1 (E) is smooth if and only if each its "coordinates" in C 1 (U
V ) is smooth. By cartesian closedness 2.8 this is the case if and only if their canonically associates are smooth
U ! V . these t together to a smooth section of the pullback bundle pr2E !
M with support in
K. 6.3. Theorem. C !-manifold structure of C ! (M
N). Let M and N be real analytic manifolds, let M be compact. Then the space C ! (M
N) of all real analytic mappings from M to N is a real analytic manifold, modeled on spaces C ! (f  TN) of real analytic sections of pullback bundles along f : M ! N over M. The proof is a variant of the proof of 6.1, using a real analytic Riemannian metric. It can also be found in Kriegl-Michor, 1990a]. 6.4. Theorem. C !-manifold structure on C 1 (M
N). Let M and N be real analytic manifolds, with M compact. Then the smooth manifold C 1 (M
N) with the structure from 6.1 is even a real analytic manifold. Proof: For a xed real analytic exponential mapping on N the charts (Uf
uf ) from 6.1 for f 2 C ! (M
N) form a smooth atlas for C 1 (M
N), since C ! (M
N) is dense in C 1 (M
N) by Grauert, 1958, Proposition 8]. The chart changings uf u;g 1 = (f;1 g ) are real analytic: this follows from a careful description of the set of real analytic curves into C 1 (f  TN). See again Kriegl-Michor, 1990a] for more details. 6.5 Remark. If M is not compact, C ! (M
N) is dense in C 1(M
N) for the Whitney-C 1-topology by Grauert, 1958, Prop. 8]. This is not the case for the D-topology from Michor, 1980], in which C 1 (M
N) is a smooth manifold. The charts Uf for f 2 C 1 (M
N) do not cover C 1 (M
N). 6.6. Theorem. Smoothness of composition. If M, N are nite dimensional smooth manifolds, then the evaluation mapping ev : C 1 (M
N)
M ! N is smooth.

20

6. Manifolds of mappings

If P is another smooth manifold, then the composition mapping 1 (P
M) ! C 1 (P
N) comp : C 1 (M
N)
Cprop 1 (P
M) denote the space of all proper smooth is smooth, where Cprop mappings P ! M (i.e. compact sets have compact inverse images). This is open in C 1 (P
M). In particular f : C 1 (M
N) ! C 1 (M
N 0) and g : C 1 (M
N) ! 1 1 (P
M). C (P
N) are smooth for f 2 C 1 (N
N 0 ) and g 2 Cprop The corresponding statement for real analytic mappings is also true, but P and M have to be compact. Proof: Using the description of smooth curves in C 1(M
N) given in 6.2 we immediately see that (ev (c1
c2)) = c^1 (t
c1(t)) is smooth for each smooth (c1
c2) : ! C 1 (M
N)
M, so ev is smooth as claimed. 1 (P
M) is open in the manifold The space of proper mappings Cprop 1 C (P
M) since changing a mapping only on a compact set does not change its property of being proper. Let (c1
c2) : ! C 1 (M
N)
1 (P
M) be a smooth curve. Then we have (comp (c1
c2))(t)(p) = Cprop c^1 (t
^c2(t
p)) and one may check that this has again property (1), so it is a smooth curve in C 1 (P
N). Thus comp is smooth. The proof for real analytic manifolds is similar. 6.7. Theorem. Let M and N be smooth manifolds. Then the in nite dimensional smooth vector bundles TC 1(M
N) and Cc1 (M
T N)  C 1 (M
TN) over C 1 (M
N) are canonically isomorphic. The same assertion is true for C ! (M
N), if M is compact. Proof: Here by Cc1(M
T N) we denote the space of all smooth mappings f : M ! TN such that f(x) = 0 f (x) for x 2= Kf , a suitable compact subset of M (equivalently, such that the associated section of the pull back bundle (M f) T N ! M has compact support). One can check directly that the atlas from 6.1 for C 1 (M
N) induces an atlas for TC 1 (M
N) which is equivalent to that for C 1(M
T N) via some natural identications in T T N. This is carried out in great detail in Michor, 1980, 10.19]. 6.8. Theorem. Exponential law. Let M be a (possibly in nite dimensional) smooth manifold, and let M and N be nite dimensional smooth manifolds. Then we have a canonical embedding M

C 1(M
C 1 (M
N))  C 1 (M
M ! N)


6. Manifolds of mappings

21

where the image in the right hand side consists of all smooth mappings f : M
M ! N which satisfy the following property (1) For each point in M there is an open neighborhood U and a compact set K  M such that the restriction f j(U
(M n K)) is constant. We have equality if and only if M is compact. If M and N are real analytic manifolds with M compact we have C ! (M
C ! (M
N)) = C ! (M
M
N) for each real analytic (possibly in nite dimensional) manifold. Proof: This follows directly from the description of all smooth curves in C 1 (M
N) given in the proof of 6.6.

22

7. Di eomorphism groups 7.1. Theorem. Dieomorphism group. For a smooth manifold

M the group Di(M) of all smooth dieomorphisms of M is an open submanifold of C 1(M
M), composition and inversion are smooth. The Lie algebra of the smooth in nite dimensional Lie group Di(M) is the convenient vector space Cc1 (TM) of all smooth vector elds on M with compact support, equipped with the negative of the usual Lie bracket. The exponential mapping Exp : Cc1(T M) ! Di 1 (M) is the ow mapping to time 1, and it is smooth. For a compact real analytic manifold M the group Di ! (M) of all real analytic dieomorphisms is a real analytic Lie group with Lie algebra C ! (TM) and with real analytic exponential mapping. Proof: It is well known that the space Di(M) of all dieomorphisms of M is open in C 1 (M
M) for the Whitney C 1 -topology. Since the topology we use on C 1 (M
M) is ner, Di(M) is an open submani1 (M
M). The composition is smooth by theorem 6.6. To fold of Cprop show that the inversion inv is smooth, we consider a smooth curve c : ! Di(M)  C 1 (M
M). Then the mapping c^ :
M ! M satises 6.6.(1) and (inv c)b fullls the nite dimensional implicit equation c^(t
(inv c)b(t
m)) = m for all t 2 and m 2 M. By the nite dimensional implicit function theorem (inv c)b is smooth in (t
m). Property 6.6.(1) is obvious. So inv maps smooth curves to smooth curves and is thus smooth. This proof is by far simpler than the original one, see Michor, 1980], and shows the power of the Frolicher-Kriegl calculus. By the chart structure from 6.1, or directly from theorem 6.7we see that the tangent space Te Di(M) equals the space Cc1 (T M) of all vector elds with compact support. Likewise Tf Di(M) = Cc1 (f  T M) which we identify with the space of all vector elds with compact support along the dieomorphism f. Right translation f is given by f (g) = f  (g) = g f, thus T( f ):X = X f and for the ow FlXt of the vector eld with compact support X we have dtd FltX = X FltX = T ( Fl ):X. So the one parameter group t 7! FlXt 2 Di(M) is the integral curve of the right invariant vector eld RX : f 7! T ( f ):X = X f on Di(M). Thus the exponential mapping of the dieomorphism group is given by Exp = Fl1 : Xc (M) ! Di(M). To show that is smooth we consider a smooth curve in Xc (M), i. e. a time dependent vector eld with compact support Xt . We may view it as a complete vector eld (0t
Xt) on
M whose smooth ow respects the level surfaces ftg
M and is smooth. Thus Exp maps smooth curves to smooth curves and is smooth X t

7. Dieomorphism groups

23

itself. Again one may compare this simple proof with the original one in Michor, 1983, section 4]. Let us nally compute the Lie bracket on Xc(M) viewed as the Lie algebra of Di(M). 

Ad(Exp(sX))Y = @t@ 0 Exp(sX) Exp(tY ) Exp(;sX) = T (FlXt ) Y FlX;t = (FlX;t ) Y
thus  @  Ad(Exp(tX))Y = @  (FlX ) Y @t 0 @t 0 ;t = ;X
Y ]
the negative of the usual Lie bracket on Xc (M). To understand this contemplate 1.2.(4). 7.2. Remarks. The group Di(M) of smooth dieomorphisms does not carry any real analytic Lie group structure by Milnor, 1984, 9.2], and it has no complexication in general, see Pressley-Segal, 1986, 3.3]. The mapping Ad Exp : Cc1 (TM) ! Di(M) ! L(C 1 (TM)
C 1 (TM)) is not real analytic, see Michor, 1983, 4.11]. For x 2 M the mapping evx Exp : Cc1 (TM) ! Di(M) ! M is not real analytic since (evx Exp)(tX) = FltX (x), which is not real analytic in t for general smooth X. The exponential mapping Exp : Cc1(T M) ! Di(M) is in a very strong sense not surjective: In Grabowski, 1988] it is shown, that Di(M) contains an arcwise connected free subgroup on 2@0 generators which meets the image of Exp only at the identity. The real analytic Lie group Di ! (M) is regular in the sense of Milnor, 1984. 7.6], who weakened the original concept of Omori, 1982]. This condition means that the mapping associating the evolution operator to each time dependent vector eld on M is smooth. It is even real analytic, compare the proof of theorem 7.2. By theorem 6.6 the left action ev : Di(M)
M ! M is smooth. If dim M  2 this action is n-transitive for each n 2 , i. e. for two ordered nite n-tupels (x1
. . .
xn) and (y1
. . .
yn) of pair wise dierent elements of M there is a dieomorphism f 2 Di(M) with f(xi ) = yi for each i. To see this connect xi with yi by a simple smooth curve ci : 0
1] ! M which is an embedding, such that the images are pair wise disjoint. Using tubular neighborhoods or submanifold charts one can

24

7. Dieomorphism groups

show that then there is a vector eld with compact support X 2 Xc (M) which restricts to c0i (t) along ci(t). The dieomorphism Exp(X) = FlX1 then maps xi to yi . In contrast to this one knows from Omori, 1978] that a Banach Lie group acting eectively on a nite dimensional manifold is necessarily nite dimensional. So there is no way to model the dieomorphism group on Banach spaces as a manifold. There is, however, the possibility to view Di(M) as an ILH-group (i. e. inverse limit of Hilbert manifolds) which sometimes permits to use an implicit function theorem. See Omori, 1974] for this. 7.3. Theorem. Principal bundle of embeddings. Let M and N be smooth manifolds. Then the set Emb(M
N) of all smooth embeddings M ! N is an open submanifold of C 1 (M
N). It is the total space of a smooth principal

ber bundle with structure group Di(M), whose smooth base manifold is the space B(M
N) of all submanifolds of N of type M. The open subset Embprop (M
N) of proper (equivalently closed) embeddings is saturated under the Di(M)-action and is thus the total space of the restriction of the principal bundle to the open submanifold Bclosed (M
N) of B(M
N) consisting of all closed submanifolds of N of type M. In the real analytic case this theorem remains true provided that M is supposed to be compact, see Kriegl-Michor, 1990a], section 6. Proof: Let us x an embedding i 2 Emb(M
N). Let g be a xed Riemannian metric on N and let expN be its exponential mapping. Then let p : N (i) ! M be the normal bundle of i, dened in the following way: For x 2 M let N (i)x := (Tx i(Tx M))?  Ti(x)N be the g-orthogonal complement in Ti(x) N. Then i

N (i) ;;;;! TN ?

p? y

?? yN

M ;;;; ! N i is an injective vector bundle homomorphism over i. Now let U i = U be an open neighborhood of the zero section of N (i) which is so small that (expN #i)jU : U ! N is a dieomorphism onto its image which describes a tubular neighborhood of the submanifold i(M). Let us consider the mapping  =  i := (expN #i)jU : N (i) U ! N


7. Dieomorphism groups

25

a dieomorphism onto its image, and the open set in Emb(M
N) which will serve us as a saturated chart,

U (i) := fj 2 Emb(M
N) : j(M)   i (U i )
j ig
where j i means that j = i o some compact set in M. Then by Michor, 1980, section 4] the set U (i) is an open neighborhood of i in Emb(M
N). For each j 2 U (i) we dene 'i (j) : M ! U i  N (i)
'i (j)(x) := ( i);1 (j(x)) Then 'i = (( i );1 ) : U (i) ! C 1 (M
N (i)) is a smooth mapping which is bijective onto the open set

V (i) := fh 2 C 1 (M
N (i)) : h(M)  U i
h 0g in C 1 (M
N (i)). Its inverse is given by the smooth mapping i : h 7!  i h. We have i (h f) = i (h) f for those f 2 Di(M) which are near enough to the identity so that h f 2 V (i). We consider now the open set

fh f : h 2 V (i)
f 2 Di(M)g  C 1 ((M
U i )):

Obviously we have a smooth mapping from it into Cc1 (U i )
Di(M) given by h 7! (h (p h);1
p h), where Cc1 (U i) is the space of sections with compact support of U i ! M. So if we let Q(i) := i (Cc1 (U i) \ V (i))  Emb(M
N) we have

W (i) := U (i) Di(M)

= Q(i)
Di(M)

= (Cc1 (U i ) \V (i))
Di(M)
since the action of Di(M) on i is free. Furthermore jQ(i) : Q(i) !

Emb(M
N)= Di(M) is bijective onto an open set in the quotient. We now consider 'i (jQ(i));1 : (Q(i)) ! C 1(U i ) as a chart for the quotient space. In order to investigate the chart change let j 2 Emb(M
N) be such that (Q(i)) \ (Q(j)) 6= . Then there is an immersion h 2 W (i) \ Q(j), so there exists a unique f0 2 Di(M) (given by f0 = p 'i (h)) such that h f0;1 2 Q(i). If we consider j f0;1 instead of j and call it again j, we have Q(i) \Q(j) 6=  and consequently U (i) \ U (j) 6= . Then the chart change is given as follows: 'i (jQ(i));1  ( j ) : Cc1 (U j ) ! Cc1 (U i) s 7!  j s 7! 'i ( j s) (pi 'i ( j s));1 :

26

7. Dieomorphism groups

This is of the form s 7!  s for a locally dened dieomorphism  : N (j) ! N (i) which is not ber respecting, followed by h 7! h (pi h);1 . Both composants are smooth by the general properties of manifolds of mappings. So the chart change is smooth. We show that the quotient space B(M
N) = Emb(M
N)= Di(M) is Hausdor. Let i, j 2 Emb(M
N) with (i) 6= (j). Then i(M) 6= j(M) in N for otherwise put i(M) = j(M) =: L, a submanifold of N the mapping i;1 j : M ! L ! M is then a dieomorphism of M and j = i (i;1 j) 2 i Di(M), so (i) = (j), contrary to the assumption. Now we distinguish two cases. Case 1. We may nd a point y0 2 i(M) n j(M), say, which is not a cluster point of j(M). We choose an open neighborhood V of y0 in N and an open neighborhood W of j(M) in N such that V \ W = . Let V := fk 2 Emb(M
N) : k(M) \ V g W := fk 2 Emb(M
N) : k(M)  W g. Then V is obviously open in Emb(M
N) and V is even open in the coarser compact open topology. Both V and W are Di(M) saturated, i 2 W , j 2 V , and V \ W = . So (V ) and (W ) separate (i) and (j) in B(M
N). Case 2 Let i(M)  j(M) and j(M)  i(M). Let y 2 i(X), say. Let (V
v) be a chart of N centered at y which maps i(M) \ V into a linear subspace, v(i(M)\V )  m \v(V )  n, where n = dimM, n = dimN. Since j(M)  i(M) we conclude that we have also v((i(M)  j(M)) \ V )  m \ v(V ). So we see that L := i(M)  j(M) is a submanifold of N of the same dimension as N. Let (WL
pL
L) be a tubular neighborhood of L. Then WL ji(M) is a tubular neighborhood of i(M) and WL jj(M) is one of j(M). 7.4. Theorem. Let M and N be smooth manifolds. Then the dieomorphism group Di(M) acts smoothly from the right on the smooth manifold Immprop (M
N) of all smooth proper immersions M ! N, which is an open subset of C 1 (M
N). Then the space of orbits Immprop (M
N)= Di(M) is Hausdor in the quotient topology. Let Immfree, prop (M
N) be set of all proper immersions, on which Di(M) acts freely. Then this is open in C 1 (M
N) and is the total space of a smooth principal ber bundle Immfree,prop (M
N) ! Immfree,prop (M
N)= Di(M): This is proved in Cervera-Mascaro-Michor, 1989], where also the existence of smooth transversals to each orbit is shown and the stratication of the orbit space into smooth manifolds is given.

27

8. The Fr olicher-Nijenhuis Bracket 8.1. In this section let M be a smooth Ldim Mmanifold. We consider the graded

commutative algebra $(M) = k=0 $k (M) of dierential forms on M. The space Derk $(M) consists of all (graded) derivations of degree k, i.e. all linear mappings D : $(M) ! $(M) with D($` (M))  $k+` (M) and D(' ^ ) = D(') ^  + (;1)k` ' ^ D() for ' 2 $` (M). Lemma. Then the space Der $(M) = Lk Derk $(M) is a graded Lie algebra with the graded commutator D1
D2] := D1 D2 ; (;1)k1 k2 D2 D1 as bracket. This means that the bracket is graded anticommutative, D1
D2] = ;(;1)k1 k2 D2
D1 ], and satis es the graded Jacobi identity D1
D2
D3 ]] = D1
D2]
D3] + (;1) D2
D1
D3]] (so that ad(D1 ) = D1
] is itself a derivation). Proof: Plug in the denition of the graded commutator and compute. From the basics of dierential geometry one is already familiar with some graded derivations: for a vector eld X on M the derivation iX is of degree ;1, LX is of degree 0, and d is of degree 1. Note also that the important formula LX = d iX + iX d translates to LX = iX
d]. 8.2. A derivation D 2 Derk $(M) is called algebraic if D j $0 (M) = 0. Then D(f:!) = f:D(!) for f 2 C 1(M
R), so D is of tensorial character and induces a derivation Dx 2 Derk %Tx M for each x 2 M. It is uniquely determined by its restriction to 1-forms Dx jTx M : Tx M ! %k+1T  M which we may view as an element Kx 2 %k+1 Tx M  Tx M depending smoothly on x 2 M. To express this dependence we write D = iK = i(K), whereLK 2 C 1 (%k+1 T  M  T M) =: $k+1(M TM). M k We call $(M
T M) = dim k=0 $ (M
T M) the space of all vector valued dierential forms. Theorem. (1) For K 2 $k+1(M
T M) the formula (iK !)(X1
. . .
Xk+` ) = X = (k+1)!1(`;1)! sign !(K(X 1
. . .
X (k+1))
X (k+2)
. . .) 2xk+`

for ! 2 $` (M), Xi 2 X(M) (or Tx M) de nes an algebraic graded derivation iK 2 Derk $(M) and any algebraic derivation is of this form. (2) By i(K
L]b) := iK
iL] we get a bracket 
]bon $;1(M
T M) which de nes a graded Lie algebra structure with the grading as indicated, and for K 2 $k+1 (M
T M), L 2 $`+1 (M
T M) we have K
L]b= iK L ; (;1)k` iL K:

28

8. Frolicher-Nijenhuis bracket


]bis called the algebraic bracket or also the Nijenhuis-Richardson bracket, see Nijenhuis-Richardson, 1967].

Proof: Since %Tx M is the free graded commutative algebra generated by the vector space Tx M any K 2 $k+1 (M
TM) extends to a

graded derivation. By applying it to an exterior product of 1-forms one can derive the formula in (1). The graded commutator of two algebraic derivations is again algebraic, so the injection i : $+1 (M
T M) ! Derk ($(M)) induces a graded Lie bracket on $+1 (M
TM) whose form can be seen by applying it to a 1-form. 8.3. The exterior derivative d is an element of Der1 $(M). In view of the formula LX = iX
d] = iX d + d iX for vector elds X, we dene for K 2 $k (M T M) the Lie derivation LK = L(K) 2 Derk $(M) by LK := ik
d]. Then the mapping L : $(M
T M) ! Der $(M) is injective, since LK f = iK df = df K for f 2 C 1 (M
R). Theorem. For any graded derivation D 2 Derk $(M) there are unique K 2 $k (M TM) and L 2 $k+1 (M TM) such that D = LK + iL :

We have L = 0 if and only if D
d] = 0. D is algebraic if and only if K = 0. Proof: Let Xi 2 X(M) be vector elds. Then f 7! (Df)(X1
. . .
Xk ) is a derivation C 1 (M
R) ! C 1 (M
R), so there is a unique vector eld K(X1
. . .
Xk ) 2 X(M) such that (Df)(X1
. . .
Xk ) = K(X1
. . .
Xk )f = df(K(X1
. . .
Xk )): Clearly K(X1
. . .
Xk ) is C 1 (M
R)-linear in each Xi and alternating, so K is tensorial, K 2 $k (M TM). The dening equation for K is Df = df K = iK df = LK f for f 2 C 1(M
R). Thus D ;LK is an algebraic derivation, so D ;LK = iL by 8.2 for unique L 2 $k+1(M TM). Since we have d
d] = 2d2 = 0, by the graded Jacobi identity we obtain 0 = iK
d
d]] = iK
d]
d] + (;1)k;1d
iK
d]] = 2LK
d]. The mapping L 7! iL
d] = LK is injective, so the last assertions follow. 8.4. Applying i(IdTM ) on a k-fold exterior product of 1-forms we see that i(IdTM )! = k! for ! 2 $k (M). Thus we have L(IdTM ) = ! = i(IdTM )d! ; d i(IdTM )! = (k + 1)d! ; kd! = d!. Thus L(IdTM ) = d.

8. Frolicher-Nijenhuis bracket

29

8.5. Let K 2 $k(M TM) and L 2 $`(M T M). Then obviously LK
LL]
d] = 0, so we have L(K)
L(L)] = L(K
L]) for a uniquely dened K
L] 2 $k+`(M TM). This vector valued form K
L] is called the Frolicher-Nijenhuis bracket of K and L. M k Theorem. The space $(M TM) = Ldim k=0 1 $ (M TM) with its usual grading is a graded Lie algebra. IdTM 2 $ (M TM) is in the center, i.e. K
IdTM ] = 0 for all K. L : ($(M TM)

]) ! Der $(M) is an injective homomorphism of graded Lie algebras. For vector elds the Frolicher-Nijenhuis bracket coincides with the Lie bracket. Proof: df X
Y ] = L(X
Y ])f = LX
LY ]f. The rest is clear. 8.6. Lemma. For K 2 $k (M TM) and L 2 $`+1 (M TM) we have LK
iL] = i(K
L]) ; (;1)k` L(iL K), or iL
LK ] = L(iL K) ; (;1)k i(L
K]): This generalizes the usual formula for vector elds. Proof: For f 2 C 1(M
R) we have iL
LK ]f = iL iK df ; 0 = iL (df K) = df (iL K) = L(iL K)f. So iL
LK ] ; L(iL K) is an algebraic derivation. iL
LK ]
d] = iL
LK
d]] ; (;1)k` LK
iL
d]] = = 0 ; (;1)k` L(K
L]) = (;1)k i(L
K])
d]): Since 
d] kills the "L's" and is injective on the "i's", the algebraic part of iL
LK ] is (;1)k i(L
K]). 8.7. The space Der $(M) is a graded module over the graded algebra $(M) with the action (! ^ D)' = ! ^ D('). Theorem. Let the degree of ! be q, of ' be k, and of  be `. Let the other degrees be as indicated. Then we have: (1) (2) (3) (4)

! ^ D1
D2 ] = ! ^ D1
D2] ; (;1)(q+k1 )k2 d2(!) ^ D1 : i(! ^ L) = ! ^ i(L) ! ^ LK = L(! ^ K) + (;1)q+k;1 i(d! ^ K): ! ^ L1
L2 ]b= ! ^ L1
L2]b;

30

8. Frolicher-Nijenhuis bracket

; (;1)(q+`1 ;1)(`2 ;1)i(L2 )! ^ L1 : ! ^ K1
K2 ] = ! ^ K1
K2 ] ; (;1)(q+k1 )k2 L(K2)! ^ K1 + (;1)q+k1 d! ^ i(K1 )K2 : '  X
  Y ] = ' ^   X
Y ] ;  ; iY d' ^   X ; (;1)k`iX d ^ '  Y ;  ; d(iY ' ^ )  X ; (;1)k` d(iX  ^ ')  Y = ' ^   X
Y ] + ' ^ LX   Y ; LY ' ^   X + (;1)k (d' ^ iX   Y + iY ' ^ d  X) :

(5) (6)

Proof: For (1) , (2) , (3) write out the denitions. For (4) compute i(! ^ L1
L2 ]b). For (5) compute L(! ^ K1
K2]). For (6) use (5) . 8.8. Theorem. For K 2 $k (M TM) and ! 2 $` (M) the Lie derivative of ! along K is given by the following formula, where the Xi are vector elds on M. (LK !)(X1
. . .
Xk+`) = X = k!1`! sign L(K(X 1
. . .
X k ))(!(X (k+1)
. . .
X (k+`) ))

X + k! (;`;1 1)! sign  !(K(X 1
. . .
X k )
X (k+1)]
X (k+2)
. . .)

+



(;1)k;1 X sign  !(K(X
X ]
X
. . .)
X 1 2 3 (k+2)
. . .): (k;1)! (`;1)! 2!



Proof: It suces to consider K = '  X. Then by 8.7.3 we have L('  X) = ' ^ LX ; (;1)k;1 d' ^ iX . Now use the usual global formulas to expand this.

8.9. Theorem. Global formula for the Fr
olicher-Nijenhuis bracket. For K 2 $k (M TM) and L 2 $` (M TM) we have for the Frolicher-Nijenhuis bracket K
L] the following formula, where the Xi are vector elds on M. K
L](X1
. . .
Xk+` ) = X = k!1`! sign  K(X 1
. . .
X k )
L(X (k+1)
. . .
X (k+`) )]

+ k! (;`;1 1)! + ((k;;1)1)! `! k`

X

X

sign  L(K(X 1
. . .
X k )
X (k+1) ]
X (k+2)
. . .) sign  K(L(X 1
. . .
X ` )
X (`+1) ]
X (`+2)
. . .)

;

;1) + (k;(1)! (`;1)! 2! k 1

(k;1)`

+ (k;(;1)!1)(`;1)! 2!

X

X

8. Frolicher-Nijenhuis bracket

31

sign  L(K(X 1
X 2 ]
X 3
. . .)
X (k+2)
. . .) sign  K(L(X 1
X 2 ]
X 3
. . .)
X (`+2)
. . .):

Proof: It suces to consider K = '  X and L =   Y , then for '  X
  Y ] we may use 8.7.6 and evaluate that at (X1
. . .
Xk+`). After some combinatorial computation we get the right hand side of the above formula for K = '  X and L =   Y . There are more illuminating ways to prove this formula, see Michor,

1987].

8.10. LocalPformulas. In a local P chart (U
u) on the manifold PM we put K j U = Ki d  @i , L j U = Lj d  @j , and ! j U = ! d , where  = (1  1 < 2 < < k  dimM) is a form index, d = du1 ^ . . . ^ du , @i = @u@ and so on. k

i

Plugging Xj = @i into the global formulas 8.2, 8.8, and 8.9, we get the following local formulas: j

X

iK ! j U = Ki 1 ... !i +1 ... + ;1 d X i K
L]bj U = K1 ... Lji +1 ... + k

k

k

k `

k



k `

; (;1)(k;1)(`;1)Li1 ... Kij +1 ... + d  @j `

X i LK ! j U = K1 ...k @i !k+1 ...k+`

`

k `



+ (;1)k (@1 Ki 1 ...  ) !i +2 ... + d k

k

K
L] j U =

X i K1 ...k @i Ljk+1 ...k+`

k `

; (;1)k`Li1 ... @iKj +1 ... + ; kKj 1 ... ;1 i @ Li +1 ... +  + (;1)k``Lj1 ... ;1 i @ Ki +1 ... + d  @j `

k

`

k

`

k `

k

k `

`

`

k `

8.11. Theorem. For Ki 2 $k (M TM) and Li 2 $k +1 (M TM) we have (1)

i

i

LK1 + iL1
LK2 + iL2 ] = ;  = L K1
K2] + iL1 K2 ; (;1)k1 k2 iL2 K1 ;  + i L1
L2]b+ K1
L2] ; (;1)k1 k2 K2
L1] :

32

8. Frolicher-Nijenhuis bracket

Each summand of this formula looks like a semidirect product of graded Lie algebras, but the mappings i : $(M T M) ! End($(M T M)

]) ad : $(M T M) ! End($(M T M)

]b) do not take values in the subspaces of graded derivations. We have instead for K 2 $k (M TM) and L 2 $`+1 (M TM) the following relations: (2) (3)

iL K1
K2 ] = iL K1
K2] + (;1)k1 ` K1
iL K2 ]   ; (;1)k1 ` i(K1
L])K2 ; (;1)(k1 +`)k2 i(K2
L])K1 K
L1
L2]b] = K
L1]
L2]b+ (;1)kk1 L1
K
L2]]b;   ; (;1)kk1 i(L1 )K
L2] ; (;1)(k+k1 )k2 i(L2 )K
L1]

The algebraic meaning of the relations of this theorem and its consequences in group theory have been investigated in Michor, 1989b]. The corresponding product of groups is well known to algebraists under the name "Zappa-Szep"-product. Proof: Equation (1) is an immediate consequence of 8.6. Equations (2) and (3) follow from (1) by writing out the graded Jacobi identity, or as follows: Consider L(iL K1
K2]) and use 8.6 repeatedly to obtain L of the right hand side of (2). Then consider i(K
L1
L2]b]) and use again 8.6 several times to obtain i of the right hand side of (3). 8.12. Corollary (of 8.9). For K, L 2 $1 (M TM) we have K
L](X
Y ) = KX
LY ] ; KY
LX] ; L(KX
Y ] ; KY
X]) ; K(LX
Y ] ; LY
X]) + (LK + KL)X
Y ]:

8.13. Curvature. Let P 2 $1(M TM) be a ber projection, i.e. P P = P . This is the most general case of a (rst order) connection. We may call ker P the horizontal space and imP the vertical space of the connection. If P is of constant rank, then both are sub vector bundles of T M. If imP is some primarily xed sub vector bundle or (tangent bundle of) a foliation, P can be called a connection for it. Special cases of this will be treated extensively later on. For the general theory see Michor, 1989a] The following result is immediate from 8.12.

8. Frolicher-Nijenhuis bracket

33

Lemma. We have

# P
P] = 2R + 2R
where R, R# 2 $2 (M TM) are given by R(X
Y ) = P (Id ; P)X
(Id ; # Y ) = (Id ; P )PX
PY ]. P)Y ] and R(X
If P has constant rank, then R is the obstruction against integrability of the horizontal bundle ker P , and R# is the obstruction against integrability of the vertical bundle imP. Thus we call R the curvature and R# the cocurvature of the connection P. We will see later, that for a principal ber bundle R is just the negative of the usual curvature. 8.14. Lemma. General Bianchi identity. If P 2 $1(M T M) is a # then connection ( ber projection) with curvature R and cocurvature R, we have # =0 P
R + R] R
P ] = iR R# + iR R: Proof: We have P
P ] = 2R + 2R# by 8.13 and P
P
P]] = 0 by the graded Jacobi identity. So the rst formula follows. We have 2R = P P
P] = iP
P ] P. By 8.11.2 we get iP
P ] P
P] = 2iP
P ] P
P] ; 0 = # # = iR R+i # R R R) 4R
P]. Therefore R
P] = 41 iP
P ] P
P ] = i(R+ R)(R+ since R has vertical values and kills vertical vectors, so iR R = 0 likewise # for R. 8.15. Naturality of the Fr
olicher-Nijenhuis bracket. Let f : M ! N be a smooth mapping between manifolds. Two vector valued forms K 2 $k (M TM) and K 0 2 $k (N TN) are called f -related or f -dependent, if for all Xi 2 Tx M we have (1) Kf0 (x) (Tx f X1
. . .
Tx f Xk ) = Tx f Kx (X1
. . .
Xk ):

Theorem.

(2) If K and K 0 as above are f-related then iK f  = f  iK 0 : $(N) ! $(M). (3) If iK f  j B 1 (N) = f  iK 0 j B 1 (N), then K and K 0 are f-related, where B 1 denotes the space of exact 1-forms. (4) If Kj and Kj0 are f-related for j = 1
2, then iK1 K2 and iK10 K20 are f-related, and also K1
K2]b and K10
K20 ]b are f-related. (5) If K and K 0 are f-related then LK f  = f  LK 0 : $(N) ! $(M). (6) If LK f  j $0 (N) = f  LK 0 j $0(N), then K and K 0 are f-related. (7) If Kj and Kj0 are f-related for j = 1
2, then their FrolicherNijenhuis brackets K1
K2 ] and K10
K20 ] are also f-related.

34

8. Frolicher-Nijenhuis bracket

Proof: (2) By 8.2 we have for ! 2 $q (N) and Xi 2 Tx M: (iK f  !)x (X1
. . .
Xq+k;1 ) = X = k! (q1;1)! sign  (f  !)x (Kx (X 1
. . .
X k )
X (k+1)
. . .) 1

= k! (q;1)! = k! (q1;1)!

X

X

sign  !f (x) (Tx f Kx (X 1
. . .)
Tx f X (k+1)
. . .) sign  !f (x) (Kf0 (x) (Tx f X 1
. . .)
Txf X (k+1)
. . .)

= (f  iK 0 !)x (X1
. . .
Xq+k;1) (3) follows from this computation, since the df, f 2 C 1 (M
R) separate points. (4) follows from the same computation for K2 instead of !, the result for the bracket then follows 8.2.2. (5) The algebra homomorphism f  intertwines the operators iK and iK 0 by (2), and f  commutes with the exterior derivative d. Thus f  intertwines the commutators iK
d] = LK and iK 0
d] = LK 0 . (6) For g 2 $0(N) we have LK f  g = iK d f  g = iK f  dg and  f LK 0 g = f  iK 0 dg. By (3) the result follows. (6) The algebra homomorphism f  intertwines LK and LK 0 , so also their graded commutators which equal L(K1
K2 ]) and L(K10
K20 ]), respectively. Now use (6) . 8.16. Let f : M ! N be a local dieomorphism. Then we can consider the pullback operator f  : $(M TM) ! $(M T M), given by j

j

(1) (F K)x (X1
. . .
Xk ) = (Tx f);1 Kf (x) (Tx f X1
. . .
Txf Xk ): Clearly K and f  K are then f-related. Theorem. In this situation we have: (2) f  K
L] = f  K
f  L]. (3) f  iK L = if  K f  L. (4) f  K
L]b= f  K
f  L]b. (5) For a vector eld X 2 X(M) and K 2 $(M TM) the Lie derivative LX K = @t@ 0 (FlXt ) K satis es LX K = X
K], the FrolicherNijenhuis-bracket. We may say that the Frolicher-Nijenhuis bracket, 
]b, etc. are natural bilinear concomitants. Proof: (2) { (4) are obvious from 8.16. (5) Obviously LX is R-linear, so it suces to check this formula for K =   Y ,  2 $(M) and

8. Frolicher-Nijenhuis bracket

35

Y 2 X(M). But then

LX (  Y ) = LX   Y +   LX Y = LX   Y +   X
Y ] = X
  Y ] by 8.7.6:

8.17. Remark. At last we mention the best known application of the Frolicher-Nijenhuis bracket, which also led to its discovery. A vector valued 1-form J 2 $1 (M T M) with J J = ;Id is called a nearly complex structure if it exists, p dimM is even and J can be viewed as a ber multiplication with ;1 on TM. By 8.12 we have J
J](X
Y ) = 2(JX
JY ] ; X
Y ] ; JX
JY ] ; JJX
Y ]): The vector valued form 21 J
J] is also called the Nijenhuis tensor of J. For it the following result is true: A manifold M with a nearly complex structure J is a complex manifold (i.e., there exists an atlas for M with holomorphic chart-change mappings) if and only if J
J] = 0. See Newlander-Nirenberg, 1957].

36

9. Fiber Bundles and Connections

9.1. Denition. A (
ber) bundle (E
p
M
S) consists of manifolds E, M, S, and a smooth mapping p : E ! M furthermore each x 2 M has an open neighborhood U such that E j U := p;1(U) is dieomorphic to U
S via a ber respecting dieomorphism:

E j U ;;;;! U
S ?

p? y

?

pr1 ? y

U U E is called the total space, M is called the base space or basis, p is a surjective submersion, called the projection, and S is called standard
ber. (U
) as above is called a
ber chart. A collection of ber charts (U
 ), such that (U ) is an open cover of M, is called a (
ber) bundle atlas. If we x such an atlas, then   ;1(x
s) = (x
 (x
s)), where  : (U \U )
S ! S is smooth and  (x
) is a dieomorphism of S for each x 2 U := U \ U . We may thus consider the mappings  : U ! Di(S) with values in the group Di(S) of all dieomorphisms of S their dierentiability is a subtle question, which will be discussed in 13.1 below. In either form these mappings  are called the transition functions of the bundle. They satisfy the cocycle condition:  (x)   (x) =  (x) for x 2 U  and (x) = IdS for x 2 U . Therefore the collection ( ) is called a cocycle of transition functions. Given an open cover (U ) of a manifold M and a cocycle of transition functions ( ) we may construct a ber bundle F (E
p
M
S) similarly as in the following way: On the disjoint union  fg
U
S we consider the equivalence relation: (
x
s) (
y
t) if and only if x = y and  (y
t) = s. The quotient space then turns out to be a ber bundle, and we may use the mappings  ((
x
s)]) := (x
s) as a bundle atlas whose cocycle of transition functions is exactly the one we started from. 9.2. Lemma. Let p : N ! M be a surjective submersion (a bred manifold) such that p;1(x) is compact for each x 2 M and let M be connected. Then (N
p
M) is a ber bundle. Proof: We have to produce a ber chart at each x0 2 M. So let (U
u) be a chart centered at x0 on M such that u(U)

= Rm . For each x 2 U ; 1 let x (y) := (Ty u) :u(x), then x 2 X(U), depending smoothly on x 2 U, such that u(Fl t u;1(z)) = z + t:u(x), so each x is a complete x

9. Connections on general ber bundles

37

vector eld on U. Since p is a submersion, with the help of a partition of unity on p;1(U) we may construct vector elds x 2 X(p;1 (U)) which depend smoothly on x 2 U and are p-related to x : T p:x = x p. Thus p Flt = Fl t p and Flt is ber respecting, and since each ber is compact and x is complete, x has a global ow too. Denote p;1 (x0) by S. Then ' : U
S ! p;1(U), dened by '(x
y) = Fl1 (y), is a dieomorphism and is ber respecting, so (U
';1 ) is a ber chart. Since M is connected, the bers p;1(x) are all dieomorphic. 9.3. Let (E
p
M
S) be a ber bundle we consider the ber linear mapping Tp : TE ! TM and its kernel ker Tp =: V E which is called the vertical bundle of E. The following is special case of 8.13. Denition: A connection on the ber bundle (E
p
M
S) is a vector valued 1-form 2 $1(E V E) with values in the vertical bundle VE such that = and Im = V E so is just a projection TE ! V E. Then ker is of constant rank, so ker is a sub vector bundle of TE, it is called the space of horizontal vectors or the horizontal bundle and it is denoted by HE Clearly T E = HE  V E and Tu E = HuE  Vu E for u 2 E. Now we consider the mapping (T p
E ) : T E ! TM
M E. Then (Tp
E );1 (0p(u)
u) = Vu E by denition, so (Tp
E ) j HE : HE ! TM
M E is ber linear over E and injective, so by reason of dimensions it is a ber linear isomorphism: Its inverse is denoted by x

x

x

x

C := ((Tp
E ) j HE);1 : TM
M E ! HE ,! T E: So C : T M
M E ! TE is ber linear over E and is a right inverse for (Tp
E ). C is called the horizontal lift associated to the connection . Note the formula (u ) = u ; C(Tp:u
u) for u 2 Tu E. So we can equally well describe a connection by specifying C. Then we call the vertical projection and  := idTE ; = C (Tp
E ) will be called the horizontal projection. 9.4. Curvature. If : TE ! V E is a connection on a ber bundle (E
p
M
S), then as in 8.13 the curvature R of is given by 2R = 
] = Id ;
Id ; ] = 
] 2 $2 (E V E): We have R(X
Y ) = 21 
](X
Y ) = X
Y ], so R is an obstruction against integrability of the horizontal subbundle. Since the vertical bundle V E is integrable, by 8.14 we have the Bianchi identity 
R] = 0. 9.5. Pullback. Let (E
p
M
S) be a ber bundle and consider a smooth mapping f : N ! M. Since p is a submersion, f and p are

38

9. Connections on general ber bundles

transversal and thus the pullback N
(f
M
p) E exists. It will be called the pullback of the ber bundle E by F and we will denote it by f  E. The following diagram sets up some further notation for it: p f

f  E ;;;;! E ?

f ? y

?

p? y

N ;;;;! M f

Proposition. In the situation above we have:

(1) (f  E
f  p
N
S) is again a ber bundle, and p f is a ber wise dieomorphism. (2) If 2 $1 (E TE) is a connection on the bundle E, the vector valued form f  , given by (f  )u (X) := Tu (p f);1 : :Tu(p f):X for X 2 Tu E, is again a connection on the bundle f  E. The forms f  and are p f-related in the sense of 8.15. (3) The curvatures of f  and are also p f-related: R(f  ) = f  R( ). Proof: (1). If (U
) is a ber bundle atlas of (E
p
M
S) in the sense of 9.1, then (f ;1 (U )
(f  p
pr2  p f)) is visibly a ber bundle atlas for (f  E
f  p
N
S), by the formal universal properties of a pullback. (2) is obvious. (3) follows from (2) and 8.15.6. 9.6. Flat connections. Let us suppose that a connection on the bundle (E
p
M
S) has zero curvature. Then by 9.4 the horizontal bundle is integrable and gives rise to the horizontal foliation. Each point u 2 E lies on a unique leaf L(u) such that Tv L(u) = Hv E for each v 2 L(u). The restriction p j L(u) is locally a dieomorphism, but in general it is neither surjective nor is it a covering onto its image. This is seen by devising suitable horizontal foliations on the trivial bundle pr2 : R
S 1 ! S 1 . 9.7. Local description of connections. Let be a connection on (E
p
M
S). Let us x a ber bundle atlas (U ) with transition functions ( ), and let us consider the connection ( );1 ) 2 $1(U
S U
TS), which may be written in the form (( );1 ) )(x
y ) =: ;;(x
y) + y for x 2 Tx U and y 2 Ty S
since it reproduces vertical vectors. The ; are given by (0x
;(x
y)) := ;T ( ): :T();1 :(x
0y ):

9. Connections on general ber bundles

39

We consider ; as an element of the space $1(U  X(S)), a 1-form on U  with values in the innite dimensional Lie algebra X(S) of all vector elds on the standard ber. The ; are called the Christoel forms of the connection with respect to the bundle atlas (U
 ). Lemma. The transformation law for the Christoel forms is Ty ( (x
)):; (x
y) = ;(x
 (x
y)) ; Tx ( (
y)):x : The curvature R of satis es (;1 ) R = d; + ;
;]X(S ): The formula for the curvature is the Maurer-Cartan formula which in this general setting appears only in the level of local description. Proof: From  ( );1(x
y) = (x
 (x
y)) we get that T( ( );1):(x
y ) = (x
T(x
y)( ):(x
y )) and thus: T( ;1 ):(0x
; (x
y)) = ; (T ( ;1 )(x
0y )) = = ; (T (;1 ):T (  ;1 ):(x
0y )) = = ; (T (;1 )(x
T(x
y)( )(x
0y ))) = = ; (T (;1 )(x
0y )) ; (T(;1 )(0x
T(x
y) (x
0y )) = = T(;1 ):(0x
;(x
 (x
y))) ; T (;1 )(0x
Tx ( (
y)):x ): This implies the transformation law. For the curvature R of we have by 9.4 and 9.5.3 (;1 ) R (( 1
1)
( 2
2)) = = (;1 ) R (Id ; (;1 ) )( 1
1 )
(Id ; (;1 ) )( 2
2)] = = (;1 ) ( 1
;( 1 ))
( 2
;( 2 ))] = ;  = (;1 )  1
 2]
 1; ( 2) ;  2 ; ( 1) + ;( 1 )
; ( 2)] = = ;; ( 1
 2]) +  1 ;( 2 ) ;  2 ;( 1 ) + ;( 1 )
;( 2 )] = = d; ( 1
 2 ) + ; ( 1)
; ( 2)]X(S ):

9.8. Theorem. Parallel transport. Let be a connection on a bundle (E
p
M
S) and let c : (a
b) ! M be a smooth curve with 0 2 (a
b), c(0) = x. Then there is a neighborhood U of Ex
f0g in Ex
(a
b) and a smooth mapping Ptc : U ! E such that: (1) p(Pt(c
t
ux)) = c(t) if de ned, and Pt(c
ux
0) = ux .

40

9. Connections on general ber bundles

(2) ( dtd Pt(c
t
ux)) = 0c(t) if de ned. (3) Reparametrisation invariance: If f : (a0
b0) ! (a
b) is smooth with 0 2 (a0
b0), then Pt(c
f(t)
ux) = Pt(c f
t
Pt(c
ux
f(0))) if de ned. (4) U is maximal for properties (1) and (2). (5) The parallel transport is smooth as a mapping Pt

C 1 (
M)
(ev0
M
p) E
U ;! E
where U is its domain of de nition. (6) If X 2 X(M) is a vector eld on the base then the parallel transport along the ow lines of X is given by the ow of the horizontal lift CX of X: Pt(FlX (x)
t
ux) = FlCX t (ux ):

Proof: We rst give three dierent proofs of assertions (1) { (4). First proof: In local bundle coordinates ( dtd Pt(c
ux
t)) = 0 is an

ordinary dierential equation of rst order, nonlinear, with initial condition Pt(c
ux
0) = ux. So there is a maximally dened local solution curve which is unique. All further properties are consequences of uniqueness. Second proof: Consider the pullback bundle (c E
cp
(a
b)
S) and the pullback connection c on it. It has zero curvature, since the horizontal bundle is 1-dimensional. By 9.6 the horizontal foliation exists and the parallel transport just follows a leaf and we may map it back to E, in detail: Pt(c
ux
t) = p c((c p j L(ux ));1(t)). Third proof: Consider a ber bundle atlas (U
 ) as in 9.7. Then  (Pt(c
;1 (x
y)
t)) = (c(t)
(y
t)), where ;

0 = (;1 )

; d

  d d ;d dt c(t)
dt (y
t) = ;; dt c(t)
(y
t) + dt (y
t)


so (y
t) is the integral curve ;(evolution  line) through y 2 S of the d  time dependent vector eld ; dt c(t) on S. This vector eld visibly depends smoothly on c. Clearly local solutions exist and all properties follow. (5). By the considerations of section 6 it suces to show that Pt maps a smooth curve (c1
c2
c3) : ! U  C 1(
M)
(ev0
M
p) E


9. Connections on general ber bundles

41

to a smooth curve in E. We use the description of the smooth curves in C 1 (
M) from 6.2 and see that the ordinary dierential equation considered in the rst proof then just depends on some parameters more. By the theory of ordinary dierential equations the solution is smooth in this parameters also, which we then set equal. (6) is obvious from the denition. 9.9. Lemma. Let be a connection on a bundle (E
p
M
S) with curvature R and horizontal lift C. Let X 2 X(M) be a vector eld on the base. Then for the horizontal lift CX 2 X(E) we have   1 LCX = @t@ 0 (FlCX t ) = CX
] = ; 2 iCX R:

  Proof: From 8.16.(5) we get LCX = @t@ 0 (FlCX t ) = CX
]. From

8.11.(2) we have iCX R = iCX 
] = iCX
] ; 
iCX ] + 2i
CX ] = ;2 CX
] The vector eld CX is p-related to X and 2 $1(E T E) is p-related to 0 2 $1(M TM), so by 8.15.(7) the form CX
] 2 $1(E T E) is also p-related to 0 = X
0] 2 $1 (M T M). So Tp:CX
] = 0, CX
] has vertical values, and CX
] = CX
]. 9.10. A connection on (E
p
M
S) is called a complete connection, if the parallel transport Ptc along any smooth curve c : (a
b) ! M is dened on the whole of Ec(0)
(a
b). The third proof of theorem 9.8 shows that on a ber bundle with compact standard ber any connection is complete. The following is a sucient condition for a connection to be complete: There exists a ber bundle atlas (U
) and complete Riemannian metrics g on the standard ber S such that each Christoel form ; 2 $1 (U
X(S)) takes values in the linear subspace of gbounded vector elds on S For in the third proof of theorem 9.8 above the time dependent vector eld ; ( dtd c(t)) on S is g-bounded for compact time intervals. So by continuation the solution exists over c;1(U ), and thus globally. A complete connection is called an Ehresmann connection in Greub - Halperin - Vanstone I, p 314], where it is also indicated how to prove the following result.

42

9. Connections on general ber bundles

Theorem. Each ber bundle admits complete connections. Proof: Let dimM = m. Let (U
) be a ber bundle atlas as in 9.1. By topological dimension theory Nagata, 1965] the open cover

(U ) of M admits a renement such that any m + 2 members have empty intersection. Let U ) itself have this property. Choose a smooth partition of unity (f ) subordinated to (U ). Then the setsPV := f x : f (x) > m1+2 g  U form still an open cover of M since f (x) = 1 and at most m + 1 of the f (x) can be nonzero. By renaming assume that each V is connected. Then we choose an open cover (W ) of M such that W  V . Now let g1 and g2 be complete Riemannian metrics on M and S, respectively (see Nomizu - Ozeki, 1961] or Morrow, 1970]). For not connected Riemannian manifolds complete means that each connected component is complete. Then g1jU
g2 P is a Riemannian metric on U
S and we consider the metric g := f  (g1 jU
g2) on E. Obviously p : E ! M is a Riemannian submersion for the metrics g and g1. We choose now the connection : TE ! V E as the orthonormal projection with respect to the Riemannian metric g. Claim: is a complete connection on E. Let c : 0
1] ! M be a smooth curve. We choose a partition 0 = t0 < t1 < < tk = 1 such that c(ti
ti+1 ])  V for suitable i. It suces to show that Pt(c(ti + )
t
uc(t ) ) exists for all 0  t  ti+1 ; ti and all uc(t ) 2 Ec(t ) , for all i | then we may piece them together. So we may assume that c : 0
1] ! V for some . Let us now assume that for some (x
y) 2 V
S the parallel transport Pt(c
t
(x
y)) is dened only for t 2 0
t0) for some 0 < t0 < 1. By the third proof of 9.8 we have Pt(c
t
(x
y)) = (c(t)
(t)), where  : 0
t0) ! S is the maximally dened integral curve through y 2 S of the time dependent ;1   d vector eld P ; ( dt c(t)
) on S. We put g := ( ) g, then (g )(x
y) = (g1 )x
( f (x)  (x
) g2 )y . Since pr1 : (V
S
g) ! (V
g1jV) is a Riemannian submersion and since the connection (;1 ) is also given by orthonormal projection onto the vertical bundle, we get i

i

i

i

1 > g1 -lengtht00 (c) = g -length(c
) = =

Z t0 q 0

Z t0 0

j(c0 (t)
dtd (t))jg dt = 

jc0(t)j2g1 + P f (c(t))( (c(t)
;) g2)( dtd (t)
dtd (t)) dt  Z t0 p Z t0 p  f (c(t)) j dtd (t)jg2 dt  m + 2 j dtd (t)jg2 dt: 0

0

So g2-lenght() is nite and since the Riemannian metric g2 on S is

9. Connections on general ber bundles

43

complete, limt!t0 (t) =: (t0 ) exists in S and the integral curve  can be continued.

44

10. Principal Fiber Bundles and G-Bundles 10.1. Denition. Let G be a Lie group and let (E
p
M
S) be a ber bundle as in 9.1. A G-bundle structure on the ber bundle consists of the following data: (1) A left action ` : G
S ! S of the Lie group on the standard ber. (2) A ber bundle atlas (U
 ) whose transition functions ( ) act on S via the G-action: There is a family of smooth mappings (' : U ! G) which satises the cocycle condition ' (x)'  (x) = ' (x) for x 2 U  and '(x) = e, the unit in the group, such that  (x
s) = `(' (x)
s) = ' (x):s. A ber bundle with a G-bundle structure is called a G-bundle. A ber bundle atlas as in (2) is called a G-atlas and the family (' ) is also called a cocycle of transition functions, but now for the G-bundle. To be more precise, two G-atlases are said to be equivalent (to describe the same G-bundle), if their union is also a G-atlas. This translates as follows to the two cocycles of transition functions, where we assume that the two coverings of M are the same (by passing to the common renement, if necessary): (' ) and ('0 ) are called cohomologous if there is a family ( : U ! G) such that ' (x) = (x);1 :'0 (x): (x) holds for all x 2 U . In (2) one should specify only an equivalence class of G-bundle structures or only a cohomology class of cocycles of G-valued transition functions. From any open cover (U ) of M, some cocycle of transition functions (' : U ! G) for it, and a left G-action on a manifold S, we may construct a G-bundle by gluing, which depends only on the cohomology class of the cocycle. By some abuse of notation we write (E
p
M
S
G) for a ber bundle with specied G-bundle structure. Examples: The tangent bundle of a manifold M is a ber bundle with structure group GL(n). More general a vector bundle (E
p
M
V ) is a ber bundle with standard ber the vector space V and with GL(V )structure. 10.2. Denition. A principal (
ber) bundle (P
p
M
G) is a G-bundle with typical ber a Lie group G, where the left action of G on G is just the left translation. So by 10.1 we are given a bundle atlas (U
' : P jU ! U
G) such that we have ' '; 1 (x
a) = (x
' (x):a) for the cocycle of transition functions (' : U ! G). This is now called a principal bundle atlas. Clearly the principal bundle is uniquely specied by the cohomology class of its cocycle of transition functions.

10. Principal Fiber Bundles and G-Bundles

45

Each principal bundle admits a unique right action r : P
G ! P , called the principal right action, given by ' (r('; 1 (x
a)
g)) = (x
ag). Since left and right translation on G commute, this is well dened. As in 1.3 we write r(u
g) = u:g when the meaning is clear. The principal right action is visibly free and for any ux 2 Px the partial mapping ru = r(ux
) : G ! Px is a dieomorphism onto the ber through ux , whose inverse is denoted by u : Px ! G. These inverses together give a smooth mapping  : P
M P ! G, whose local expression is ('; 1 (x
a)
'; 1(x
b)) = a;1:b. This mapping is also uniquely determined by the implicit equation r(ux
(ux
vx)) = vx , thus we also have (ux :g
u0x:g0) = g;1:(ux
u0x):g0 and (ux
ux) = e. 10.3. Lemma. Let p : P ! M be a surjective submersion (a bred manifold), and let G be a Lie group which acts freely on P such that the orbits of the action are exactly the bers p;1 (x) of p. Then (P
p
M
G) is a principal ber bundle. Proof: Let the action be a right one by using the group inversion if necessary. Let s : U ! P be local sections (right inverses) for p : P ! M such that (U ) is an open cover of M. Let '; 1 : U
G ! P jU be given by '; 1 (x
a) = s (x):a, which is obviously injective with invertible tangent mapping, so its inverse ' : P jU ! U
G is a ber respecting dieomorphism. So (U
') is already a ber bundle atlas. Let  : P
M P ! G be given by the implicit equation r(ux
(ux
u0x)) = u0x , where r is the right G-action.  is smooth by the implicit function theorem and clearly (ux
u0x :g) = (ux
u0x):g. Thus we have ' '; 1 (x
g) = ' (s (x):g) = (x
(s (x)
s (x):g)) = (x
(s(x)
s (x)):g) and (U
') is a principal bundle atlas. x

x

10.4. Remarks. In the proof of Lemma 10.3 we have seen, that a

principal bundle atlas of a principal ber bundle (P
p
M
G) is already determined if we specify a family of smooth sections of P, whose domains of denition cover the base M. Lemma 10.3 can serve as an equivalent denition for a principal bundle. But this is true only if an implicit function theorem is available, so in topology or in innite dimensional dierential geometry one should stick to our original denition. From the Lemma itself it follows, that the pullback f  P over a smooth mapping f : M 0 ! M is again a principal ber bundle. 10.5. Homogeneous spaces. Let G be a Lie group with Lie algebra g. Let K be a closed subgroup of G, then by a well known theorem K is a closed Lie subgroup whose Lie algebra will be denoted by K. There is a

46

10. Principal Fiber Bundles and G-Bundles

unique structure of a smooth manifold on the quotient space G=K such that the projection p : G ! G=K is a submersion, so by the implicit function theorem p admits local sections. Theorem. (G
p
G=K
K) is a principal ber bundle. Proof: The group multiplication of G restricts to a free right action  : G
K ! G, whose orbits are exactly the bers of p. By lemma 10.3 the result follows. For the convenience of the reader we discuss now the best know homogeneous spaces. The group SO(n) acts transitively on S n;1  Rn. The isotropy group of the "north pole" (0
. . .
0
1) is the subgroup 

1 0 0 SO(n ; 1)



which we identify with SO(n ; 1). So S n;1 = SO(n)=SO(n ; 1) and (SO(n)
p
S n;1
SO(n ; 1)) is a principal ber bundle. Likewise (O(n)
p
S n;1
O(n ; 1)), (SU(n)
p
S 2n;1
SU(n ; 1)), (U(n)
p
S 2n;1
U(n ; 1)), and (Sp(n)
p
S 4n;1
Sp(n ; 1)) are principal ber bundles. The Grassmann manifold G(k
n R) is the space of all k-planes containing 0 in Rn. The group O(n) acts transitively on it and the isotropy group of the k-plane Rk
f0g is the subgroup 



O(k) 0 0 O(n ; k)


therefore G(k
n R) = O(n)=O(k)
O(n ; k) is a compact manifold and we get the principal ber bundle (O(n)
p
G(k
n R)
O(k)
O(n ; k)). Likewise ~ n
R)
SO(k)
SO(n ; k)), (SO(n)
p
G(k
(U(n)
p
G(k
n
C)
U(k)
U(n ; k)), and (Sp(n)
p
G(k
n
H)
Sp(k)
Sp(n ; k)) are principal ber bundles. The Stiefel manifold V (k
n
R) is the space of all orthonormal kframes in Rn. Clearly the group O(n) acts transitively on V (k
n
R) and the isotropy subgroup of (e1
. . .
ek ) is 1
O(n ; k), so V (k
n
R) = O(n)=O(n ; k) is a compact manifold and we get a principal ber bundle (O(n)
p
V (k
n
R)
O(n ; k)) But O(k) also acts from the right on V (k
n
R), its orbits are exactly the bers of the projection p : V (k
n
R) ! G(k
n
R). So by lemma 10.3 we get a principal ber

10. Principal Fiber Bundles and G-Bundles

47

bundle (V (k
n
R)
p
G(k
n
R)
O(k)). Indeed we have the following diagram where all arrows are projections of principal ber bundles, and where the respective structure groups are written on the arrows: O(n;k)

;;;;;! V (k
n
R)

O(n)

(a)

?

O(k)? y

?

O(k)? y

V (n ; k
n
R) ;;;;;! G(k
n
R)
O(n;k)

It is easy to see that V (k
n) is also dieomorphic to the space f A 2 L(Rk
Rn) : At :A = 1k g, i.e. the space of all linear isometries Rk ! Rn . There are furthermore complex and quaternionic versions of the Stiefel manifolds. More examples will be given in sections on jets below.

10.6. Homomorphisms. Let  : (P
p
M
G) ! (P 0
p0
M 0
G) be a principal
ber bundle homomorphism, i.e. a smooth G-equivariant mapping  : P ! P 0. Then obviously the diagram  P ;;;;! P 0

(a)

?

p? y

?

p0 ? y

M ;;;; ! M0 

commutes for a uniquely determined smooth mapping # : M ! M 0. For each x 2 M the mapping x := jPx : Px ! P0 (x) is G-equivariant and therefore a dieomorphism, so diagram (a) is a pullback diagram. But the most general notion of a homomorphism of principal bundles is the following. Let : G ! G0 be a homomorphism of Lie groups.  : (P
p
M
G) ! (P 0
p0
M 0
G0) is called a homomorphism over of principal bundles, if  : P ! P 0 is smooth and (u:g) = (u): (g) holds in general. Then  is ber respecting, so diagram (a) makes again sense, but it is no longer a pullback diagram in general. If  covers the identity on the base, it is called a reduction of the structure group G0 to G for the principal bundle (P 0
p0
M 0
G0) | the name comes from the case, when is the embedding of a subgroup. By the universal property of the pullback any general homomorphism  of principal ber bundles over a group homomorphism can be written

48

10. Principal Fiber Bundles and G-Bundles

as the composition of a reduction of structure groups and a pullback homomorphism as follows, where we also indicate the structure groups: (P
G) ;;;;! (# P 0
G0) ;;;;! (P 0
G0) (b)

p? y

?

?? y

M

M

?

p0 ? y

;;;; ! M0 

10.7. Associated bundles. Let (P
p
M
G) be a principal bundle and let ` : G
S ! S be a left action of the structure group G on a manifold S. We consider the right action R : (P
S)
G ! P
S, given by R((u
s)
g) = (u:g
g;1:s). Theorem. In this situation we have: (1) The space P
G S of orbits of the action R carries a unique smooth manifold structure such that the quotient map q : P
S ! P
G S is a submersion. (2) (P
G S
p#
M
S
G) is a G-bundle, where p# : P
G S ! M is given by q

P
S ;;;;! P
G S

(a)

?

?

pr1 ? y

p? y

P

;;;; ! M: p

In this diagram qu : fug
S ! (P
G S)p(u) is a dieomorphism for each u 2 P . (3) (P
S
q
P
G S
G) is a principal ber bundle with principal action R. (4) If (U
' : P jU ! U
G) is a principal bundle atlas with cocycle of transition functions (' : U ! G), then together with the left action ` : G
S ! S this cocycle is also one for the G-bundle (P
G S
p#
M
S
G). Notation: (P
G S
p#
M
S
G) is called the associated bundle for the action ` : G
S ! S. We will also denote it by P S
`] or simply P S] and we will write p for p# if no confusion is possible. We also dene the smooth mapping  : P
M PS
`] ! S by (ux
vx ) := qu;1 (vx ). It satises (u
q(u
s)) = s, q(ux
(ux
vx )) = vx , and (ux:g
vx ) = g;1 :(ux
vx). In the special situation, where S = G and the action is left translation, so that PG] = P , this mapping coincides with  considered in 10.2. x

10. Principal Fiber Bundles and G-Bundles

49

Proof: In the setting of the diagram in (2) the mapping p pr1 is constant on the R-orbits, so p# exists as a mapping. Let (U
' : P jU ! U
G) be a principal bundle atlas with transition functions (' : U ! G). We dene ;1 : U
S ! p#;1(U )  P
G S by ;1 (x
s) = q('; 1 (x
e)
s), which is ber respecting. For each orbit in p#;1(x)  P
G S there is exactly one s 2 S such that this orbit passes through ('; 1 (x
e)
s), since the principal right action is free. Thus ;1 (x: ) : S ! p#;1 (x) is bijective. Furthermore  ;1 (x
s) = q('; 1 (x
e)
s) = = q('; 1 (x
' (x):e)
s) = q('; 1 (x
e):' (x)
s) = = q('; 1 (x
e)
' (x):s) = ;1 (x
' (x):s)
so  ;1 (x
s) = (x
' (x):s) So (U
) is a G-atlas for P
G S and makes it into a smooth manifold and a G-bundle. The dening equation for  shows that q is smooth and a submersion and consequently the smooth structure on P
G S is uniquely dened, and p# is smooth by the universal properties of a submersion. By the denition of  the diagram ' Id

(b)

p;1(U )
S ;;;;! U
G
S ?? ? ? qy

Id`y

p#;1 (U )

U
S

commutes since its lines are dieomorphisms we conclude that qu : fug
S ! p#;1 (p(u)) is a dieomorphism. So (1), (2), and (4) are checked. (3) We rewrite the last diagram in the following form: (c)

p;1(U )
S



q;1(V ) ;;;;! V
G ?

q? y

p#;1 (U )

?

pr1 ? y

V

Here V := p#;1 (U )  P
G S and the dieomorphism  is dened by ; 1(;1 (x
s)
g) := (' (x
g)
g;1:s). Then we have ; 1(;1 (x
s)
g) = ; 1( ;1 (x

' (x):s)
g) = = ('; 1 (x
g)
g;1:'  (x):s) = ('; 1 (x
' (x):g)
g;1:' (x);1:s) =

50

10. Principal Fiber Bundles and G-Bundles

= (; 1 (;1 (x
s)
' (x):g)
so  ; 1 (;1 (x
s)
g) = (;1 (x
s)
' (x):g) and we conclude that (P
S
q
P
G S
G) is a principal bundle with structure group G and the same cocycle (' ) we started with.

10.8. Corollary. Let (E
p
M
S
G) be a G-bundle, speci ed by a

cocycle of transition functions (' ) with values in G and a left action ` of G on S. Then from the cocycle of transition functions we may glue a unique principal bundle (P
p
M
G) such that E = P S
`]. This is the usual way a dierential geometer thinks of an associated bundle. He is given a bundle E, a principal bundle P, and the G-bundle structure then is described with the help of the mappings  and q.

10.9. Equivariant mappings and associated bundles. 1. Let (P
p
M
G) be a principal ber bundle and consider two left actions of G, ` : G
S ! S and `0 : G
S 0 ! S 0 . Let furthermore f : S ! S 0 be a G-equivariant smooth mapping, so f(g:s) = g:f(s) or f `g = `0g f. Then IdP
f : P
S ! P
S 0 is equivariant for the actions R : (P
S)
G ! P
S and R0 : (P
S 0 )
G ! P
S 0 , so there is an induced mapping (a)

Idf

P
S ;;;;! P
S 0 ?

q? y

IdG f

?

q0 ? y

P
G S ;;;;! P
G S 0
which is ber respecting over M, and a homomorphism of G-bundles in the sense of the denition 10.10 below. 2. Let  : (P
p
M
G) ! (P 0
p0
M 0
G) be a principal ber bundle homomorphism as in 10.6. Furthermore we consider a smooth left action ` : G
S ! S. Then 
IdS : P
S ! P 0
S is G-equivariant and induces a mapping 
G IdS : P
G S ! P 0
G S, which is ber respecting over M, ber wise a dieomorphism, and again a homomorphism of Gbundles in the sense of denition 10.10 below. 3. Now we consider the situation of 1 and 2 at the same time. We have two associated bundles P S
`] and P 0S 0
`0 ]. Let  : (P
p
M
G) ! (P 0
p0
M 0
G) be a principal ber bundle homomorphism and let f : S ! S 0 be an G-equivariant mapping. Then 
f : P
S ! P 0
S 0 is

10. Principal Fiber Bundles and G-Bundles

51

clearly G-equivariant and therefore induces a mapping 
G f : PS
`] ! P 0S 0
`0] which again is a homomorphism of G-bundles. 4. Let S be a point. Then PS] = P
G S = M. Furthermore let y 2 S 0 be a xed point of the action `0 : G
S 0 ! S 0 , then the inclusion i : fyg ,! S 0 is G-equivariant, thus IdP
i induces IdP
G i : M = Pfyg] ! PS 0 ], which is a global section of the associated bundle PS 0]. If the action of G on S is trivial, so g:s = s for all s 2 S, then the associated bundle is trivial: PS] = M
S. For a trivial principal ber bundle any associated bundle is trivial. 10.10. Denition. In the situation of 10.8, a smooth ber respecting mapping  : PS
`] ! P 0S 0
`0] covering a smooth mapping # : M ! M 0 of the bases is called a homomorphism of G-bundles, if the following conditions are satised: P is isomorphic to the pullback # P 0, and the local representations of  in pullback-related ber bundle atlases belonging to the two G-bundles are ber wise G-equivariant. Let us describe this in more detail now. Let (U0
0 ) be a G-atlas for P 0S 0
`0] with cocycle of transition functions ('0 ), belonging to the principal ber bundle atlas (U0
'0 ) of (P 0
p0
M 0
G). Then the pullback-related principal ber bundle atlas (U = #;1 (U0 )
) for P = # P 0 as described in the proof of 9.5 has the cocycle of transition functions (' = '0 # ) it induces the G-atlas (U
) for PS
`]. Then (0  ;1 )(x
s) = (# (x)
(x
s)) and  (x
) : S ! S 0 should be G-equivariant for all  and all x 2 U . Lemma. Let  : P S
`] ! P 0S0
`0] be a homomorphism of associated bundles and G-bundles. Then there is a principal bundle homomorphism  : (P
p
M
G) ! (P 0
p0
M 0
G) and a G-equivariant mapping f : S ! S 0 such that  = 
G f : PS
`] ! P 0S 0
`0 ]. Proof: The homomorphism  : (P
p
M
G) ! (P 0
p0
M 0
G) of principal ber bundles is already determined by the requirement that P = # P 0, and we have # = #. The G-equivariant mapping f : S ! S 0 can be read o the following diagram  P
M P S] ;;;;! S ? ?   ? f? y y (a) P 0
M 0 P 0S 0 ] ;;;; ! S0
0 which by the assumptions is seen to be well dened in the right column. So a homomorphism of associated bundles is described by the whole triple ( : P ! P 0
f : S ! S 0 (G-equivariant)
 : E ! E 0 ), such that diagram (a) commutes. M

52

10. Principal Fiber Bundles and G-Bundles

10.11. Associated vector bundles. Let (P
p
M
G) be a principal ber bundle, and consider a representation : G ! GL(V ) of G on a

nite dimensional vector space V . Then PV
] is an associated ber bundle with structure group G, but also with structure group GL(V ), for in the canonically associated ber bundle atlas the transition functions have also values in GL(V ). So by section 6 PV
] is a vector bundle. If (E
p
M) is a vector bundle with n-dimensional bers we may consider the open subset GL(Rn
E)  L(M
Rn
E), consisting of all invertible linear mappings, which is a ber bundle over the base M. Composition from the right by elements of GL(n) gives a free right action on GL(Rn
E) whose orbits are exactly the bers, so by lemma 10.3 we have a principal ber bundle (GL(Rn
E)
p
M
GL(n)). The associated bundle GL(Rn
E)Rn] for the banal representation of GL(n) on Rn is isomorphic to the vector bundle (E
p
M) we started with, for the evaluation mapping ev : GL(Rn
E)
Rn ! E is invariant under the right action R of GL(n), and locally in the image there are smooth sections to it, so it factors to a ber linear dieomorphism GL(Rn
E)Rn] = GL(Rn
E)
GL(n) Rn ! E. The principal bundle GL(Rn
E) is called the linear frame bundle of E. Note that local sections of GL(Rn ) are exactly the local frame elds of the vector bundle E. To illustrate the notion of reduction of structure group, we consider now a vector bundle (E
p
M
Rn) equipped with a Riemannian metric g, that is a section g 2 C 1(S 2 E  ) such that gx is a positive definite inner product on Ex for each x 2 M. Any vector bundle admits Riemannian metrics: local existence is clear and we may glue with the help of a partition of unity on M, since the positive denite sections form an open convex subset. Now let s0 = (s01
. . .
s0n ) 2 C 1 (GL(Rn
E)jU) be a local frame eld of the bundle E over U  M. Now we may apply the Gram-Schmidt orthonormalization procedure to the basis (s1 (x)
. . .
sn (x)) of Ex for each x 2 U. Since this procedure is real analytic, we obtain a frame eld s = (s1
. . .
sn) of E over U which is orthonormal with respect to g. We call it an orthonormal frame
eld. Now let (U ) be an open cover of M with orthonormal frame elds s = (s1
. . .
sn), where s is dened on U . We consider the vecn tor bundle charts (U
 : E jU ! the orthonormal P U
Ri ) given ; 1 1 n  (x):v. For x 2 U

frame elds:  (x
v
. . .
v ) = s (x):v =: s i P we have si (x) = s j (x):g  ji (x) for C 1 -functions g : U ! R. Since s (x) and s (x) are both orthonormal bases of Ex , the matrix g (x) = (g ji (x)) is an element of O(n
R). We write s = s :g  for short. Then we have  ;1 (x
v) = s (x):v = s (x):g (x):v =

10. Principal Fiber Bundles and G-Bundles

53

;1 (x
g (x):v) and consequently  ;1 (x
v) = (x
g (x):v). So the (g : U ! O(n
R)) are the cocycle of transition functions for the vector bundle atlas (U
). So we have constructed an O(n
R)-structure on E. The corresponding principal ber bundle will be denoted by O(Rn
(E
g)) it is usually called the orthonormal frame bundle of E. It is derived from the linear frame bundle GL(Rn
E) by reduction of the structure group from GL(n) to O(n). The phenomenon discussed here plays a prominent role in the theory of classifying spaces. 10.12. Sections of associated bundles. Let (P
p
M
G) be a principal ber bundle and ` : G
S ! S a left action. Let C 1 (P
S)G denote the space of all smooth mappings f : P ! S which are G-equivariant in the sense that f(u:g) = g;1 :f(u) holds for g 2 G and u 2 P. Theorem. The sections of the associated bundle P S
`] correspond exactly to the G-equivariant mappings P ! S we have a bijection C 1 (P
S)G

= C 1 (PS]). Proof: If f 2 C 1(P
S)G we construct sf 2 C 1(P S]) in the following way: graph(f) = (Id
f) : P ! P
S is G-equivariant, since (Id
f)(u:g) = (u:g
f(u:g)) = (u:g
g;1:f(u)) = ((Id
f)(u)):g. So it induces a smooth section sf 2 C 1 (PS]) as seen from (Id
f )

P ;;;;! P
S

(a)

?

?

p? y

q? y

M ;;;;! P S] fs

If conversely s 2 C 1 (P S]) we dene fs 2 C 1(P
S)G by fs :=  (IdP
M f) : P = P
M M ! P
m PS] ! S. This is G-equivariant since fs (ux :g) = (ux :g
f(x)) = g;1 :(ux
f(x)) = g;1:fs (ux ) by 10.4. The two constructions are inverse to each other since we have fs(f ) (u) = (u
sf (p(u))) = (u
q(u
f(u))) = f(u) and ss(f ) (p(u)) = q(u
fs(u)) = q(u
(u
f(p(u))) = f(p(u)). 10.13. Theorem. Consider a principal ber bundle (P
p
M
G) and a closed subgroup K of G. Then the reductions of structure group from G to K correspond bijectively to the global sections of the associated bundle PG=K
#], where # : G
G=K ! G=K is the left action on the homogeneous space. Proof: By theorem 10.12 the section s 2 C 1 (PG=K]) corresponds to fs 2 C 1 (P
G=K)G, which is a surjective submersion since the action # : G
G=K ! G=K is transitive. Thus Ps := fs;1 (#e) is a submanifold

54

10. Principal Fiber Bundles and G-Bundles

of P which is stable under the right action of K on P. Furthermore the K-orbits are exactly the bers of the mapping p : Ps ! M, so by lemma10.3 we get a principal ber bundle (Ps
p
M
K). The embedding Ps ,! P is then a reduction of structure groups as required. If conversely we have a principal ber bundle (P 0
p0
M
K) and a reduction of structure groups  : P 0 ! P , then  is an embedding covering the identity of M and is K-equivariant, so we may view P 0 as a sub ber bundle of P which is stable under the right action of K. Now we consider the mapping  : P
M P ! G from 10.2 and restrict it to P
M P 0. Since we have (ux
vx :k) = (ux
vx):k for k 2 K this restriction induces f : P ! G=K by P
M P 0 ?? y



;;;;! G ?

p? y

P = P
M P 0=K ;;;;! G=K f

and from (ux:g
vx ) = g;1:(ux
vx) it follows that f is G-equivariant as required. Finally f ;1 (#e) = fu 2 P : (u
P 0 )  K g = P 0, so the two constructions are inverse to each other.

p(u)

10.14. The bundle of gauges. If (P
p
M
G) is a principal ber bundle we denote by Aut(P) the group of all G-equivariant dieomorphisms  : P ! P. Then p  = # p for a unique dieomorphism # of M, so there is a group homomorphism from Aut(P ) into the group Di(M) of all dieomorphisms of M. The kernel of this homomorphism is called Gau(P), the group of gauge transformations. So Gau(P ) is the space of all  : P ! P which satisfy p  = p and (u:g) = (u):g. Theorem. The group Gau(P) of gauge transformations is equal to the space C 1 (P
(G
conj))G

= C 1 (PG
conj]). Proof: We use again the mapping  : P
M P ! G from 10.2. For  2 Gau(P) we dene f 2 C 1 (P
(G
conj))G by f :=  (Id
). Then f (u:g) = (u:g
(u:g)) = g;1 :(u
(u)):g = conjg;1 f (u), so f is indeed G-equivariant. If conversely f 2 C 1 (P
(G
conj))G is given, we dene f : P ! P by f (u) := u:f(u). It is easy to check that f is indeed in Gau(P) and that the two constructions are inverse to each other.

10.15 The tangent bundles of homogeneous spaces. Let G be a Lie group and K a closed subgroup, with Lie algebras g and K, respectively. We recall the mapping AdG : G ! AutLie (g) from 1.1 and

10. Principal Fiber Bundles and G-Bundles

55

put AdG
K := AdG jK : K ! AutLie (g). For X 2 K and k 2 K we have AdG
K (k)X = AdG (k)X = AdK (k)X 2 K, so K is an invariant subspace for the representation AdG
K of K in g,and we have the factor representation Ad? : K ! GL(g=K. Then (a)

0 ! K ! g ! g=K ! 0

is short exact and K-equivariant. Now we consider the principal ber bundle (G
p
G=K
K) and the associated vector bundles Gg=K
Ad? ] and GK
AdG
K ]. Theorem. In these circumstances we have T(G=K) = Gg=K
Ad? ] = (G
K G =K
p
G=K
g=K): The left action g 7! T (#g ) of G on T(G=K) coincides with the left action of G on G
K g=K. Furthermore Gg=K
Ad? ]  GK
AdG
K ] is a trivial vector bundle. Proof: For p : G ! G=K we consider the tangent mapping Tep : g ! Te(G=K) which is linear and surjective and induces a linear isomorphism Te p : g=K ! Te(G=K). For k 2 K we have p conjk = p k k;1 = #k p and consequently Te p AdG
K (k) = Te p Te (conjk ) = Te#k Te p. Thus the isomorphism Te p : g=K ! Te(G=K) is K-equivariant for the representations Ad? and Te# : k 7! Te#k . Now we consider the associated vector bundle GTe(G=K)
Te#] = (G
K Te(G=K)
p
G=K
Te(G=K)), which is isomorphic to the vector bundle Gg=K
Ad? ], since the representation spaces are isomorphic. The mapping T # : G
Te(G=K) ! T(G=K) is K-invariant and therefore induces a mapping  as in the following diagram: T 

G
Te(G=K) ;;;;! T(G=K) ?

(b)

q? y



G
K Te(G=K) ;;;;! T(G=K) ? p? y

G=K



?

G=K ? y

G=K

This mapping  is an isomorphism of vector bundles. It remains to show the last assertion. The short exact sequence (a) induces a sequence of vector bundles over G=K: G=K
0 ! GK
AdK ] ! GG
AdG
K ] ! Gg=K
Ad? ] ! G=K
0 This sequence splits ber wise thus also locally over G=K, so we obtain

56

10. Principal Fiber Bundles and G-Bundles

Gg=K
Ad? ]  GK
AdG
K ]

= Gg
AdG
K ] and it remains to show that Gg
AdG
K ] is a trivial vector bundle. Let ' : G
g ! G
g be given by '(g
X) = (g
AdG (g)X). Then for k 2 K we have '((g
X):k) = '(gk
AdG
K (g;1 )X) = (ak
AdG (g:k:k;1)X) = (gk
AdG (g)X). So ' is K-equivariant from the "joint" K-action to the "on the left" K-action and therefore induces a mapping '# as in the diagram: '

G
g ;;;;! G
g ?

? ? y

q? y

(c)

G
K g ;;;; ! G=K
g ' ? ? p? y

pr1 ? y

G=K

G=K

The map '# is a vector bundle isomorphism.

10.16 Tangent bundles of Grassmann manifolds. From 10.5 we know that (V (k
n) = O(n)=O(n ; k)
p
G(k
n)
O(k)) is a principal ber bundle. Using the banal representation of O(k) we consider the associated vector bundle (Ek := V (k
n)Rk ]
p
G(k
n)). It is called the

universal vector bundle over G(k
n) for reasons we will discuss below in

chapter 11. Recall from 10.5 the description of V (k
n) as the space of all linear isometries Rk ! Rn we get from it the evaluation mapping ev : V (k
n)
Rk ! Rn. The mapping (p
ev) in the diagram (p
ev)

V (k
n)
Rk ;;;;! G(kn)
Rn (a)

?



q? y

V (k
n)
O(k) Rk ;;;;! G(kn)
Rn

is O(k)-invariant for the action R and factors therefore to an embedding of vector bundles  : Ek ! G(k
n)
Rn. So the ber (Ek )W over the k-plane W in Rn is just the linear subspace W. Note nally that the ber wise orthogonal complement Ek ? of Ek in the trivial vector bundle G(k
n)
Rn with its standard Riemannian metric is isomorphic to the universal vector bundle En;k over G(n ; k
n), where the isomorphism covers the dieomorphism G(k
n) ! G(n ; k
n) given also by the orthogonal complement mapping.

10. Principal Fiber Bundles and G-Bundles

57

Corollary. The tangent bundle of the Grassmann manifold is

T G(k
n) = L(Ek
Ek ? ). Proof: We have G(k
n) = O(n)=(O(k)
O(n ;k)), so by theorem 10.15 we get TG(k
n) = O(n)
so(n)=(so(k)
so(n ; k)): O(k)O(n;k)

On the other hand we have V (k
n) = O(n)=O(n ; k) and the right action of O(k) commutes with the right action of O(n ; k) on O(n), therefore V (k
n)Rk] = O(n)=O(n ; k)
Rk = O(n)
Rk
where

O(k)O(n;k)

O(k) O(n ; k) acts trivially on Rk . Finally



L(Ek
Ek? ) = L O(n) = O(n)




O(k)O(n;k)




O(k)O(n;k)

Rk
O(n)




O(k)O(n;k)

Rn;k



L(Rk
Rn;k)


where the left action of O(k)
O(n ; k) on L(Rk
Rn;k) is given by (A
B)(C) = B:C:A;1. Finally we have an O(k)
O(n ; k) - equivariant linear isomorphism L(Rk
Rn;k) ! so(n)=(so(k)
so(n ; k)), as follows: so(n)=(so(k);
so(n ; k)) =   skew   0 A  k
Rn;k)  = : A 2 L( R ;A 0 skew 0 0 skew

10.17. The tangent group of a Lie group. Let G be a Lie group

with Lie algebra g. We will use the notation from 1.1. First note that TG is also a Lie group with multiplication T and inversion T, given by the expressions T(a
b) :(a
b) = Ta ( b ):a + Tb (a ):b and Ta :a = ;Te (a;1 ):Ta ( a;1 ):a . Lemma. Via the isomomorphism T : g
G ! TG, T :(X
g) = Te ( g ):X, the group structure on TG looks as follows: (X
a):(Y
b) = (X + Ad(a)Y
a:b) and (X
a);1 = (;Ad(a;1)X
a;1 ). So T G is isomorphic to the semidirect product g G. Proof: T(a
b):(T a:X
T b :Y ) = T b :T a:X + T a :T b :Y = = T ab :X + T b :T a :T a;1 :Ta :Y = T ab (X + Ad(a)Y ). Ta :T a :X = ;T a;1 :Ta;1 :T a :X = ;T a;1 :Ad(a;1)X.

58

10. Principal Fiber Bundles and G-Bundles

Remark. In the left trivialization T  : G
g ! TG, T :(g
X) = Te (g ):X, the semidirect product structure looks somewhat awkward: (a
X):(b
Y ) = (ab
Ad(b;1)X + Y ) and (a
X);1 = (a;1
; Ad(a)X). 10.18. Tangent and vertical bundles. For a ber bundle (E
p
M
S) the subbundle V E = f  2 TE : T p: = 0 g of TE is called the vertical bundle and is denoted by (V E
E
E). Theorem. Let (P
p
M
G) be a principal ber bundle with principal right action r : P
G ! P . Let ` : G
S ! S be a left action. Then

the following assertions hold: (1) (TP
Tp
TM
TG) is again a principal ber bundle with principal right action Tr : TP
TG ! TP . (2) The vertical bundle (V P

P
g) of the principal bundle is trivial as a vector bundle over P : V P = P
g. (3) The vertical bundle of the principal bundle as bundle over M is again a principal bundle: (V P
p 
M
TG). (4) The tangent bundle of the associated bundle P S
`] is given by T (PS
`]) = T PTS
T`]. (5) The vertical bundle of the associated bundle P S
`] is given by V (PS
`]) = PTS
T2`] = P
G TS. Proof: Let (U
' : P jU ! U
G) be a principal ber bundle atlas with cocycle of transition functions (' : U ! G). Since T is a functor which respects products, (TU
T' : TP jTU ! T U
TG) is again a principal ber bundle atlas with cocycle of transition functions (T' : TU ! TG), describing the principal ber bundle (TP
Tp
TM
TG). The assertion about the principal action is obvious. So (1) follows. For completeness sake we include here the transition formula for this atlas in the right trivialization of T G: T (' '; 1)(x
Te( g ):X) = = (x
Te ( ' (x):g):(' (x ) + Ad(' (x))X))


where ' 2 $1(U  g) is the right logarithmic derivative of ' which is given by ' (x ) = T ( ' (x);1 ):T (' ):x . (2) The mapping (u
X) 7! Te (ru):X = T(u
e) r:(0u
X) is a vector bundle isomorphism P
g ! V P over P . (3) Obviously Tr : TP
TG ! TP is a free right action which acts transitive on the bers of T p : T P ! TM. Since V P = (T p);1 (0M ), the bundle V P ! M is isomorphic to T P j0M and Tr restricts to a free right action, which is transitive on the bers, so by lemma 10.3 the result follows. 

10. Principal Fiber Bundles and G-Bundles

59

(4) The transition functions of the ber bundle PS
`] are given by the expression ` ('
IdS ) : U
S ! G
S ! S. Then the transition functions of T PS
`] are T (` ('
IdS )) = T ` (T'
IdTS ) : TU
TS ! TG
TS ! TS, from which the result follows. (5) Vertical vectors in TP S
`] have local representations (0x
s) 2 TU
TS. Under the transition functions of T PS
`] they transform as T(` ('
IdS )):(0x
s) = T`:(0' (x)
s) = T(`' (x) ):s = T2 `:(x
s) and this implies the result 



60

11. Principal and Induced Connections 11.1. Principal connections. Let (P
p
M
G) be a principal ber bundle. Recall from 9.3 that a (general) connection on P is a ber projection : TP ! V P , viewed as a 1-form in $1 (P TP ). Such a connection is called a principal connection if it is G-equivariant for the principal right action r : P
G ! P, so that T (rg ): = :T (rg ) and is rg -related to itself, or (rg ) = in the sense of 8.16, for all g 2 G. By theorem 8.15.6 the curvature R = 21 :
] is then also rg -related to itself for all g 2 G. Recall from 10.18.2 that the vertical bundle of P is trivialized as vector bundle over P by the principal action. So !(Xu ) := Te (Ru);1 : (Xu) 2 g and in this way we get a g-valued 1-form ! 2 $1 (P  g), which is called the (Lie algebra valued) connection form of the connection . Recall from 1.3. the fundamental vector eld mapping : G ! X(P ) for the principal right action. Lemma. If 2 $1(P  V P) is a principal connection on the principal

ber bundle (P
p
M
G) then the connection form has the following two properties: (1) ! reproduces the generators of fundamental vector elds, so we have !( X (u)) = X for all X 2 g. (2) ! is G-equivariant, so we have ((rg ) !)(Xu ) = !(Tu (rg ):Xu ) = Ad(g;1):!(Xu ) for all g 2 G and Xu 2 Tu P. Consequently we have for the Lie derivative L ! = ;ad(X):!. Conversely a 1-form ! 2 $1(P
g) satisfying (1) de nes a connection on P by (Xu ) = Te (ru):!(Xu ), which is a principal connection if and only if (2) is satis ed. Proof: (1). Te (ru):!( X (u)) = ( X (u)) = X (u) = Te (ru ):X. Since Te (ru ) : G ! Vu P is an isomorphism, the result follows. (2). From Te (rug ):!(Tu (rg ):Xu ) = !(T (r ):X ) (ug) = (Tu (rg ):Xu ) and Te (rug ):Ad(g;1):!(Xu ) = Ad(g;1 ):!(X ) (ug) = Tu (rg ): !(X ) (u) = Tu (rg ): (Xu ) both directions follow. X

g

u

u

u

u

11.2. Curvature. Let be a principal connection on the principal ber bundle (P
p
M
G) with connection form ! 2 $1 (P g). We already noted in 11.1 that the curvature R = 12 
] is then also G-invariant, (rg ) R = R for all g 2 G. Since R has vertical values we may again dene a g-valued 2-form $ 2 $2 (P  g) by $(Xu
Yu) := ;Te (ru );1 :R(Xu
Yu), which is called the (Lie algebra-valued) curvature form of the connection. We take the negative sign here to get the usual connection form as in Kobayashi-Nomizu I, 1963].

11. Principal bundles and induced connections

61

We equip the space $(P g) of all g-valued forms on P in a canonical way with the structure of a graded Lie algebra by )
*](X1
. . .
Xp+q ) = X = p!1q! sign )(X 1
. . .
X p )
*(X (p+1)
. . .
X (p+q) )]g

or equivalently by   X
  Y ] :=  ^   X
Y ]. In particular for ! 2 $1 (P g) we have !
!](X
Y ) = 2!(X)
!(Y )]. Theorem. The curvature form $ of a principal connection with connection form ! has the following properties: (1) $ is horizontal, i.e. it kills vertical vectors. (2) $ is G-equivariant in the following sense: (rg ) $ = Ad(g;1 ):$. Consequently L $ = ;ad(X):$. (3) The Maurer-Cartan formula holds: $ = d! + 12 !
!]g. Proof: (1) is true for R by 9.4. For (2) we compute as follows: X

Te (rug ):((rg ) $)(Xu
Yu) = Te (rug ):$(Tu (rg ):Xu
Tu(rg ):Yu ) = = ;Rug (Tu (rg ):Xu
Tu(rg ):Yu ) = ;Tu (rg ):((rg ) R)(Xu
Yu) = = ;Tu (rg ):R(Xu
Yu) = Tu (rg ): (X
Y ) (u) = = Ad(g;1 ):(X
Y ) (ug) = = Te (rug ):Ad(g;1):$(Xu
Yu)
by 1.3. u

u

u

u

(3). For X 2 g we have i R = 0 by (1) and i (d! + 21 :!
!]g) = i d! + 12 i !
!] ; 21 !
i !] = = L ! + X
!] = ;ad(X)! + ad(X)! = 0 X

X

X

X

X

X

So the formula holds for vertical vectors, and for horizontal vector elds X
Y 2 C 1 (H(P )) we have R(X
Y ) = X ; X
Y ; Y ] = X
Y ] = !(X
Y ]) (d! + 12 !
!])(X
Y ) = X!(Y ) ; Y !(X) ; !(X
Y ]) = ;!(X
Y ])

11.3. Lemma. Any principal ber bundle (P
p
M
G) (with paracompact basis) admits principal connections. Proof: Let (U
' : P jU ! U
G) be a principal ber bundle atlas. Let us dene  (T'; 1 (x
Teg :X)) := X for x 2 Tx U and X 2 g.

62

11. Principal bundles and induced connections

An easy computation involving lemma 1.3 shows that  2 $1 (P jU g) satises the requirements of lemma 11.1 and thus is a principal connection on P jU. Now let (f ) be a smooth partition of unity P on M which is subordinated to the open cover (U ), and let ! :=  (f p) . Since both requirements of lemma 11.1 are invariant under convex linear combinations, ! is a principal connection on P . 11.4. Local descriptions of principal connections. We consider a principal ber bundle (P
p
M
G) with some principal ber bundle atlas (U
 : P jU ! U
G) and corresponding cocycle (' : U ! G) of transition functions. We consider the sections s 2 C 1 (P jU) which are given by ' (s (x)) = (x
e) and satisfy s :' = s . (1) Let * 2 $1(G
g) be the left logarithmic derivative of the identity, i.e. *(g ) := Tg (g;1 ):g . We will use the forms * := '  * 2 $1(U  g). Let = ! 2 $1 (P  V P) be a principal connection, ! 2 $1 (P  g). We may associate the following local data to the connection: (2) ! := s  ! 2 $1 (U  g), the physicists version of the connection. (3) The Christoel forms ; 2 $1 (U  X(G)) from 9.7, which are given by (0x
;(x
g)) = ;T(' ): :T (');1 (x
0g ). (4)  := ('; 1) ! 2 $1(U
G g), the local expressions of !. Lemma. These local data have the following properties and are related by the following formulas. (5) The forms ! 2 $1(U  g) satisfy the transition formulas ! = Ad('; 1 )! + * 
and any set of forms like that with this transition behavior determines a unique principal connection. (6)  (x
Tg :X) =  (x
0g ) + X = Ad(g;1 )! (x ) + X. (7) ;(x
g) = ;Te (g ):(x
0g ) = ;Te (g ):Ad(g;1)! (x ). Proof: From the denition of the Christoel forms we have ; (x
g) = ;T(' ): :T (');1(x
0g ) = ;T(' ):Te(r' (x
g) )!:T(' );1 (x
0g ) = ;Te (' r' (x
g) )!:T (' );1 (x
0g ) = ;Te (g )!:T (' );1 (x
0g ) = ;Te (g ) (x
0g ): This is the rst part of (7). The second part follows from (6).  (x
Tg :X) = !(T (' );1 (x
0g )) = =  (x
0g ) + !( X ('; 1(x
g))) =  (x
0g ) + X: 



11. Principal bundles and induced connections

63

So the rst part of (6) holds. The second part is seen from  (x
0g ) =  (x
Te ( g )0e) = (! T (');1 T( g ))(x
0e) = = (! T(rg '; 1 ))(x
0e) = Ad(g;1 )!(T('; 1 )(x
0e)) = Ad(g;1 )(s  !)(x ) = Ad(g;1 )! (x ): Via (7) the transition formulas for the ! are easily seen to be equivalent to the transition formulas for the Christoel forms in lemma 9.7. 11.5. The covariant derivative. Let (P
p
M
G) be a principal ber bundle with principal connection = !. We consider the horizontal projection  = IdTP ; : TP ! HP, cf. 9.3, which satises   = , im = HP, ker  = V P , and  T(rg ) = T(rg ) for all g 2 G. If W is a nite dimensional vector space, we consider the mapping  : $(P W ) ! $(P  W) which is given by ( ')u (X1
. . .
Xk ) = 'u ((X1 )
. . .
(Xk )): The mapping  is a projection onto the subspace of horizontal dierential forms, i.e. the space $hor (P W) := f 2 $(P W ) : iX  = 0 for X 2 V P g. The notion of horizontal form is independent of the choice of a connection. The projection  has the following properties:  ('^) =  '^ , if one of the two forms has real values   =    (rg ) = (rg )  for all g 2 G  ! = 0 and  L( X ) = L( X )  . They follow easily from the corresponding properties of , the last property uses that Flt (X ) = rexp tX . Now we dene the covariant exterior derivative d! : $k (P  W ) ! k $ +1(P  W) by the prescription d! :=  d. Theorem. The covariant exterior derivative d! has the following properties. (1) d! (' ^ ) = d! (') ^   + (;1)deg '  ' ^ d! (). (2) L( X ) d! = d! L( X ) for each X 2 g. (3) (rg ) d! = d! (rg ) for each g 2 G. (4) d! p = d p = p d : $(M W) ! $hor (P W). (5) d! ! = $, the curvature form. (6) d! $ = 0, the Bianchi identity. (7) d!  ; d! =  i(R) d, where R is the curvature. (8) d! d! =  i(R) d. Proof: (1) through (4) follow from the properties of .

64

11. Principal bundles and induced connections

(5). We have (d! !)(
) = ( d!)(
) = d!(
) = ()!() ; ()!(ch) ; !(
]) = ;!(
]) and R(
) = ; ($(
)) = 
]: (6). We have d! $ = d! (d! + 21 !
!]) =  dd! + 12  d!
!] = 21  (d!
!] ; !
d!]) =  d!
!] = d!
 !] = 0
since  ! = 0: (7). For ' 2 $(P W) we have (d!  ')(X0
. . .
Xk ) = (d ')((X0 )
. . .
(Xk )) X di )
. . .
(Xk ))) = (;1)i (Xi )(( ')((X0 )
. . .
(X 0 i k

+ =

X i<j

(;1)i+j ( ')((Xi )
(Xj )]
(X0 )
. . . di )
. . .
(X dj )
. . .) . . .
(X di )
. . .
(Xk ))) (;1)i (Xi )('((X0 )
. . .
(X

X 0 i k

+

X i<j

(;1)i+j '((Xi )
(Xj )] ; (Xi )
(Xj )]
(X0)
. . .

di )
. . .
(X dj )
. . .) . . .
(X = (d')((X0 )
. . .
(Xk )) + (iR ')((X0 )
. . .
(Xk )) = (d! +  iR)(')(X0
. . .
Xk ):

(8). d! d! =  dd = ( iR +  d)d by (7) =  iR :

11. Principal bundles and induced connections

65

11.6 Theorem. Let (P
p
M
G) be a principal ber bundle with prin-

cipal connection !. Then the parallel transport for the principal connection is globally de ned and g-equivariant. In detail: For each smooth c : ! M there is a smooth mapping Ptc :
Pc(0) ! P such that the following holds: (1) Pt(c
t
u) 2 Pc(t), Pt(c
0) = IdP (0) , and !( dtd Pt(c
t
u)) = 0. (2) Pt(c
t) : Pc(0) ! Pc(t) is G-equivariant, i.e. Pt(c
t
u:g) = Pt(c
t
u):g. (3) For smooth f : ! we have Pt(c
f(t)
u) = Pt(c f
t
Pt(c
f(0)
u)). Proof: The Christoel forms ; 2 $1 (U
X(G)) of the connection ! with respect to a principal ber bundle atlas (U
' ) take values in the Lie subalgebra XL (G) of all left invariant vector elds on G, which are bounded with respect to any left invariant Riemannian metric on G. Each left invariant metric on a Lie group is complete. So the connection is complete by the remark in 9.10. Properties (1) and (3) follow from theorem 9.8, and (2) is seen as follows: !( dtd Pt(c
t
u):g) = Ad(g;1 )!( dtd Pt(c
t
u)) = 0 implies that Pt(c
t
u):g = Pt(c
t
u:g). 11.7. Holonomy groups. Let (P
p
M
G) be a principal ber bundle with principal connection = !. We assume that M is connected and we x x0 2 M. In view of developments which we make later in section 12 we dene the holonomy group Hol(
x0)  Di(Px0 ) as the group of all Pt(c
1) : Px0 ! Px0 for c any piecewise smooth closed loop through x0 . (Reparametrizing c by a function which is at at each corner of c we may assume that any c is smooth.) If we consider only those curves c which are nullhomotopic, we obtain the restricted holonomy group Hol0 (
x0). Now let us x u0 2 Px0 . The elements (u0
Pt(c
t
u0)) 2 G form a subgroup of the structure group G which is isomorphic to Hol(
x0) we denote it by Hol(!
u0) and we call it also the holonomy group of the connection. Considering only nullhomotopic curves we get the restricted holonomy group Hol0 (!
u0) a normal subgroup Hol(!
u0). Theorem. 1. We have Hol(!
u0:g) = Ad(g;1 ) Hol(!
u0) and Hol0 (!
u0:g) = Ad(g;1) Hol0 (!
u0). 2. For each curve c in M we have Hol(!
Pt(c
t
u0)) = Hol(!
u0 ) and Hol0 (!
Pt(c
t
u0)) = Hol0(!
u0 ). 3. Hol0 (!
u0) is a connected Lie subgroup of G and the quotient group Hol(!
u0)= Hol0 (!
u0) is at most countable, so Hol(!
u0) is also a Lie subgroup of G. c

66

11. Principal bundles and induced connections

4. The Lie algebra hol(!
u0)  g of Hol(!
u0) is linearly generated by f$(Xu
Yu ) : Xu
Yu 2 Tu P g. It is isomorphic to the Lie algebra hol(
x0) we considered in 9.9. 5. For u0 2 Px0 let P(!
u0) be the set of all Pt(c
t
u0) for c any (piecewise) smooth curve in M with c(0) = x0 and for t 2 . Then P(!
u0) is a sub ber bundle of P which is invariant under the right action of Hol(!
u0) so it is itself a principal ber bundle over M with structure group Hol(!
u0) and we have a reduction of structure group, cf. 10.6 and 10.13. The pullback of ! to P (!
u0) is then again a principal connection form i ! 2 $1 (P (!
u0) hol(!
u0)). 6. P is foliated by the leaves P(!
u), u 2 Px0 . 7. If the curvature $ = 0 then Hol0 (!
u0) = feg and each P (!
u) is a covering of M. 8. If one uses piecewise C k -curves for 1  k < 1 in the de nition, one gets the same holonomy groups. In view of assertion 5 a principal connection ! is called irreducible if Hol(!
u0) equals the structure group G for some (equivalently any) u0 2 Px0 . Proof: 1. This follows from the properties of the mapping  from 10.2 and from the from the G-equivariancy of the parallel transport. The rest of this theorem is a compilation of well known results, and we refer to Kobayashi-Nomizu I, 1963, p. 83] for proofs.

11.8. Inducing principal connections on associated bundles. Let (P
p
M
G) be a principal bundle with principal right action r : P
G ! P and let ` : G
S ! S be a left action of the structure group G on some manifold S. Then we consider the associated bundle PS] = PS
`] = P
G S, constructed in 10.7. Recall from 10.18 that its tangent and vertical bundle are given by T (PS
`]) = T PT S
T`] = TP
TG TS and V (P S
`]) = PTS
T2`] = P
G TS. Let = ! 2 $1 (P  T P) be a principal connection on the principal bundle P . We construct the induced connection # 2 $1(PS]
T(PS])) by the following diagram:

Id

T P
TS ;;;;! TP
TS

T(P
S)

TP
TG T S ;;;; ! TP
TG TS 

T(P
G S):

? Tq=q0 ? y

? q0 ? y

?

Tq ? y

Let us rst check that the top mapping
Id is T G-equivariant. For g 2 G and X 2 g the inverse of Te (g )X in the Lie group TG is denoted

11. Principal bundles and induced connections

67

by (Te (g )X);1 , see lemma 10.17. Furthermore by 1.3 we have T r(u
Te(g )X) = Tu (rg )u + Tr((OP
LX )(u
g)) = Tu (rg )u + Tg (ru)(Te (g )X) = Tu (rg )u + X (ug): We may compute (
Id)(Tr(u
Te(g )X)
T`((Te (g )X);1
s)) = ( (Tu (rg )u + X (ug))
T`((Te (g )X);1
s)) = ( (Tu (rg )u ) + ( X (ug))
T`((Te (g )X);1
s)) = ( (Tu (rg )u ) + X (ug)
T`((Te (g )X);1
s)) = (Tr((
Id)u
Te (g )X)
T`((Te (g )X);1
s)) So the mapping
Id factors to # as indicated in the diagram, and we have # # = # from (
Id) (
Id) =
Id. The mapping # is berwise linear, since
Id and q0 = Tq are. The image of # is q0 (V P
T S) = q0 (ker(Tp : T P
TS ! T M)) = ker(T p : TP
TG TS ! TM) = V (P S
`]): Thus # is a connection on the associated bundle PS]. We call it the induced connection. From the diagram it also follows, that the vector valued forms
Id 2 $1 (P
S TP
TS) and # 2 $1 (P S] T (PS])) are (q : P
S ! P S])related. So by 8.15 we have for the curvatures RId = 21 
Id

Id] = 21 
]
0 = R
0
R = 12  #
# ]


that they are also q-related, i.e. Tq (R
0) = R (Tq
M Tq). By uniqueness of the solutions of the dening dierential equation we also get that Pt (c
t
q(u
s)) = q(Pt (c
t
u)
s): 11.9. Recognizing induced connections. We consider again a principal ber bundle (P
p
M
G) and a left action ` : G
S ! S. Suppose that ) 2 $1 (P S] T (P S])) is a connection on the associated bundle PS] = P S
`]. Then the following question arises: When is the connection ) induced from a principal connection on P ? If this is the case, we say that ) is compatible with the G-structure on P S]. The answer is given in the following

68

11. Principal bundles and induced connections

Theorem. Let ) be a (general) connection on the associated bundle PS]. Let us suppose that the action ` is in nitesimally eective, i.e. the fundamental vector eld mapping : g ! X(S) is injective.

Then the connection ) is induced from a principal connection ! on P if and only if the following condition is satis ed: In some (equivalently any) ber bundle atlas (U
) of P S] belonging to the G-structure of the associated bundle the Christoffel forms ; 2 $1(U  X(S)) have values in the sub Lie algebra Xfund(S) of fundamental vector elds for the action `. Proof: Let (U
' : P jU ! U
G) be a principal ber bundle atlas for P . Then by the proof of theorem 10.7 the induced ber bundle atlas (U
 : P S]jU ! U
S) is given by (1) ;1 (x
s) = q('; 1 (x
e)
s)
(2) ( q)('; 1 (x
g)
s) = (x
g:s): Let = ! be a principal connection on P and let # be the induced connection on the associated bundle P S]. By 9.7 its Christoel symbols are given by ; (x
s) = ;(T( ) # T(;1 ))(x
0s) = ;(T( ) # Tq (T('; 1 )
Id))(x
0e
0s) by (1) = ;(T( ) Tq (
Id))(T ('; 1 )(x
0e)
0s) by 11.8 = ;(T( ) Tq)( (T ('; 1 )(x
0e))
0s) = (T( ) T q)(T('; 1 )(; (x
e))
0s) by 11.4,(3) ; 1 = ;T ( q ('
Id))(0x
!(x )
0s) by 11.4,(7) s = ;Te (` )! (x ) by (2) = ; ! ( ) (s): So the condition is necessary. Now let us conversely suppose that a connection ) on P S] is given such that the Christoel forms ; with respect to a ber bundle atlas of the G-structure have values in Xfund(S). Then unique g-valued forms ! 2 $1 (U  g) are given by the equation ; (x ) = (! (x ))
since the action is innitesimally eective. From the transition formulas 9.7 for the ; follow the transition formulas 11.4.(5) for the ! , so that they give a unique principal connection on P , which by the rst part of the proof induces the given connection ) on PS]. 

x

11. Principal bundles and induced connections

69

11.10. Inducing principal connections on associated vector bundles. Let (P
p
M
G) be a principal ber bundle and let : G !

GL(W ) be a representation of the structure group G on a nite dimensional vector space W. We consider the associated vector bundle (E := PW
]
p
M
W), which was treated in some detail in 10.11. The tangent bundle T(E) = T P
TG TW has two vector bundle structures with the projections E : T(E) = T P
TG T W ! P
G W = E
Tp pr1 : T (E) = TP
TG TW ! TM the rst one is the vector bundle structure of the tangent bundle, the second one is the derivative of the vector bundle structure on E. Now let = ! 2 $1 (P TP ) be a principal connection on P . We consider the induced connection # 2 $1 (E T(E)) from 11.8. Inserting the projections of both vector bundle structures on T(E) into the diagram in 11.8 one easily sees that the induced connection is linear in both vector bundle structures: the new aspect is that it is a linear endomorphism of the vector bundle (TE
T p
TM). We say that it is a linear connection on the associated bundle. Recall now from 11.8 the vertical lift vlE : E
M E ! V E, which is an isomorphism, pr1{E {berwise linear and also p{T p{berwise linear. Now we dene the connector K of the linear connection # by K := pr2 (vlE );1 # : TE ! V E ! E
M E ! E:

Lemma. The connector K : TE ! E is E {p{ berwise linear and Tp{p{ berwise linear and satis es K vlE = pr2 : E
M E ! T E ! E. Proof: This follows from the berwise linearity of the composants of K and from its denition.

11.11. Linear connections. If (E
p
M) is a vector bundle, a connection ) 2 $1 (E TE) such that ) : TE ! V E ! TE is also Tp{Tp{

berwise linear is called a linear connection. An easy check with 11.9 or a direct construction shows that ) is then induced from a unique principal connection on the linear frame bundle GL( n
E) of E (where n is the ber dimension of E). Equivalently a linear connection may be specied by a connector K : TE ! E with the three properties of lemma 11.10. For then HE := fu : K(u ) = 0p(u)g is a complement to V E in TE which is Tp{berwise linearly chosen.

70

11. Principal bundles and induced connections

11.12. Covariant derivative on vector bundles. Let (E
p
M) be a vector bundle with a linear connection, given by a connector K : T E ! E with the properties in lemma 11.10. For any manifold N, smooth mapping s : N ! E, and vector eld X 2 X(N) we dene the covariant derivative of s along X by (1) rX s := K T s X : N ! T N ! TE ! E: If f : N ! M is a xed smooth mapping, let us denote by Cf1 (N
E) the vector space of all smooth mappings s : N ! E with p s = f {

they are called sections of E along f. From the universal property of the pullback it follows that the vector space Cf1 (N
E) is canonically linearly isomorphic to the space C 1 (f  E) of sections of the pullback bundle. Then the covariant derivative may be viewed as a bilinear mapping (2)

r : X(N)
Cf1 (N
E) ! Cf1 (N
E):

Lemma. This covariant derivative has the following properties: (3) rX s is C 1 (N
)-linear in X 2 X(N). (4) rX s is -linear in s 2 Cf1 (N
E). So for a tangent vector Xx 2 Tx N the mapping rX : Cf1(N
E) ! Ef (x) makes sense and (rX s)(x) = rX (x) s. (5) rX (h:s) = dh(X):s + h:rX s for h 2 C 1 (N
), the derivation property of rX . (6) For any manifold Q and smooth mapping g : Q ! N and Yy 2 Ty Q we have rTg:Y s = rY (s g). If Y 2 X(Q) and X 2 X(N) are g-related, then we have rY (s g) = (rX s) g. Proof: All these properties follow easily from the denition (1). Remark: Property (6) is not well understood in some dierential geometric literature. See e.g. the clumsy and unclear treatment of it in Eells-Lemaire, 1983]. For vector elds X, Y 2 X(M) and a section s 2 C 1 (E) an easy x

y

y

computation shows that

RE (X
Y )s : = rX rY s ; rY rX s ; rX
Y ] s = (rX
rY ] ; rX
Y ] )s is C 1 (M
)-linear in X, Y , and s. By the method of 7.4 it follows that RE is a 2-form on M with values in the vector bundle L(E
E), i.e. RE 2 $2(M L(E
E)). It is called the curvature of the covariant derivative.

11. Principal bundles and induced connections

71

For f : N ! M, vector elds X, Y 2 X(N) and a section s 2 Cf1 (N
E) along f we obtain

rX rY s ; rY rX s ; rX
Y ] s = (f  RE )(X
Y )s:

11.13. Covariant exterior derivative. Let (E
p
M) be a vector bundle with a linear connection, given by a connector K : TE ! E. For a smooth mapping f : N ! M let $(N f  E) be the vector space

of all forms on N with values in the vector bundle f  E. We can also view them as forms on N with values along f in E, but we do not introduce an extra notation for this. The graded space $(N f  E) is a graded $(N)-module via (' ^ )(X1
. . .
Xp+q ) = X = p!1q! sign() '(X 1
. . .
X p ) (X (p+1)
. . .
X (p+q) ):

It can easily be shown that the graded module homomorphisms H : $(N f  E) ! $(N f  E) (so that H(' ^ ) = (;1)deg H: deg ' ' ^ H( )) are exactly the mappings (K) for K 2 $q (N f  L(E
E)), which are given by ((K) )(X1
. . .
Xp+q ) = X = p!1q! sign() K(X 1
. . .
X p )( (X (p+1)
. . .
X (p+q) )):

The covariant exterior derivative dr : $p (N : f  E) ! $p+1 (N f  E) is dened by (where the Xi are vector elds on N) (dr )(X0
. . .
Xp ) =

p X

ci
. . .
Xp ) (;1)i rX (X0
. . .
X

i=0 X + (;1)i+j 0 i<j p

i

ci
. . .
X cj
. . .
Xp ): (Xi
Xj ]
X0
. . .
X

Lemma. The covariant exterior derivative is well de ned and has the

following properties. (1) For s 2 C 1 (f  E) = $0(N f  E) we have (dr s)(X) = rX s. (2) dr(' ^ ) = d' ^ + (;1)deg '' ^ dr . (3) For smooth g : Q ! N and 2 $(N f  E) we have dr(g ) = g (dr ). (4) drdr = (f  RE ) .

72

11. Principal bundles and induced connections

Proof: It suces to investigate decomposable forms = '  s for ' 2 $p (N) and s 2 C 1 (f  E). Then from the denition we have dr('  s) = d'  s + (;1)p ' ^ drs. Since by 11.12,(3) dr s 2 $1 (N f  E), the mapping dr is well dened. This formula also implies (2) immediately. (3) follows from 11.12,(6). (4) is checked as follows: dr dr('  s) = dr (d'  s + (;1)p ' ^ dr s) by (2) = 0 + (;1)2p ' ^ drdr s = ' ^ (f  RE )s by the denition of RE = (f  RE )(' ^ s): 11.14. Let (P
p
M
G) be a principal ber bundle and let : G !

GL(W) be a representation of the structure group G on a nite dimensional vector space W. Theorem. There is a canonical isomorphism from the space of P W
]valued dierential forms on M onto the space of horizontal G-equivariant W-valued dierential forms on P : q] : $(M PW
]) ! $hor (P  W)G = f' 2 $(P W) : iX ' = 0 for all X 2 V P
(rg ) ' = (g;1 ) ' for all g 2 Gg: In particular for W = with trivial representation we see that p : $(M) ! $hor (P)G = f' 2 $hor (P) : (rg ) ' = 'g is also an isomorphism. The isomorphism q] : $0(M PW ]) = C 1(P W]) ! $0hor (P  W )G = C 1 (P
W)G is a special case of the one from 10.12. Proof: Recall the smooth mapping  : P
M P ! G from 10.2, which satises r(ux
(ux
vx)) = vx , (ux :g
u0x:g0 ) = g;1 :(ux
u0x):g0 , and (ux
ux) = e. Let ' 2 $khor (P W)G, X1
. . .
Xk 2 TuP , and X10
. . .
Xk0 2 Tu0 P such that Tu p:Xi = Tu0 p:Xi0 for each i. Then we have for g = (u
u0), so that ug = u0 : q(u'u (X1
. . .
Xk )) = q(ug
(g;1 )'u (X1
. . .
Xk )) = q(u0
((rg ) ')u (X1
. . .
Xk )) = q(u0
'ug (Tu (rg ):X1
. . .
Tu(rg ):Xk )) = q(u0
'u0 (X10
. . .
Xk0 ))
since Tu (rg )Xi ; Xi0 2 Vu0 P:

11. Principal bundles and induced connections

73

By this prescription a vector bundle valued form 2 $k (M P W]) is uniquely determined. For the converse recall the smooth mapping # : P
M PW
] ! W from 10.7, which satises #(u
q(u
w)) = w, q(ux
#(ux
vx)) = vx , and #(ux g
vx) = (g;1 )# (ux
vx ). For 2 $k (M PW]) we dene q] 2 $k (P  W) as follows. For Xi 2 Tu P we put (q] )(X1
. . .
Xk ) := #(u
p(u)(Tu p:X1
. . .
Tup:Xk )): Then q] is smooth and horizontal. For g 2 G we have ((rg ) (q] ))u (X1
. . .
Xk ) = (q] )ug (Tu (rg ):X1
. . .
Tu(rg ):Xk ) = #(ug
p(ug) (Tug p:Tu (rg ):X1
. . .
Tug p:Tu(rg ):Xk )) = (g;1 )# (u
p(u)(Tu p:X1
. . .
Tu p:Xk )) = (g;1 )(q] )u (X1
. . .
Xk ): Clearly the two constructions are inverse to each other. 11.15. Let (P
p
M
G) be a principal ber bundle with a principal connection = !, and let : G ! GL(W) be a representation of the structure group G on a nite dimensional vector space W. We consider the associated vector bundle (E := PW
]
p
M
W), the induced connection # on it and the corresponding covariant derivative. Theorem. The covariant exterior derivative d! from 11.5 on P and the covariant exterior derivative for P W]-valued forms on M are connected by the mapping q] from 11.14, as follows: q] dr = d! q] : $(M PW ]) ! $hor (P  W)G:

Proof: Let rst f 2 $0hor (P W)G = C 1(P
W)G , then f = q] s for s 2 C 1 (PW]) and we have f(u) = (u
s(p(u))) and s(p(u)) = q(u
f(u)) by 11.14 and 10.12. Therefore Ts:T p:Xu = Tq(Xu
Tf:Xu), where Tf:Xu = (f(u)
df(Xu )) 2 TW = W
W . If  : T P ! HP is the horizontal projection as in 11.5, we have Ts:T p:Xu = Ts:T p::Xu = Tq(:Xu
Tf::Xu). So we get (q] drs)(Xu ) = (u
(drs)(Tp:Xu )) = (u
rTp:X s) = (u
K:Ts:Tp:Xu) = (u
K:Tq(:Xu
T f::Xu)) u

by 11.13,(1) by 11.12.(1) from above

74

11. Principal bundles and induced connections

= (u
pr2:vlP;1W ] : # :Tq(:Xu
Tf::Xu)) by 11.10 ; 1 = (u
pr2:vlP W ] :T q:(
Id)(:Xu
Tf::Xu))) by 11.8 = (u
pr2:vlP;1W ] :T q(0u
Tf::Xu))) since : = 0 = (u
q:pr2:vlP;1 W :Tq(0u
Tf::Xu))) since q is ber linear  = (u
q(u
df::Xu)) = ( df)(Xu ) = (d! q] s)(Xu ): Now we turn to the general case. It suces to check the formula for a decomposable P W]-valued form ) =   s 2 $k (M
PW ]), where  2 $k (M) and s 2 C 1 (P W]). Then we have d! q] (  s) = d! (p  q] s) = d! (p ) q] s + (;1)k  p  ^ d! q] s by 11.5,(1) =  p d q] s + (;1)k p  ^ q] drs from above  ] k  ] = p d q s + (;1) p  ^ q drs = q] (d  s + (;1)k  ^ dr s) = q] dr(  s):

11.16. Corollary. In the situation of theorem 11.15 above we have for the curvature form $ 2 $2hor (P  g) and the curvature RP W ] 2 $2 (M L(PW]
P W])) the relation

qL] (P W ]
P W ]) RP W ] = 0 $
where 0 = Te : g ! L(W
W) is the derivative of the representation .

Proof: We use the notation of the proof of theorem 11.15. By this theorem we have for X, Y 2 Tu P (d! d! qP] W ] s)u (X
Y ) = (q] dr drs)u(X
Y ) = (q] RP W ] s)u (X
Y ) = (u
RP W ] (Tu p:X
Tup:Y )s(p(u))) = (qL] (P W ]
P W ]) RP W ] )u (X
Y )(qP] W ] s)(u):

11. Principal bundles and induced connections

On the other hand we have by theorem 11.5.(8) (d! d! q] s)u (X
Y ) = ( iR dq] s)u (X
Y ) = (dq] s)u (R(X
Y )) since R is horizontal = (dq] s)(; (X
Y ) (u)) by 11.2  = @t@ 0 (q] s)(Fl;(t ) (u))  = @t@ 0 (u: exp(;t$(X
Y ))
s(p(u: exp(;t$(X
Y )))))  = @t@ 0 (u: exp(;t$(X
Y ))
s(p(u)))  = @t@ 0 (exp t$(X
Y ))(u
s(p(u))) by 11.7 = 0 ($(X
Y ))(q] s)(u): XY

75

76

12. Holonomy 12.1. Holonomy groups. Let (E,p,M,S) be a ber bundle with a complete connection , and let us assume that M is connected. We choose a xed base point x0 2 M and we identify Ex0 with the standard ber S. For each closed piecewise smooth curve c : 0
1] ! M through

x0 the parallel transport Pt(c

1) =: Pt(c
1) (pieced together over the smooth parts of c) is a dieomorphism of S. All these dieomorphisms form together the group Hol(
x0), the holonomy group of at x0, a subgroup of the dieomorphism group Di(S). If we consider only those piecewise smooth curves which are homotopic to zero, we get a subgroup Hol0 (
x0), called the restricted holonomy group of the connection at x0. 12.2. Holonomy Lie algebra. In the setting of 12.1 let C : TM
M E ! T E be the horizontal lifting as in 9.3, and let R be the curvature (9.4) of the connection . For any x 2 M and Xx 2 Tx M the horizontal lift C(Xx ) := C(Xx
) : Ex ! TE is a vector eld along Ex. For Xx and Yx 2 Tx M we consider R(CXx
CYx) 2 X(Ex). Now we choose any piecewise smooth curve c from x0 to x and consider the dieomorphism Pt(c
t) : S = Ex0 ! Ex and the pullback Pt(c
1) R(CXx
CYx) 2 X(S). Let us denote by hol(
x0) the closed linear subspace, generated by all these vector elds (for all x 2 M, Xx , Yx 2 Tx M and curves c from x0 to x) in X(S) with respect to the compact C 1-topology, and let us call it the holonomy Lie algebra of at x0 . 12.3. Lemma. hol(
x0) is a Lie subalgebra of X(S). Proof: For X 2 X(M) we consider the local ow FlCX t of the horizontal lift of X. It restricts to parallel transport along any of the ow lines of X in M. Then for vector elds on M the expression CX  CY  CX  CZ  d dt j0(Fls ) (Flt ) (Fl;s ) (Flz ) R(CU
CV ) CX  CZ   = (FlCX s ) CY
(Fl;s ) (Flz ) R(CU
CV )] CZ   = (FlCX s ) CY
(Flz ) R(CU
CV )] Ex0

Ex0 Ex0

is in hol(
x0), since it is closed in the compact C 1-topology and the derivative can be written as a limit. Thus CZ   (FlCX s ) CY1
CY2]
(Flz ) R(CU
CV )] Ex0 2 hol(
x0)

by the Jacobi identity and CZ   (FlCX s ) CY1
Y2]
(Flz ) R(CU
CV )] Ex0 2 hol(
x0)


12. Holonomy

77

so also their dierence CZ   (FlCX s ) R(CY1
CY2)
(Flz ) R(CU
CV )] Ex0 is in hol(
x0). 12.4. The following theorem is a generalization of the theorem of Ambrose and Singer on principal connections. Theorem. Let be a connection on the ber bundle (E
p
M
S) and let M be connected. Suppose that for some (hence any) x0 2 M the holonomy Lie algebra hol(
x0) is nite dimensional and consists of complete vector elds on the ber Ex0 Then there is a principal bundle (P
p
M
G) with nite dimensional structure group G, an irreducible connection ! on it and a smooth action of G on S such that the Lie algebra g of G equals the holonomy Lie algebra hol(
x0), the ber bundle E is isomorphic to the associated bundle P S], and is the connection induced by !. The structure group G equals the holonomy group Hol(
x0). P and ! are unique up to isomorphism. By a theorem of Palais, 1957] a nite dimensional Lie subalgebra of X(Ex0 ) like Hol(
x0) consists of complete vector elds if and only if it is generated by complete vector elds as a Lie algebra. Proof: Let us again identify Ex0 and S. Then g := hol(
x0) is a nite dimensional Lie subalgebra of X(S), and since each vector eld in it is complete, there is a nite dimensional connected Lie group G0 of dieomorphisms of S with Lie algebra g, see Palais, 1957]. Claim 1: G0 equals Hol0(
x0), the restricted holonomy group. Let f 2 Hol0 (
x0), then f = Pt(c
1) for a piecewise smooth closed curve c through x0, which is nullhomotopic. Since the parallel transport is essentially invariant under reparametrization, 9.8, we can replace c by c g, where g is smooth and at at each corner of c. So we may assume that c itself is smooth. Since c is homotopic to zero, by approximation we may assume that there is a smooth homotopy H : 2 ! M with H1j0
1] = c and H0 j0
1] = x0. Then ft := Pt(Ht
1) is a curve in Hol0 (
x0) which is smooth as a mapping
S ! S. Claim 2: ( dtd ft) ft;1 =: Zt is in g for all t. To prove claim 2 we consider the pullback bundle H  E ! 2 with the induced connection H . It is sucient to prove claim 2 there. Let X = dsd and Y = dtd be the constant vector elds on 2, so X
Y ] = 0. Then Pt(c
s) = FlCX s jS and so on. We put CY CX CY ft
s = FlCX ;s Fl;t Fls Flt : S ! S


78

12. Holonomy

so ft
1 = ft . Then we have in the vector space X(S) ;1 = ;(FlCX ) CY + (FlCX ) (FlCY ) (FlCX ) CY
( dtd ft
s ) ft
s s s t ;s

Z1

d ;( d ft
s) f ;1  ds t
s ds dt 0 Z 1 CX  CY  CX   = ;(FlCX s ) CX
CY ] + (Fls ) CX
(Flt ) (Fl;s ) CY ]

;1 = ( dtd ft
s ) ft
s 0



 CY  CX  ;(FlCX s ) (Flt ) (Fl;s ) CX
CY ] ds:

Since X
Y ] = 0 we have CX
CY ] = CX
CY ] = R(CX
CY ) and 



X   (FlCX t ) CY = C (Flt ) Y +

= CY +

Z1

d dt 0



 (FlCX t ) CY

 (FlCX t ) CY dt = CY +

= CY +

Z1

Z1 0

0



 (FlCX t ) CX
CY ] dt

 (FlCX t ) R(CX
CY ) dt:

;1 is in Thus all parts of the integrand above are in g and so ( dtd ft
s) ft
s g for all t and claim 2 follows. Now claim 1 can be shown as follows. There is a unique smooth curve g(t) in G0 satisfying Te ( g(t) )Zt = Zt :g(t) = dtd g(t) and g(0) = e via the action of G0 on S the curve g(t) is a curve of dieomorphisms on S, generated by the time dependent vector eld Zt , so g(t) = ft and f = f1 is in G0. So we get Hol0(
x0)  G0. Claim 3: Hol0(
x0) equals G0. In the proof of claim 1 we have seen that Hol0 (
x0) is a smoothly arcwise connected subgroup of G0 , so it is a connected Lie subgroup by the results cited in 5.6. It suces thus to show that the Lie algebra g of G0 is contained in the Lie algebra of Hol0(
x0), and for that it is enough to show, that for each  2 g there is a smooth mapping f : ;1
1]
S ! S such that the associated curve f lies in Hol0 (
x0) with f0 (0) = 0 and f00 (0) = . By denition we may assume  = Pt(c
1)R(CXx
CYx) for Xx , Yx 2 Tx M and a smooth curve c in M from x0 to x. We extend Xx and Yx to vector elds X and Y 2 X(M) with X
Y ] = 0 near x. We may also suppose that Z 2 X(M) is a vector eld which extends c0 (t) along c(t): make c simple piecewise smooth by deleting any loop, reparametrize it in such a way that it is again smooth (see the beginning of this proof)

12. Holonomy

79

and approximate it by an embedding. The vector elds  thus obtained still generate the nite dimensional vector space g. So we have CZ    = (FlCZ 1 ) R(CX
CY ) = (Fl1 ) CX
CY ] since X
Y ](x) = 0 CY CX CY CX  1 d2 = (FlCZ 1 ) 2 dt2 jt=0 (Fl;t Fl;t Flt Flt ) 2 CY CX CY CX CZ = 12 dtd 2 jt=0(FlCZ ;1 Fl;t Fl;t Flt Flt ) Fl1 )


where we used the well known formula expressing the Lie bracket of two vector elds as the second derivative of the (group theoretic) commutator of the ows. The parallel transport in the last equation rst follows c from x0 to x, then follows a small closed parallelogram near x in M (since X
Y ] = 0 near x) and then follows c back to x0. This curve is clearly nullhomotopic, so claim 3 follows. Step 4: Now we make Hol(
x0) into a Lie group which we call G, by taking Hol0(
x0) = G0 as its connected component of the identity. Then Hol(
x0)= Hol0 (
x0) is a countable group, since the fundamental group 1(M) is countable (by Morse Theory M is homotopy equivalent to a countable CW-complex). Step 5: Construction of a cocycle of transition functions with values in G. Let (U
u : U ! m ) be a locally nite smooth atlas for M such that each u : U ! m ) is surjective. Put x := u; 1 (0) and choose smooth curves c : 0
1] ! M with c (0) = x0 and c (1) = x. For each x 2 U let cx : 0
1] ! M be the smooth curve t 7! u; 1(t:u(x)), then cx connects x and x and the mapping (x
t) 7! cx(t) is smooth U
0
1] ! M. Now we dene a ber bundle atlas (U
 : E jU ! U
S) by ;1 (x
s) = Pt(cx
1) Pt(c
1) s. Then  is smooth since Pt(cx
1) = FlCX for a local vector eld Xx depending smoothly on x. 1 Let us investigate the transition functions. x

;



 ;1 (x
s) = x
Pt(c
1);1 Pt(cx
1);1 Pt(cx
1) Pt(c
1) s ;  = x
Pt(c :C x:(cx);1 :(c);1
4) s =: (x
 (x) s)
where   : U  ! G:

Clearly   : U 
S ! S is smooth which implies that   : U  ! G is also smooth. ( ) is a cocycle of transition functions and we use it to glue a principal bundle with structure group G over M which we call (P
p
M
G). From its construction it is clear that the associated bundle PS] = P
G S equals (E
p
M
S). Step 6: Lifting the connection to P . For this we have to compute the Christoel symbols of with respect

80

12. Holonomy

to the atlas of step 5. To do this directly is quite dicult since we have to dierentiate the parallel transport with respect to the curve. Fortunately there is another way. Let c : 0
1] ! U be a smooth curve. Then we have (Pt(c
t);1 (c(0)
s)) =   = c(t)
Pt(c; 1
1) Pt((cc(t) );1
1) Pt(c
t) Pt(cc(t)
1) Pt(c
1)s = (c(t)
(t):s)
where (t) is a smooth curve in the holonomy group G. Now let ; 2 $1 (U
X(S)) be the Christoel symbol of the connection with respect to the chart (U
). From the third proof of theorem 9.8 we have  (Pt(c
t);1 (c(0)
s)) = (c(t)
#(t
s)
where # (t
s) is the integral curve through s of the time dependent vector eld ; ( dtd c(t)) on S. But then we get ; ( dtd c(t))(s) = dtd # (t
s) = dtd ((t):s) = ( dtd (t)):s
where dtd (t) 2 g. So ; takes values in the Lie sub algebra of fundamental vector elds for the action of G on S. By theorem 11.9 the connection is thus induced by a principal connection ! on P . Since by 11.8 the principal connection ! has the "same" holonomy group as and since this is also the structure group of P, the principal connection ! is irreducible, see 11.7.

81

13. The nonlinear frame bundle of a ber bundle

13.1. Let now (E
p
M
S) be a ber bundle and let us x a ber bundle atlas (U ) with transition functions  : U
S ! S. By 6.8 we have C 1 (U
C 1 (S
S))  C 1(U
S
S) with equality if and only if S is compact. Let us therefore assume from now on that S is compact. Then we assume that the transition functions  : U ! Di(S
S). Now we dene the nonlinear frameSbundle of (E
p
M
S) as follows. We consider the set Di fS
E g := x2M Di(S
Ex) and give it the innite dimensional dierentiable structure which one gets by applying the functor Di(S
) to the cocycle of transition functions ( ). Then the resulting cocycle of transition functions for Di fS
E g gives it the structure of a smooth principal bundle over M with structure group Di(M). The principal action is just composition from the right. We can consider now the smooth action ev : Di(S)
S ! S and fS
E gS . The mapping the associated bundle Di fS
E gS
ev] = DiDi( S) ev : Di fS
E g
S ! E is invariant under the Di(S)-action and factors therefore to a smooth mapping Di fS
E gS
ev] ! E as in the following diagram: pr fS
E gS Di fS
E g
S ;;;;! DiDi( S) ?

ev? y



E ;;;; Di fS
E gS
ev]: The bottom mapping is easily seen to be a dieomorphism. Thus the bundle Di fS
E g is in full right the (nonlinear) frame bundle of E. 13.2. Lemma. In the setting of 13.1 the in nite dimensional smooth manifold Di fS
E g is a splitting smooth submanifold of Emb(S
E), with the obvious embedding. 13.3. Let now 2 $1(E TE) be a connection on E. We want to lift to a principal connection on Di fS
E g, and for this we need a good description of the tangent space T Di fS
E g. With the methods of Michor, 1980] one can show that T Di fS
E g =



x2M

ff 2 C 1 (S
TE jEx) : Tp f = one point in Tx M and E f 2 Di(S
Ex)g:

Starting from the connection we can then consider !(f) := T(E f);1 f : S ! TE ! V E ! TS for f 2 T Di fS
E g. Then !(f) is a vector eld on S and we have

82

13. The nonlinear frame bundle of a ber bundle

Lemma. ! 2 $1(Di fS
Eg X(S)) is a principal connection and the induced connection on E = Di fS
E gS
ev] coincides with . Proof: This follows directly from 11.9. But we also give a direct proof. The fundamental vector eld X on Di fS
E g for X 2 X(S) is given by X (g) = Tg X. Then !( X (g)) = T g;1 Tg X = X since Tg X has vertical values. So ! reproduces fundamental vector elds. Now let h 2 Di(S) and denote by rh the principal right action. Then we have

((rh ) !)(f) = !(T (rh )f) = !(f h) = T (E f h);1 f. h = Th;1 !(f) h = AdDi(S ) (h;1)!(f):

13.4 Theorem. Let (E
p
M
S) be a ber bundle with compact standard ber S. Then connections on E and principal connections on Di fS
E g correspond to each other bijectively, and their curvatures are related as in 11.8. Each principal connection on Di fS
E g admits a global parallel transport. The holonomy groups and the restricted holonomy groups are equal as subgroups of Di(S). Proof: This follows directly from 11.8 and 11.9. Each connection on E is complete since S is compact, and the lift to Di fS
E g of its parallel transport is the global parallel transport of the lift of the connection, so the two last assertions follow. 13.5. Remark1 on the holonomy Lie algebra. Let M be connected, let = ;d! ; 2 !
!]X(S ) be the usual X(S)-valued curvature of the lifted connection ! on Di fS
E g. Then we consider the -linear span of all elements (f
f ) in X(S), where f , f 2 Tf Di fS
E g are arbitrary (horizontal) tangent vectors, and we call this span hol(!). Then by the Di(S)-equivariance of the vector space hol(!) is an ideal in the Lie algebra X(S). 13.6. Lemma. Let f : S ! Ex0 be a dieomorphism in Di fS
Egx0 . Then f : X(S) ! X(Ex0 ) induces an isomorphism between hol(!) and the -linear span of all g R(CX
CY ), X, Y 2 Tx M, and g : Ex) ! Ex any dieomorphism. The proof is obvious. Note that the closure of f (hol(!)) is (a priori) larger than hol(
x0) of 12.2. Which one is the right holonomy Lie algebra?

83

14. Gauge theory for ber bundles We x the setting of section 13. In particular the standard ber S is supposed to be compact. 14.1. We consider the bundle Di fE
E g := Sx2M Di(Ex
Ex) which bears the smooth structure described by the cocycle of transition func;1
 ) = ( ) (  ) , where ( ) is a cocycle of trantions Di(

sition functions for the bundle (E
p
M
S). 14.2. Lemma. The associated bundle Di fS
EgDi(S)
conj] is isomorphic to the ber bundle Di fE
E g. Proof: The mapping A : Di fS
Eg
Di(S) ! Di fE
E g, given by A(f
g) := f g f ;1 : Ex ! S ! S ! Ex for f 2 Di(S
Ex ), is Di(S)invariant, so it factors to a smooth mapping Di fS
E gDi(S)] ! Di fE
E g. It is bijective ans admits locally over M smooth inverses, so it is a ber respecting dieomorphism. 14.3. The gauge group Gau(E) of the bundle (E
p
M
S) is by denition the group of all principal bundle automorphisms of the Di(S)bundle (Di fS
E g which cover the identity of M. The usual reasoning gives that Gau(E) equals the space of all smooth sections of the associated bundle Di fS
E gDi(S)
conj] which by 14.2 equals the space of sections of the bundle Di fE
E g ! M. We equip it with the topology and dierentiable structure as a space of smooth sections, see 6.1. Since only the image space is innite dimensional, but admits exponential mappings (induced from the nite dimensional ones), this makes no diculties. Since the ber S is compact we see from a local (on M) application of the exponential law 6.8 that C 1 (Di fE
E g ! M) ,! Di(E) is an embedding of a splitting closed submanifold. 14.4. Theorem. The gauge group Gau(E) = C 1(Di fE
E g) is a splitting closed subgroup of Di(E), if S is compact. It has an exponential mapping which is not surjective on any neighborhood of the identity. Its Lie algebra consists of all vertical vector elds with compact support on E, with the negative of the usual Lie bracket. Proof: The rst statement has already been shown before the theorem. A curve through the identity of principal bundle automorphisms of Di fS
E g ! M is a smooth curve through the identity in Di(E) consisting of ber respecting mappings. the derivative of such a curve is thus an arbitrary vertical vector eld with compact support. The space of all these is therefore the Lie algebra of the gauge group, with the negative of the usual Lie bracket.

84

14. Gauge theory for dieomorphism groups

The exponential mapping is given by the ow operator of such vector elds. Since on each ber it is just isomorphic the exponential mapping of Di(S), it has all the properties of the latter. 14.5. Remark. If S is not compact we may circumvent the nonlinear frame bundle, and we may dene the gauge group Gau(E) directly as the splitting closed subgroup of Di(E) which consists of all ber respecting dieomorphisms which cover the identity of M. The Lei algebra of Gau(E) consists then of all vertical vector elds on E with compact support on E. 14.6 The space of connections. Let J 1(E) ! E be the ane bundle of 1-jets of sections of E ! M. We have J 1(E) = f` 2 L(Tx M
Tu E) : Tp ` = IdT M
u 2 E
p(u) = xg. Then a section of J 1 (E) ! E is just a horizontal lift mapping TM
M E ! TE which is ber linear over E, so it describes a connection as treated in 9.2 and we may view the space of sections C 1 (J 1 (E) ! E) as the space of all connections. 14.7. Theorem. The action of the gauge group Gau(E) on the space of connections C 1 (J 1 (E)) is smooth. Proof: This follows from 6.6 14.8. We will now give a dierent description of the action. We view now a connection again as a linear ber wise projection TE ! V E, So the space of connections is now Conn(E) := f 2 $1 (E TE) : =
(TE) = V E g. Since S is compact the canonical isomorphism Conn(E) ! C 1 (J 1 (E)) is even a dieomorphism. Then the action of f 2 Gau(E)  Di(E) on 2 Conn(E) is given by f = (f ;1 ) = Tf T f ;1 . Now it is very easy to describe the innitesimal action. Let X be a vertical vector eld with compact support on E and consider its global ow FlXt . Then we have dtd j0(FlXt ) = LX = X
], the Frolicher Nijenhuis bracket, by 8.16(5). The tangent space of Conn(E) at is the space T Conn(E) = f) 2 $1 (E TE) : )jV E = 0g. The "innitesimal orbit" at in T Conn(E) is fX
] : X 2 Cc1 (V E)g. The isotropy subgroup of a connection is ff 2 Gau(E) : f  = g. Clearly this just the group of all those f which respect the horizontal bundle HE = ker . The most interesting object is of course the orbit space Conn(E)= Gau(E). If the base manifold M is compact then one can show that the orbit space is stratied into smooth manifolds, each one corresponding to a conjugacy class of holonomy groups in Di(S). Those strata whose holonomy group groups are up to conjugacy contained in a xed compact group (like SU(2)) acting on S are dieomorphic to the strata of the usual nite dimensional gauge theory. The x

14. Gauge theory for dieomorphism groups

85

proof of this assertions is quite complicated and will be dealt with in another paper, see Gil-Medrano Michor, 1989] for the starting idea.

86

15. A classifying space for the di eomorphism group 15.1. Let `2 be the Hilbert space of square summable sequences and

let S be a compact manifold. By a slight generalization of theorem 7.3 (we use a Hilbert space instead of a Riemannian manifold N) the space Emb(S
`2 ) of all smooth embeddings is an open submanifold of C 1 (S
`2 ) and it is also the total space of a smooth principal bundle with structure group Di(S) acting from the right by composition. The base space B(S
`2 ) := Emb(S
`2 )= Di(S) is a smooth manifold modeled on Frechet spaces which are projective limits of Hilbert spaces. B(S
`2 ) is a Lindelof space in the quotient topology and the model spaces admit bump functions, thus B(S
`2 ) admits smooth partitions of unity. We may view B(S
`2 ) as the space of all submanifolds of `2 which are dieomorphic to S, a nonlinear analogy of the innite dimensional Grassmanian. 15.2. Lemma. The total space Emb(S
`2 ) is contractible. So by the general theory of classifying spaces the base space B(S
`2 ) is a classifying space of Di(S). I will give a detailed description of the classifying process in 15.4 below. Proof: We consider the continuous homotopy A : `2
0
1] ! `2 through isometries which is given by A0 = Id and by At(a0
a1
a2
. . .) = (a0
. . .
an;2
an;1 cos n (t)
an;1 sin n (t)
an cos n (t)
an sin n (t)
an+1 cos n (t)
an+1 sin n (t)
. . .) 1  t  1 , where  (t) = '(n((n + 1)t ; 1))  for a xed smooth for n+1 n n 2 function ' : ! which is 0 on (;1
0], grows monotonically to 1 in 0
1], and equals 1 on 1
1)]. Then A1=2 (a0
a1
a2
. . .) = (a0
0
a1
0
a2
0
. . .) is in `2even and on the other hand A1 (a0
a1
a2
. . .) = (0
a0
0
a1
0
a2
0
. . .) is in `2odd . The same homotopy makes sense as a mapping A : 1
! 1 , and here it is easily seen to be smooth: a smooth curve in 1 is locally bounded and thus takes locally values in a nite dimensional subspace N  1 . The image under A then has values in 2N  1 and the expression is clearly smooth as a mapping into 2N . This is a variant of a homotopy constructed by Ramadas, 1982]. Given two embeddings e1 and e2 2 Emb(S
`2 ) we rst deform e1 through embeddings to e01 2 Emb(S
`2even ), and e2 to e02 2 Emb(S
`2odd ) then we connect them by te01 + (1 ; t)e02 which is a smooth embedding for all t since the values are always orthogonal.

15. Classifying spaces for dieomorphism groups

87

15.3. We consider the smooth action ev : Di(S)
S ! S and the associated bundle Emb(S
`2 )S
ev] = Emb(S
`2 )
Di(S ) S which we

call E(S
`2 ), smooth ber bundle over B(S
`2 ) with standard ber S. In view of the interpretation of B(S
`2 ) as the nonlinear Grassmannian we may visualize E(S
`2 ) as the "universal S-bundle" as follows: E(S
`2 ) = f(N
x) 2 B(S
`2 )
`2 : x 2 N g with the dierentiable structure from the embedding into B(S
`2 )
`2 . The tangent bundle TE(S
`2 ) is then the space of all (N
x

v) where N 2 B(S
`2 ), x 2 N,  is a vector eld along and normal to N in `2 , and v 2 Tx `2 such that the part of v normal to Tx N equals (x). This follows from the description of the principal ber bundle Emb(S
`2 ) ! B(S
`2 ) given in 7.3 combined with 6.7. Obviously the vertical bundle V E(S
`2 ) consists of all (N
x
v) with x 2 N and v 2 Tx N. The orthonormal projection p(N
x) : `2 ! Tx N denes a connection class : TE(S
`2 ) ! V E(S
`2 ) which is given by class (N
x

v) = (N
x
p(N
x)v). It will be called the classifying connection for reasons to be explained in the next theorem. 15.4. Theorem. Classifying space for Di(S). The ber bundle (E(S
`2 )
pr
B(S
`2)
S) is classifying for S-bundles and class is a classifying connection: For each nite dimensional bundle (E
p
M
S) and each connection on E there is a smooth (classifying) mapping f : M ! B(S
`2 ) such that (E
) is isomorphic to (f  E(S
`2 )
f  class ). Homotopic maps pull back isomorphic S-bundles and conversely (the homotopy can be chosen smooth). The pulled back connection is invariant under a homotopy H if and only if i(C class T(x
t)H:(0x
dtd ))Rclass = 0 where Rclass is the curvature of class . Since S is compact the classifying connection class is complete and its parallel transport Ptclass has the following classifying property:  class f~ Ptf  (c
t) = Ptclass (f c
t) f
~

where f~ : E

= F E(S
`2 ) ! E(S
`2 ) is the berwise dieomorphic which covers the classifying mapping f : M ! B(S
`2 ). Proof: We choose a Riemannian metric g1 on the vector bundle V E ! E and a Riemannian metric g2 on the manifold M. We can combine these two into the Riemannian metric g := (T pj ker ) g2  g1 on the manifold E, for which the horizontal and vertical spaces are orthogonal. By the theorem of Nash there is an isometric imbedding h : E ! N for N large enough. We then embed N into the Hilbert space `2 and

88

15. Classifying spaces for dieomorphism groups

consider f : M ! B(S
`2 ), given by f(x) = h(Ex). Then f~=(f
h)

E ;;;;;! E(S
`2 ) ?

? ? ypr

p? y

M ;;;; ! B(S
`2 ) f is berwise a dieomorphism, so this diagram is a pullback and we have f  E(S
`2 ) = E. Since T(f
h) maps horizontal and vertical vectors to orthogonal ones, (f
h) class = . If Pt denotes the parallel transport of the connection and c : 0
1] ! M is a (piecewise) smooth curve we have for u 2 Ec(0))  class @  f(Pt(c
~ t
u)) = class :T f:~ @t@ 0 Pt(c
t
u) @t 0  = T f:~ : @t@ 0 Pt(c
t
u) = 0
so ~ ~ f(Pt(c
t
u)) = Ptclass (f c
t
f(u)): Now let H be a continuous homotopy M
I ! B(S
`2 ). Then we may approximate H by smooth mappings with the same H0 and H1 , if they are smooth, see Br
ocker-J
anich, 1973], where the infnite dimensionality of B(S
`2 ) does not disturb. Then we consider the bundle H  E(S
`2 ) ! M
I, equipped with the connection H  class , whose curvature is H  Rclass . Let @t be the vector eld tangential to all fxg
I on M
I. Parallel transport along the lines t 7! (x
t) with respect H  class is given by the ow of the horizontal lift (H  C class )(@t ) of @t . Let us compute its action on the connection H  class whose curvature is H  Rclass by 9.5.(3). By lemma 9.9 we have @ @t

 (H  C class )(@ )  t Flt H  class = ; 12 i(H  C class )(@t ) (H  Rclass )  ; = ; 1 H  i(C class T(x
t) H:(0x
dtd ))Rclass

2

which implies the result. 15.5. Theorem. Ebin-Marsden, 1970] Let S be a compact orientable manifold, let 0 be a positive volume form on S with total mass 1. Then Di(S) splits smoothly into Di(S) = Di 0 (S)
Vol(S) where Di 0 (S) is the Lie group of all 0 -preserving dieomorphisms and Vol(S) is the space of all volume forms of total mass 1. Proof: We show rst that there exists a smooth mapping  : Vol(S) ! Di(S) such that () 0 = .

15. Classifying spaces for dieomorphism groups

89

We put t = 0 +t( ; 0 ). We want a smooth curve t 7! ft 2 Di(S) with ft t = 0 . We have @t@ ft = Xt ft for a time dependent vector @  eld Xt on S. Then 0 = @t@ ft t = ft LX t +f R t @t t = fRt (LX t +( = 0 )), so LX t = 0 ;  and consequently S LX t = S (0 ; ) = 0. So the cohomology class of LX t in H dim S (S) is zero, and we have LX t = diX t + iX 0 = d! for some ! 2 $dim S ;1(S). Now we choose ! such that d! = 0 ;  and we choose it smoothly depending on  by the theorem of Hodge. Then the time dependent vector eld Xt is uniquely determined by iX t = ! since t is nowhere 0. Let ft be the evolution operator of Xt and put () = f1;1 . Now we may prove the theorem proper. We dene a mapping ) : Di(S) ! Di 0 (S)
Vol(S) by )(f) := (f (f  0 );1
f  0), which is smooth by sections 6 and 7. An easy computation shows that the inverse is given by the smooth mapping );1 (g
) = g (). 15.6. A consequence of theorem 15.5 is that the classifying spaces of Di(S) and Di 0 (S) are homotopy equivalent. So their classifying spaces are also homotopy equivalent. We now sketch a smooth classifying space for Di 0 . Consider the space B1 (S
`2 ) of all submanifolds of `2 of type S and total volume 1 in the volume form induced from the inner product on `2 . It is a closed splitting submanifold of codimension 1 of B(S
`2 ) by the NashMoser inverse function theorem, see Hamilton, 1982]. This theorem is applicable if we use `2 as image space, because the modeling spaces are then tame Frechet spaces in his sense. It is not applicable directly for 1 as image space. 15.7. Theorem. Classifying space for Di !!(S). Let S be a compact real analytic manifold. Then the space Emb (S
`2 ) of real analytic embeddings of S into the Hilbert space `2 is the total space of a real analytic principal ber bundle with structure group Di ! (S) and real analytic base manifold B ! (S
`2 ), which is a classifying space for the Lie group Di ! (S). It carries a universal Di ! (S)-connection. In other words: t

t

t

t

t

t

t

t

t

(Emb! (S
N)
Di (S ) S ! B ! (S
`2 ) !

classi es ber bundles with typical ber S and carries a universal (generalized) connection. The proof is similar to that of 15.4 with the appropriate changes to C!.

90

16. A characteristic class 16.1. Bundles with ber volumes. Let (E
p
M
S) be a ber bundle and let  be a smooth section of the line bundle %nV  E ! E where n = dimS, such that for each x 2 M the n-form x := jEx is a positive

volume form of total mass 1. Such a form will be called a
ber volume on E. Such forms exist on any oriented bundle with compact oriented ber type. We may plug n vertical vector elds Xi 2 C 1 (V E) into  to get a function (X1
. . .
Xn ) on E, but we cannot treat  as a dierential form on E. 16.2. Lemma. Let 0 be a xed volume form on S and let  be a ber volume on the bundle E. Then there is a bundle atlas (U
 : E jU ! U
S) such that ( jEx) 0 = x for all x 2 U . Proof: We start with any bundle atlas (U
0 : EjU ! U
S). By theorem 15.5 there is a dieomorphism '
x 2 Di(S) such that ('
x ) ((0 );1 jfxg
S) x = 0, and '
x depends smoothly on x. Then  := (IdU
';
x1 ) 0 : E jU ! U
S is the desired bundle chart. 16.3. Let now be a connection on (E
p
M
S) and let C : T M
M E ! TE be its horizontal lifting. Then for X 2 X(M) the ow FlCX t of its horizontal lift CX respects bers, so for the ber volume  the  pullback (FlCX t )  makes sense and is again a ber volume on E. We can now dene a kind of covariant derivative by 

 rX  := dtd j0 (FlCX t ) :

(a)

In fact rS is a covariant derivative on the vector bundle $n(p) := n ver (E) := x2M $ (Ex) with its obvious smooth vector bundle structure, whose standard ber is the nuclear Frechet space $n (S). We shall need a description of the convariant derivative r in terms of Lie derivatives. So let j : V E ! TE be the embedding of the vertical bundle and let us consider j  : $k (E) ! C 1 (%k V  E). Likewise we consider  : C 1 (%k V  E) ! $k (E). These are algebraic mappings, algebra homomorphisms for the wedge product, and satisfy j   = IdC 1 ( V  E ) . The other composition  j  : $k (E) ! $k (E) is a projection onto the image of  . If X 2 X(M) is a vector eld on M, then the horizontal lift CX 2 X(E) and X are p-related, i.e. Tp CX = X p. Thus the ows are X CX p-conjugated: p FlCX t = Flt p. So Flt respects the vertical bundle $n

k

16. Characteristic classes for general ber bundles

91

V E and we have j FltCX = FlCX t j. Therefore we have (b)  CX     @ rX  = @t@ 0 (FlCX t )  = @t 0 (Flt ) j         = j  @t@ 0 (FlCX t )  = j LCX  = 0 + j iCX d 
where LCX is the usual Lie derivative. This formula also shows that rX  is C 1 (M
)-linear in X, as asserted. Now let (U
) be a ber bundle atlas for (E
p
M
S). Put  := (;1 ) , a connection on U
S ! U . Its horizontal lift is given by C (X) = (X
;(X)). Let  be a ber k-form on U
S, so  : U ! $k (S) is just a smooth mapping. Thus we have for x 2 U 

(rX ) = @t@ 0 ((Fl; (X ) ) ( FlXt )) = dU (X) + LS; (X )  = X() + LS; (X ) 

(c)











where dU is the exterior derivative of $k (S)-valued forms on U , and where LS is the Lie derivative on S. 16.4. Lemma. Let be a connection on the bundle (E
p
M
S) with curvature R 2 $2(E V E). Then the curvature of the linear covariant derivative r on the vector bundle ($k (p)
M
$k (S)) is given by R(r)(X
Y ) = LvR(CX
CY ) , where Lv denotes the vertical Lie derivative, on each ber separately. Proof: The easiest proof is local. Let (U
) be a ber bundle atlas for (E
p
M
S). Put again  := (;1 ) , a connection on U
S ! U with horizontal lift is given by C  (X) = (X
;(X)). Let  be a ber k-form on U
S, so  : U ! $k (S) is just a smooth mapping. Thus we have for X, Y 2 X(U ), where we write r for r , R(r)(X
Y ) = (rX rY ; rY rX ; rX
Y ] ) = rX (Y () + LS;(Y ) ) ; rY (X() + LS;(Y ) ) ; X
Y ]() ; LS;X
Y ]  = X(Y ()) + LSX ;(Y )  + LS;(Y ) X() + LS;(X ) Y () + LS;(X ) LS;(Y )  ; Y (X()) ; LSY ;(X )  ; LS;(X )Y () ; LS;(Y ) X() ; LS;(Y ) LS;(X )  ; X
Y ]() ; LS;X
Y ]  = LS (X;(Y ) ; Y ;(X) ; ;(X
Y ]))  + LS;(X )
LS;(Y ) ] ;  = LS dU ;(X
Y ) + ;(X)
;(Y )]  = LS (((;1 )  R)(X
Y )): 





92

16. Characteristic classes for general ber bundles

In this computation we used that LS : X(S)
$k (S) ! $k (S) is bilinear and continuous so that the product rule of dierential calculus applies. The last formula comes from lemma 9.7. 16.5. Denition. The connection on E is called -respecting connection if rX  = 0 for all X 2 X(M).

Lemma.

(1) A connection is -respecting if and only if for some (any) bundle atlas (U
) as in lemma 16.2 the corresponding Christoel forms ; take values in the Lie algebra X0 (S) of divergence free vector elds on S (i.e. LX 0 = 0). (2) There exist many -respecting connections. (3) If is -respecting and if R is its curvature, then for Xx and Yx 2 Tx M the form i(R(CXx
CYx)) is closed in !n;1(Ex ). Proof: 1. Put  := (;1 ) , a connection on U
S ! U . Its horizontal lift is given by C (X) = (X
; (X)). Thus we have rX 0 = L; (X ) 0, and this is zero for all X 2 X(U ) if and only if ; 2 $1 (U  X0 (S)). 2. By the rst part -respecting connections exist locally and since rX is C 1 (M
)-linear in X 2 X(M) we may glue them via a partition of unity on M. 3. We compute locally in a chart (U
) belonging to a ber bundle atlas satisfying lemma 16.2. Then by 1. we have ; 2 $1(U  X0 (S)), so by lemma 9.7 the local expression of the curvature R = (;1 ) R = dU ; + ;
;] is an element of $2 (U  X0 (S)). But then 





dS iSR (X
Y ) 0 = LSR (X
Y ) 0 = 0: 



16.6. We consider the cohomology class i(R(CX
CY ))] 2 H n;1(Ex ) for X, Y 2 Tx M and the nite dimensional vector bundle H n;1(p) := S n;1 x2M H (Ex) ! M. It is described by a cocycle of transition functions U ! GL(H n;1(S)), x 7! H n;1(  (x
)), which are locally constant by the homotopy invariance of cohomology. So the vector bundle H n;1(p) ! M admits a unique at linear connection r respecting the resulting discrete structure group, and the induced covariant exterior derivative dr : $( M H n;1(p)) ! $k+1 (M H n;1(p)) satises d2r = 0 and denes the De Rham cohomology of M with twisted coecient domain H n;1(p). We may view iR ] : Xx
Yx 7! i(R(CXx
CYx))] as an element of $2 (M H n;1(p)).

16. Characteristic classes for general ber bundles

93

16.7. Lemma. (1) If  2 C 1 (%k V  E ! E) induces a section ] : M ! H k (p) then we have rX ] = rX ], for any connection on E.

(2) If is a -respecting connection on E then dr iR ] = 0. Proof: (1). For any bundle atlas (U
) the parallel sections of the vector bundle H k (p) for the unique at connection respecting the discrete structure group are exactly those which in each local bundle chart U
H k (S) are given by the (locally) constant mappings: U ! H k (S). Obviously rX ] := rX ] denes a connection and in the local bundle chart U ! H k (S) we have rX   = dtd j0(Fl(tX
; (X )) )   = dU   (X) + LS; (X )  
where dU is the exterior derivative on U of $k -valued mappings. But if ] is parallel for the at connection respecting the discrete structure group, so   ] is locally constant, then dU   takes values in the space B k (S) of exact forms. And since   (x) is closed for all x 2 U the second summand LS; (X )   = dS i; (X )  + i; (X ) 0 takes values in the space of exact forms anyhow. So the two connections have the same local parallel sections, so they coincide. (2). Let X0 , X1 , X2 2 X(M). Then we have dr iR ](X0
X1
X2) = X = (rX0 (iR ](X1
X2 )) ; iR ](X0
X1 ]
X2)) 













= = =

cyclic

X

cyclic







rX0 (iR(CX1
CX2 ) ) ; iR(C X0
X1 ]
CX2 ) 

X   j LCX0 iR(CX1
CX2 )   ; iR(C X0
X1 ]
CX2 )  cyclic X  

j iCX0
R(CX1
CX2 )] +  + j  iR(CX1
CX2 ) LCX0   ; iR(C X0
X1 ]
CX2 ) 

X  ;   = j i CX0
R(CX1
CX2)] + 0 ; R(CX0
X1]
CX2)  
cyclic

where we used iCX
LZ ] = iCX
Z ] , that j   = Id, and that R has vertical values, which implies j  iR(CX1
CX2 ) LCX0   = iR(CX1
CX2 ) j  LCX0   = iR(CX1
CX2 ) rX0  = 0:

94

16. Characteristic classes for general ber bundles

Now we need the Bianchi identity R
] = 0 from 9.4. Writing out the global formula 8.9 for the horizontal vector elds CXi we get 0 = R
](CX0
CX1
CX2 ) X = (; CX0
R(CX1
CX2 )] ; R(CX0
X1 ]
CX2)): cyclic

From this it follows that driR ] = 0. 16.8. Denition. So we may dene a characteristic class k(E
) of the bundle (E
p
M
S) with ber volume  as the class k(E
) := iR ]H ;1 (p) ]H 2 (M H ;1 (p)) : 16.9. Lemma. The class k(E
) is independent of the choice of the -respecting connection. Proof: Let and 0 be two  preserving connections on (E
p
M), and let C and C 0 be their horizontal liftings. We put D = C ; C 0 : TM
M E ! V E, which is ber linear over E, then Ct = C + tD = (1 ; t)C + tC 0 is a curve of horizontal lifts which preserve . In fact we have LD(X ) jEx = 0 for each Xx 2 Tx M. Let t be the connection corresponding to Ct and let Rt be its curvature. Then we have Rt(CtX
CtY ) = t CtX
Ct Y ] = CtX
CtY ] ; CtX
Y ] = R(CX
CY ) + tCX
DY ] + tDX
CY ] + t2DX
DY ] ; tDX
Y ]: @  Rt(CtX
CtY ) = CX
DY ] + DX
CY ] + 2tDX
DY ] ; DX
Y ]: @t 0    ; @  @ @t 0 i(R;t(Ct X
CtY )) = j @t 0 i(Rt(Ct X
CtY ))   = j  iCX
DY ] ; iCY
DX ] + 2tiDX
DY ] ; iDX
Y ]   ;  = j  LCX
iDY ] ; LCY
iDX ] + 2tLDX
iDY ] ; iDX
Y ]  : ;  = j  LCX iDY ; 0 ; LCY iDX ; iDX
Y ] + 2t(diDX iDY ; 0)  : Here we used that D has vertical values which implies j  iDY LCX   = iDY j  LCX   = iDY rX  = 0. Next we use lemma 16.7.1 to compute the dierential of the of the 1-form iD ] 2 $1 (M H n;1(p)). driD ](X
Y ) = rX iD(Y ) ] ; rY iD(X ) ] ; iD(X
Y ]) ] = rX iD(Y )  ; rY iD(X )  ; iD(X
Y ]) ] = j  (LCX iD(Y ) ; LCY iD(X ) ; iD(X
Y ]) )  ]: n

n

x

 So we get @t@ 0 iR ] = driD ] in the space $2 (M H n;1(p)). t

16. Characteristic classes for general ber bundles

95

16.10. Remarks. The idea of this class is originally due to Kainz, 1985]. From the point of view of algebraic topology it is not very interesting since it coincides with the second dierential in the Serre spectral sequence of the bration E ! M of the class corresponding to .

16.11. Review of linear connections on vector bundles. Let (V
p
M
n) be a vector bundle equipped with a ber metric g. Let r be a linear covariant derivative on E which respects g. Let s = (si ) be a local frame eld over U  M of E: so si 2 C 1 (E jU) and the si (x) are a linear basis of Ex for each x 2 U. n Then s denes P a vector bundle chart  : E jU ! U
by the

prescription (P si ui) = (ui) or (s:u) = u. If s0 is another frame eld, we have s0i = j sj hji or s0 = s:h for a function h 2 C 1 (U \ U 0
GL(n)). For the vector bundle charts we have then 0 ;1 (x
u) = (x
h(x):u). We also have T : T(E jU) ! T U
n
n, and for the chart change we have T(0 ;1 ) : T (U \ U 0 )
n
n ! T(U \ U 0)
n
n T (0 ;1 )(Xx
v
w) = (Xx
h(x):v
dh(Xx):v + h(x):w) P

The connection form is dened by rX si = j sj !ij (X) or rs = s:! for ! 2 $1 (U
gl(n)) in general and ! 2 $1 (U
o(n)) if r respects g and the frame eld s is orthonormal. If rs0 = s0 :!0 is the connection form for 0 another frame s0 = s:h, we have P thej transition formula h:! = !:h + dh. For a general section s:u = j sj u we have r(s:u) = s:!:u + s:du. Now we want to express the connection = r 2 $1(E V E), the horizontal lift C : TM
M E ! TE and the connector K = vpr : TE ! E locally in terms of the connection form (compare with 11.10 and following). Since for a local section x 7! (x
u(x)) of U
n ! U we have

rX (s:u) = s:(!(X):u + du(X)) = K T (s:u) X by 11.12,

(T() T(s:u))(Xx ) = (Xx
u(x)
du(Xx)) we see that

( K T(;1 ))(Xx
v
w) = (Xx
!(Xx ):v + w) (T T (;1 ))(Xx
v
w) = (0x
v
w + !(Xx ):v) (T C (Id
))(Xx
v) = (Xx
v
;!(Xx ):v):

96

16. Characteristic classes for general ber bundles

The Christoel form ; of with respect to the bundle chart  is given by 9.7 (0x
;(Xx
v)) = ;T() T (;1 )(Xx
v
0) = (0x
v
;!(Xx ):v) ;(Xx
v) = ;!(Xx ):v Now the curvature Rr (X
Y ) = rX
rY ] ;rX
Y ] . If we put Rr:s = s:$, then we have $ 2 $2(U
gl(n)) and $ 2 $2(U
o(n)) if r respects g and if s is an orthonormal frame eld. Then the curvature form $ is given by $ = d! + ! ^ !. The curvature R = 21 
] 2 $2 (E V E) can be written in terms of the Christoel forms by 9.7 (;1 ) R(X
Y ) = dU ;(X
Y ) + ;(X)
;(Y )]X( ) = d(;!)(X
Y ) ; (;!(X)) ^ (;!(Y )) = ;$(X
Y )
since GL(n) acts from the left on n , so the fundamental vector eld mapping is a Lie algebra anti homomorphism. Let us consider now the unit sphere bundle SE = fu 2 E : g(u
u) = 1g in the vector bundle E. If we use an orthonormal frame then  : SE jU ! U
S n;1 is a ber bundle chart. We have T(U
S n;1 ) = f(Xx
v
w) 2 TU
n
n : jvj = 1
v?w = 0g local(Xx
v
w) = (0x
v
w + !(Xx ):v) hv
w + !(Xx ):vi = hv
!(Xx ):vi = 0
since !(Xx ) 2 o(n). So induces a connection on SE with the same Christoel forms (acting on a submanifold now). If we assume that E is ber orientable and if we consider the ber volume  on SE induced by g, then these Christoel forms take values in the Lie algebra of divergence free vector elds, so the induced connection SE is -respecting. The local expression of the curvature of SE is again ;$. 16.12. Lemma. 1. For a ber orientable vector bundle (E
p
M
n) with ber dimension n > 2 and a ber Riemannian metric g we consider the induced ber volume  on the unit sphere bundle SE. Then the characteristic class k(SE
) = 0. 2. For a complex line bundle (E
p
M) with smooth hermitian metric g. We consider again the induced ber volume on the unit sphere bundle (SE
p
S 1 ). Then for the characteristic class we have k(SE
) = ;c1 (E)
n

16. Characteristic classes for general ber bundles

97

the negative of the rst Chern class of E. Proof: 1. This follows from H n;2(S n;1 ) = 0. 2. This follows since H 0(p) = M
, so from 16.8 the class k(SE
) is induced by the negative of the curvature form $, which also denes the rst Chern class.

98

17. Self duality 17.1. Let again (E
p
M
S) be a ber bundle with compact standard ber S. By a
berwise symplectic form ! on E we mean a smooth section of the vector bundle %2V  E ! E such that !x := ! Ex 2 $2(Ex )

is a symplectic form on each ber Ex . So as in 16.1 we may plug two vertical vector elds X and Y into ! to get a function !(X
Y ) on E. If dimS = 2n, let !xn = !x ^ ^ !x be the Liouville volume form of !x on Ex. We will suppose that each ber Ex has total mass 1 for this ber volume: we may multiply ! by a suitable positive smooth function on M. I do not know whether lemma 16.2 remains true for berwise symplectic structures. 17.2. Example. For a compact Lie group G let Ad : G ! GL(g ) be the coadjoint action. Then the union of all orbits of principal type (maximal dimension) is the total space of a ber bundle which bears a canonical berwise symplectic form. 17.3. Let now be a connection on (E
p
M
S) and let r be the associated covariant derivative on the (innite dimensional) vector bundle S $2ver (E) := x2M $2 (Ex) over M which is given by  rX ! := dtd j0(FlCX t )!

as in 16.3.

Denition: Let ! be a berwise symplectic form on E. The connection on E is called !-respecting if rX ! = 0 for all X 2 X(M). 17.4. Lemma. If dimH 2(S ) = 1 then there are !-respecting con-

nections on E and they form a convex set in the space of all connections. Proof: Let us rst investigate the trivial situation. So e = M
S and the berwise symplectic form is given by a smooth mapping ! : M ! RZ 2 (S)  $2(S) such that !x is a symplectic form for each x. Since S !xn = 1 for each x and since  7! n is a covering mapping H 2 (S) n 0 = n 0 ! H 2n n 0 = n 0, we see that the cohomology class !x] 2 H 2(S) is locally constant on M, so ! : M ! !x0 + B 2 (S) if M is connected. Thus dM ! : TM ! B 2 (M) is a 1-form on M with values in the nuclear vector space B 2 (S) of exact forms, since it is a closed subspace of $2 (S) by Hodge theory. Let G : $k (S) ! $k (S) be the Green operator of Hodge theory for some Riemannian metric on S and dene ; 2 $1 (M : X(S)) by i(;(Xx ))!x = ;d G(dM !)x (Xx ). Since !x

17. Self duality

99

is non degenerate, ; is uniquely determined by this procedure and we have L;(X ) !x = dS i;(X ) !x + 0 = ;dd G(dM !)x (Xx ) = ;(dM !)x (Xx )
since it is in B 2 (S). For the covariant derivative r induced by the connection with Christoel form ; we have rX ! = dtd j0(Fl(tX
;(X )) ) ! = (dM !)(X) + L;(X )! = 0
so the Christoel form ; denes an !-respecting connection. For the general situation we know now that there are local solutions, and we may use a partition of unity on M to glue the horizontal lifts to get a global !-respecting connection. The second assertion is obvious. 17.5. Let now be an !-respecting connection on (E
p
M
S) where ! is a berwise symplectic form on E. Let R = R( ) be the curvature of and let C be the horizontal lifting of . Then i(R(CXx
CYx))!x 2 $1 (Ex) and i(R(CXx
CYx))!xn 2 $2n;1(Ex ). As in lemma 16.5 we may prove that both are closed forms. So i(R(CXx
CYx))!x )] 2 H 1(Ex ) and i(R(CXx
CYx))!xn ] 2 H 2n;1(Ex). Let us also x an auxiliary Riemannian metric on the base manifold M and let us consider the associated Hodge --operator: $k (M) ! $m;k (M), where m = dimM. Let us suppose that the dimension of M is 4 and let us consider the forms iR !] 2 $2 (M H 1(p)) and iR !n ] 2 $2 (M H 2n;1(p)) with values in the at vector bundles H i (p) from 16.6. Both are dr -closed forms (before applying  to the second one) since the proof of 16.7 applies without changes. Let Dx : H 2n;1(Ex) ! H 1(Ex ) be the Poincare duality operator for the compact oriented manifold Ex  these t together to a smooth vector bundle homomorphism D : H 2n;1(p) ! H 1(p). Denition: An !-respecting connection is called self dual or anti self dual if D  iR !n ] = iR !] or = ;iR !] respectively. This notion makes sense also for not ! -respecting connections. 17.6. For the notion of self duality of 17.5 there is also a Yang-Mills functional if M 4 is supposed to be compact. Let be any connection. For Xi 2 Tx M we consider the 4-form x

x

'x ( )(X1
. . .
X4 ) := 41 Alt

Z

Ex

(iR !)x (X1
X2) ^ (iR !n )x (X3
X4 )


100

17. Self duality

where Alt is the alternator of the indices of the Xi . The Yang-Mills functional is then Z F( ) := '( ): M

17.7. A typical example of a ber bundle with compact symplectic bers is the union of coadjoint orbits of a xed type of a compact Lie group. Closely related to this are open subsets of Poisson manifolds. !respecting connections exist if the orbit type has second Betti number 1. The Yang-Mills functional and the self duality notion exist even without this requirement. It would be nice to investigate the characteristic class of section 16 for the berwise symplectic structure instead of a ber volume form for examples of coadjoint actions. 17.8. Finally we sketch another notion of self duality. We consider a berwise contact structure  on the ber bundle (E
p
M
S) with compact standard ber S: So x 2 $1(Ex ) depends smoothly on x and is a contact structure on Ex, i. e. x ^ (dx)n is a positive volume form on Ex , where dimS = 2n + 1 now. A connection on the bundle E is now called -respecting if in analogy with 16.2 we have   rX  := @t@ 0 (FlCX t ) =0

for each vector eld X on M. If R is the curvature of such a connection then iR(CX
CY ) is berwise locally constant by the analogue of 16.5.(3). If S is connected then (X
Z) 7! iR(CX
CY ) denes a closed 2-form on M with real coecients and it denes also a cohomology class in H 2 (M ). But it is not clear that -respecting connections exist.

101

References Brocker, Theodor Janich, Klaus, \Einfuhrung in die Dierentialtopologie," Springer-Verlag, Heidelberg, 1973. Cervera, Vicente Mascaro, Francisca Michor, Peter W., The orbit structure of the action of the dieomorphism group on the space of immersions, Preprint. De La Harpe, Pierre Omori, Hideki, About interaction between Banach Lie groups and
nite dimensional manifolds, J. Math. Kyoto Univ. 12 (1972),. Ebin, D.G. Marsden, J.E., Groups of dieomorphisms and the motion of an incompressible uid, Ann. Math. 92 (1970), 102{163. Eells, James, A setting for global analysis, Bull AMS 72 (1966), 571{807. Ehresmann, Charles, Les connexions in
nitesimales dans un espace
bre dierentiable, in \Colloque Topologie, Bruxelles 1950," Liege, 1951, pp. 29{55. Frolicher, Alfred Kriegl, Andreas, \Linear spaces and dierentiationtheory," Pure and Applied Mathematics, J. Wiley, Chichester, 1988. Frolicher, A. Nijenhuis, A., Theory of vector valued dierential forms. Part I., Indagationes Math 18 (1956), 338{359. Gil-Medrano, Olga Peter W. Michor, The Riemannian manifold of all Riemannian metrics, to appear, Quaterly Journal of Math. (Oxford). Grabowski, Janusz, Free subgroups of dieomorphism groups, Fundamenta Math. 131 (1988), 103{121. Greub, Werner Halperin, Steve Vanstone, Ray, \Connections, Curvature, and Cohomology I," Academic Press, 1972. Hamilton, Richard S., The inverse function theorem of Nash and Moser, Bull. Am. Math. Soc. 7 (1982), 65{222. Hirsch, Morris W., \Dierential topology," GTM 33, Springer-Verlag, New York, 1976. Kainz, Gerd, Eine charakteristische Klasse fur F -Faserbundel, handwritten manuscript 1985. Kobayashi, S. Nomizu, K., \Foundations of Dierential Geometry. Vol. I.," J. Wiley, 1963. Kriegl, Andreas Michor, Peter W., A convenient setting for real analytic mappings, Acta Mathematica 165 (1990a), 105{159. Kriegl, A. Michor, P. W., Aspects of the theory of in
nite dimensional manifolds, Dierential Geometry and Applications 1(1) (1990b). A. Kriegl, Michor, P. W., \Foundations of Global Analysis," A book in the early stages of preparation. Lecomte, P. B. A. Michor, P. W. Schicketanz, H., The multigraded Nijenhuis-Richardson Algebra, its universal property and application, Preprint 1990, submitted to J. Pure Applied Algebra.. Marathe, Kishore B. Martucci, G., The geometry of gauge
elds, J. Geometry Physics 6 (1989), 1{106. Michor, Peter W., \Manifolds of dierentiable mappings," Shiva Mathematics Series 3, Orpington, 1980. Michor, P. W., Manifolds of smooth mappings IV: Theorem of De Rham, Cahiers Top. Geo. Di. 24 (1983), 57{86.

102

List of Symbols

Michor, Peter W., Remarks on the Frolicher-Nijenhuis bracket, in \Proceedings of the Conference on Dierential Geometry and its Applications, Brno 1986," D. Reidel, 1987, pp. 197{220. Michor, P. W., Gauge theory for dieomorphism groups, in \Proceedings of the Conference on Dierential Geometric Methods in Theoretical Physics, Como 1987, K. Bleuler and M. Werner (eds.)," Kluwer, Dordrecht, 1988, pp. 345{371. Michor, P. W., Graded derivations of the algebra of dierential forms associated with a connection, in \Dierential Geometry, Proceedings, Peniscola 1988,," Springer Lecture Notes 1410, 1989a, pp. 249{261. Michor, P. W., Knit products of graded Lie algebras and groups, Proceedings of the Winter School on Geometry and Physics, Srni 1989, Suppl. Rendiconti Circolo Matematico di Palermo, Ser. II 22 (1989b), 171{175. Michor, Peter W., The moment mapping for a unitary representation, J. Global Analysis and Geometry 8, No 3 (1990), 299{313. Michor, Peter W. Neuwirther, Martin, Pseudoriemannian metrics on spaces of bilinear structures, preprint 1990. Modugno, Marco, Systems of vector valued forms on a
bred manifold and applications to gauge theories,, in \Lecture Notes in Math.," Springer-Verlag, 1987. Morrow, J., The denseness of complete Riemannian metrics, J. Di. Geo. 4 (1970), 225{226. Nagata, J., \Modern dimension theory," North Holland, 1965. Newlander, A. Nirenberg, L., Complex analytic coordinates in almost complex manifolds, Ann. of Math. 65 (1957), 391{404. Nijenhuis, A. Richardson, R., Deformation of Lie algebra structures, J. Math. Mech. 17 (1967), 89{105. Nomizu, K. Ozeki, H., The existence of complete Riemannian metrics, Proc. AMS 12 (1961), 889{891. Omori, Hideki, On Banach Lie groups acting on
nite dimensional manifolds, Tohoku Math. J. 30 (1978), 223{250. Omori, Hideki, \Innite dimensional Lie transformation groups," Lecture Notes in Math. 427, Springer-Verlag, Berlin etc., 1974. Palais, Richard S., A global formulation of the Lie theory of transformation groups, Mem. AMS 22 (1957). Ramadas, T. R., On the space of maps inducing isomorphic connections, Ann Inst. Fourier 32 (263-276), 1982.

List of Symbols

C 1(E), also C 1(E ! M) the space of smooth sections of a bre bundle C 1(M
R) the space of smooth functions on M d : $p (M) ! $p+1 (M) usually the exterior derivative (E
p
M
S), also simply E usually a bre bundle with total space E, base M, and standard bre S FlXt the ow of a vector eld X k , short for the k
k-identity matrix Id . LX Lie derivative k

Index

103

G usually a general Lie group with multiplication  : G
G ! G, left translation , and right translation ` : G
S ! S usually a left action M usually a (base) manifold natural numbers 0 nonnegative integers real numbers r : P
G ! P usually a right action, in particular the principal right action of a principal bundle TM the tangent bundle of a manifold M with projection M : TM ! M Integers

Index
-respecting connection, 100 -respecting connection, 92 !-respecting connection, 98

A

algebraic bracket, 28 algebraic derivation, 27 anti self dual, 99 approximation property (bornological) , 15 associated bundle, 48 associated vector bundles, 52

B

base space, 36 basis, 36 Bianchi identity, 37 bundle atlas, 36 bundle of gauges, 54 bundle, 36

C

cartesian closedness, 8, 11, 13 characteristic class, 94 Chern class, 97 Christoel forms, 39 classifying connection, 87 classifying space, 53, 87, 89 cocurvature, 33 cocycle condition, 36 cocycle of transition functions, 36 cohomologous, 44 complete connection, 41

connection form (Lie algebra valued), 60 connection form, 95 connection, 32, 37, 92 connector, 69 continuous derivation over eva , 14 convenient vector space, 7 covariant derivative on vector bundles, 70 covariant derivative, 63, 70 covariant exterior derivative, 63, 71 curvature form (Lie algebra valued), 60 curvature form, 96 curvature, 32, 33, 37, 70 c1 -topology, 6 C ! -manifold structure of C ! (M N ), 19 C ! -manifold structure on C 1 (M N ), 19

D

De Rham cohomology of M with twisted coecient domain H n;1 (p), 92 derivations (graded) , 27 dieomorphism group, 22

E

Ehresmann connection, 41 equivariant mappings and associated bundles, 50 exponential law, 20

F

f -dependent, 33

104 f -related, 33

ber bundle atlas, 36 ber bundle, 36 ber chart, 36 ber volume, 90 berwise symplectic form, 98 at connections, 38 frame bundle, 81 Frolicher-Nijenhuis bracket, 29 fundamental vector eld, 5 fundamental vector eld, 5

G

G-atlas, 44 G-bundle structure, 44 G-bundle, 44

Index

L

Lie derivation, 28 Lie group, 5 linear connection, 69 linear frame bundle, 52 local description of connections, 38 local descriptions of principal connections, 62 local formulas for the FrolicherNijenhuis bracket, 31

M

manifold structure of C 1 (M N ), 17 Maurer-Cartan formula, 39

N

gauge group of a ber bundle, 83 gauge transformations, 54 general Bianchi identity, 33 generalization of the theorem of Ambrose and Singer, 77 global formula for the FrolicherNijenhuis bracket, 30 Grassmann manifold, 46

n-transitive, 23 natural bilinear concomitants, 34 naturality of the Frolicher-Nijenhuis bracket, 33 nearly complex structure, 35 Nijenhuis tensor, 35 Nijenhuis-Richardson bracket, 28 nonlinear frame bundle, 81

Hodge-
-operator, 99 holomorphic curves, 10 holomorphic, 10 holonomy group, 65, 76 holonomy groups for principal connections, 65 holonomy Lie algebra, 76 homogeneous spaces, 45 homomorphism of G-bundles, 51 homomorphism over  of principal bundles, 47 horizontal bundle, 37 horizontal dierential forms, 63 horizontal foliation, 38 horizontal lift, 37 horizontal projection, 37 horizontal space, 32 horizontal vectors, 37

operational tangent vector, 14 orthonormal frame bundle, 53 orthonormal frame eld, 52

H

I

induced connection, 66, 67 inducing principal connections on associated vector bundles, 69 irreducible principle connection, 66

K

kinematic tangent vector, 14

O O

parallel transport, 39 principal (ber) bundle, 44 principal bundle atlas, 44 principal bundle of embeddings, 24 principal connection, 60 principal ber bundle homomorphism, 47 principal right action, 45 projection, 36 proper, 20 pullback operator, 34 pullback, 38 pullback of ber bundles, 37

R

real analytic curves, 12 real analytic mappings, 12 real analytic, 12 recognizing induced connections, 67 reduction of the structure group, 47 regular, 23 restricted holonomy group, 65, 76

Index Riemannian metric , 52 right logarithmic derivative, 58

S

sections of associated bundles, 53 self dual, 99 smooth curves in C 1 (M N ), 18 smooth mappings, 7 smooth partitions of unity, 14 smooth, 6, 7 smoothness of composition, 19 space of connections, 84 standard ber, 36 Stiefel manifold, 46

T

105

tangent group of a Lie group, 57 tangent vectors as derivations, 16 theorem of Hartogs, 10 topologically real analytic, 12 total space, 36 transformation law for the Christoel forms, 39 transition functions, 36

U

universal vector bundle, 56

V

vector valued dierential forms, 27 vertical bundle, 37 vertical bundle, 58 vertical projection, 37 vertical space, 32

tangent and vertical bundles of principal and associated bundles, 58 tangent bundles of Grassmann manifolds, 56 tangent bundles of homogeneous spaces, Y 54 Yang-Mills functional, 100