L A Cordero (Elitor) University of Santiago.de Compostela
Differential geometry
Pitman Advanced Publishing Program LONDON MELBOURNE
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© L A Cordero 1985 First published 1985 53C, 57R30
AMS Subject ISSN 0743-0337
in Publication Data Library of Congress Cataloging Main entry under title:
Differential geometry. (Research notes in mathematics; 131) Proceedings of a colloquium held at Santiago de Compostela, Spain, September 1984. Bibliography: p. 1. Geometry, Differential. 1. Cordero. 1. A 11. Series.
QA641.D383
1985
516.36
85-6350
ISBN 0-273-08708-8 British L'lbraty Cataloguing in Publication Data
Differential geometry.—(Research notes in mathematics, ISSN (1743-0337;131) 1. Geometry, Differential I. Cordero, L.A. II. Series 516.3'6 0A641
iSBN (1-273-08708-8
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Contents
Preface Participants in the Conference PART ONE
:
RIEMANNIAN GEOMETRY
D.E. BLAIR
Critical points of functio'is of the curvature on spaces of associated metrics
1
F.E. BURSTALL
Twistor fibrations of flag manifolds and harmonic maps of a 2-sphere into a
7
J.L. CABRERIZO, M. FERNANDEZ ANDRES
CR—subinanifolds of a locally conformal Kihier manIfold
17
R. CADOED
The Almansi property of the distance function In the Riemannian context
33
C. CURRAS-BOSCH
Infinitesimal transformations with bounded norm
41
EELLS
Certain variational principles in Riemannian geometry
46
F. GOMEZ, M.
The topological dimension of Jhe cut locus
61
GRAY
Ersatz Chern polynomial and Weyl 's tube formula
66
J.F.T. LOPERA
Conformal vector fields on a certain class of nilpotent groups
76
A. MACHADO, I. SALAVESSA
Grassman manifolds as subsets of Euclidean spaces
85
5.14. SALMON
Minimal surfaces and symetric spaces
103
U. SIMON
Eigenvalues of the Laplacian and minimal
J. G-deformations
into spheres
115
and Weyl's tube theorem
121
of surfaces in R3 and R1
138
oF a harmonic morphism
149
S.c.
)LIATION THEORY NT HAL ,
fibre spaces and product decompositions
J. GIRBQJ, M. NICOLALJ
of holoinorphic foliations dnC transversely holomorphic 162
A.
.JGER
of local isometries
174
A.
Generalized foliations and local Lie algebras of Kirillov
198
X. MASA
Cohomology of Lie foliations
211
J. PRADINES
Graph and holonoiny of singular foliations
215
H. SARALEGtJI
The Euler class for flows of Isometries
220
H. SUZUKI
An interpretation of the Well operator x(y1)
228
I. VAISMAN
Lagrangian foliations and characteristic classes
24S
E. VOGT
Examples of circle foliations on open 3-manifolds
257
R.A. WOLAK
Some remarks on V-6—foliations
276
PART THREE
OTHER TOPICS
C.T.J. DODSON
Fibt-ilations and group actions
290
M. FERRARIS, M. FRANCAVIGLIA
The theory of formal connections and fibred connections in fibred manifolds
297
J. GANCARZEWICZ
Horizontal lift of connections to a natural vector bundle
318
Preface
This volume includes a large number of the lectures given at the Fifth International Colloquium on Differential Geometry, held in Santiago de Compostela, Spain, in September 1984. The colloquium was organized by the Departamento de Geometrfa y Topologla of the Universidad de Santiago do Compostela. The main themes were Riernannian geometry and foliation theory. Lectures of one hour were given by Professors J. Eells, J. Girbau, A. Gray, A. Haefliger, A. Lichnerowicz, B.L. Reinhart, L. Vanhecke and T.J. Wilimore.
like to express our On behalf of the organizing comittee gratitude to the sponsors of the congress, namely the Direcci6n Xeral de Universidades (Xunta de Galicia), the Universidad de Santiago de Coinpostela and the Comisión Asesora de Investigacidn Cientcfica y Tecnica (Madrid), for the generous funding that made the congress possible. Thanks are due also to Pitman Publishing for publishing this volume and to Professors Eells and
Gray for aiding in the preparation. Finally, I want to express the deep appreciation of the mathematicians in Santiago to all the participants in the congress for their enthusiasm and new ideas which contributed to the success of the colloquium. Santiago de Compostela
April, 1985
Luis A. Cordero
Participants in the conference
E. ALBERTO BANQUE
Barcelona, SPAIN F. ALCALDE
Lugo, SPAIN E.M. AMORES
Madrid, SPAIN
J.M. CARBALLES
Santiago, SPAIN J. CASTANHEIRA DA COSTA Braga, PORTUGAL R. CASTRO
Santiago, SPAIN
Clausthal, GERMANY
I. CATTANEO GASPARIN! Roma, ITALY
L.C. de ANDRES
0. CHINEA
S.!. ANDERSON
Pais Vasco, SPAIN M. BARROS
Granada, SPAIN D. BERNARD
Strasbourg, FRANCE
D.E. BLAIR
Michigan, U.S.A.
La Laguna, SPAIN L.A. CORDERO
Santiago, SPAIN C. CUARTERO
Madrid, SPAIN C. CURRAS-BOSCH
Barcelona, SPAIN
R.A. BLUMENTHAL
I.M. DA COSTA SALAVESSA Lisboa, PORTUGAL
A. BONOME
C.T.J. DODSON Lancaster, UNITED KINGDOM
St. Louis, U.S.A Santiago, SPAIN A. BUCK! Lublin, POLAND F. BURSTALL Bath, UNITED KINGDOM
J.L. CABRERIZO
Th. DUCHAMP
Seattle, U.S.A. F.J. ECHARTE
Sevilla, SPAIN * J. EELLS
Sevilla, SPAIN
Warwick, UNITED KINGDOM
R. CADDEO
J.J. ETAYO Madrid, SPAIN
Cagliari, ITALY A. CANDEL
Santiago, SPAIN
L. FERNANDEZ
Sevllla,SPAIN N. FERNANDEZ
Santiago. SPAIN
M.J.T. FERREIRA
* A. LICHNEROWICZ
Lisboa, PORTUGAL
College de France, FRANCE
£. FOSSAS
M. LLABRES
Barcelona, SPAIN
Barcelona Put., SPAIN
M.
Torino, ITALY
J.T. LOPEZ RAVA Granada, SPAIN
E. GALLEGO
J.L. LOPEZ ROSENDO
Barcelona Aut., SPAIN
Santiago, SPAIN
J. GANCARZEWICZ
E. MACLAS
Krakow, POLAND
Lugo, SPAIN
O.J. GARAY
A. MACHADO
Granada, SPAIN
Lisboa, PORTUGAL
L. GEATTI
A. MARTINEZ LOPEZ
Pisa, ITALY
Granada, SPAIN
J. GETINO
A.A. MARTINEZ SEVILLA Granada, SPAIN
Oviedo, SPAIN
* J. GIRBAU
X. MASA
Barcelona Put., SPAIN
Santiago, SPAIN
J.R. GOMEZ
a. MENCIA
Sevilla, SPAIN
Pals Vasco, SPAIN
A. GOMEZ TATO
M.D. MONAR
Santiago, SPAIN
La Laguna, SPAIN
* A.
M.C. MUROZ
Maryland, U.S.A. * A. HAEFLIGER SWITZERLAND
Mallorca, SPAIN M. NICOLAU
Barcelona Aut., SPAIN
G. HECTOR
J.A.
hue, FRANCE
Santiago, SPAIN
L. HERVELLA
J.O. PEREZ JIMENEZ Granada, SPAIN
Santiago, SPAIN M.T. IGLESIAS Santiago, SPAIN
PHAM MAU QUAN
II. de LEON Santiago, SPAIN
liP. de PRADA Pals Vasco, SPAIN
P. LIBERMANN
J. PRADINES Toulouse, FRANCE
Paris, FRANCE
Paris, FRANCE
A. RAS
Barcelona, SPAIN * B.L. QEINHART
* L. VANHECKE Leuven, BELGIUM E. VAZQUEZ ABAL
Maryland. U.S.A.
Santiago, SPAIN
A. REVENTOS
M.C.S. VIANA FERREIRA Lisboa, PORTUGAL
Barcelona kit., SPAIN G.B. RIZZA Parma, ITALY
E. VIDAL ABASCAL
Santiago, SPAIN
B. RODRIGUEZ
c. VILLAVERDE
Santiago. SPAIN
Lugo, SPAIN
A. ROMERO
J.L. VIVIENTE Zaragoza, SPAIN
I
Granada, SPAIN I. ROZAS
Pals Vasco, SPAIN S.M. SALAMON Oxford, UNITED KINGQQM M. SALGADO
Santiago, SPAiN N. SARALEGU!
Pals Vasco, SPAIN
E. YOGI Berlin, GERMANY
* T.J. Durhøm, UNITED KINGDOM R.A. WOLACK Krakow, POLAND
J.C. WOOD Leeds, UNITED KINGDOM
J.M. SIERRA La Laguna, SPAIN U.. SDK)N
Berlin, GERRANY K. SPERA Roma, ITALY
H. SUZUKI
Hokkaldo Univ., JAPAN D.
Lille, FRANCE E. de la TORRE Santiago, SPAIN J.F. TORRES LOPERA
Santiago, SPAIN I. VAISMAN
Haifa, ISRAEL
(*)
Invited Lecturer
DEBLAIR
Critical points of functions of the curvature on spaces of associated metrics The
study of the integral of the scalar curvature, 1(g)
R =
J
dV9, as a
function or' the set of all Riemannian metrics of the same total volume on a compact manifold is now classical and the critical points are the Einstein metrics; moreover, other functions of the curvature have been taken as integrands and studied [2,10,11,123. Two questions arise: (1) Given the
function 1(g) restricted to a smaller set of metrics, what is the critical point condition; one would expect a weaker one. The sets of metrics we have in mind are the metrics associated to a symplectic or contact structure. (2) Given these sets of metrics, are there other natural integrands depending on the structure as well as the curvature? Before giving some affirmative results we begin with a brief review of symplectic and contact manifolds. By a BynlpleOtic manifold a manifold M of dimension 2n together ç2fl with a closed 2-form such that 0. By a contact manifold we mean a manifold M of dimension 2n + I together with a 1-form ri such that A
there exists a unique vector field ç such that = 0; is called the characteristic vector field of the I and contact structure We say that M is a regular contact manifold if ç is a regular vector field, i.e. each point of Il has a neighbourhood such that every integral curve of passing through the neighbourhood passes through It is well known that given
only once. On a symplectic manifold a Rietnannian metric g is said to be an as8ociated metric if g(X,JY) = o(X,Y) where J is an almost complex structure. Thus the
set of all associated metrics is the set of all almost Kahier metrics on M which have o as their fundamental 2-form. On a contact manifold a Riemannian metric g is an associated metric if there exists a tensor field of type •2 (1,1) such that and = -Id + ® The = dTi(X,Y), =
point is that these metrics and the almost complex structures (resp. the •'s) are constructed simultaneously by the polarization of (resp. drj restricted on a local ortho— to the contact distribution (sub-bundle) = 0}) normal basis of an arbitrary metric (on (q 0.1) (sec or 14]). In both the set A of all associated metrics is infinite dh.iensional and is
totally geodesic in the set of all Riemannian metrics on N (4,5). Finally we note that all associated metrics have the same volume element, that is, a constant depending on n times
or
A
Our first result will be along the line of the second question on a contact manifold. Let h = req, where £ denotes Lie differentiation; h is a syniiietric operator which anti-coninutes with •;
If and only if h = 0, and we refer to the pair
is a Killing vector field as a K-contact
Moreover
2n — tr h2
denotes the Ricci curvature in the direction of
where
(3, p. 67].
Clearly K—contact metrics are maxima for the function L(g) = JM
and we ask if these are the only critical points. Theorem-i
form
(5). Let M be a compact regular contact manifold with contact Then an associated metric in the set of all metrics associated to
is a critical point of the function L if and only If it is a K—contact metric. While one might conjecture this result in the non—regular case, regular contact manifolds form a natural context for this question, since regular contact manifolds always carry a K-contact metric, whereas a non—regular contact manifold may or may not carry one; e.g. the 3-dimensional torus with tt&'usual contact structure is not regular and carries no K—contact metric —(see e.g. (3, pp. 7-8 and Ch. IV)). to theorems are joint work with Stere Janus (6] in the symplectic The case.
—
(6]. Let M be a compact symplectic manifold and let A be the set of metrics associated to the symplectic form. Then a metric 9 A is a R dV on A if and only If the Ricci critical point of the function 1(g) = JN opirator of g comeutes with the almost complex structure corresponding to g Theorem 2
manifold one has the
On an almost =
R
2
- R*
-
curvature defined
by
(see e.g. (9]) and hence
with equality holding if and only if the metric is Kthleridn. Consider the function
K(g)=JR_R*dV defined on A; clearly K8hler metrics are maxima and we ask for the general
critical point condition. (6].
Let M be a compact symplectic manifold and let A be the set of metrics associated to the symplectic form. Then a metric g EA is a
Theorem 3
critical point of the function K(g) if and only if the RIcci operator of g conunutes with the almost complex structure corresponding to g.
These theorems raise the interesting and seemingly difficult question of whether or not an almost KThler metric satisfying JQ = QJ, Q being the case almost Ricci operator, is It is perhaps doubtful, In metrics satisfying JQ QJ become an interesting class of metrics to study, but we do make the following remarks. S.1. Goldberg (83 showed that if J conunutes with the curvature operator, the metric is Recently a compact almost Kähler, Einstein metric is Einstein K. Sekigawa [14] showed that a compact 4-dimensional almost manifold of non—negative scalar curvature is K8hlerian. W. Thurston [15] gave an example of a compact 4-dimensional symplectic manifold with no metric on this manifold metric and E. Abbena [1] gave a natural almost and computed its curvature. While the Ricci operator has a nice form, it with the almost complex structure. does not A.J. Ledger and 1 [7) have considered the problems of the last two theorems
for contact manifolds.
[7). Let M be a compact contact manifold and let Abe the set of metrics associated to the contact form. Then a metric g e A Is a critical R dV if and only if the Ricci operator and • point of the function 1(g) Theorem 4
JN
comute when restricted to the contact distribution, i.e. for X E {n = 0), the projection of - Q4)X to {n = 0) vanishes.
.3
ow'uc*la's Is defined by Ona contact metric manifold the and It was shown by 1. Olszak (13] that R_R* — 4n2
-
IIV$112
+ 2n - tr
h2
0
with equality holding if and only If the metric is Sasakian.
(7]. Let M be a compact contact manifold and let A be the set of metrics associated to the contact form. Then a metric g A is a critical Theorem 5
point of the function K(g)
R
=
-
- 4n2dV if and only If the operators
JN
Q - 2nh and • consnute when restricted to the contact distribution. Since the critical point conditions in Theorems 4 and 5 are different, If a metric g satisfies both conditions it is easy to show that h vanishes and hence that
Is a Killing vector field.
Thus we have the
Corollary (73. If g is a critical point of both I and K, then g is a Kcontact metric. This Is a reasonable corollary since Sasakian metrics both satisfy and are K-contact. In a similar way to the question raised In the symplectic case, one can ask whether or not a K-contact structure satisfying is • Sasakian. •A metric being ,just an associated metric satisfying Q$ • $Q need not be Sasakian, e.g. agaIn,the standard contact structure on the torus carries a flat associated metric which Is not K—contact. The proofs of the theorems stated are given in the papers cited. We close with one remark about the proofs. Let g(t) be a smooth curve in A and let
to a curve in A through g
OJ+JD=0
g(O), satisfies (A)
in the syinplectic case (4] and (B)
in the contact case (4,5] where we have also denoted by 0 the corresponding
(1,1) tenser. Differentiation of the functions In the theorems gives rise to an Integral of the form I ii dY for some tensor 1. Thus the following JM
lenina Is central In the proofs of the theorems. Let T be a second order
Lenina.
tensor field on M.
Then
dV = 0
JM
for all D satisfying (A) in the symplectic case, (B) in the contact case, if and only if T coniuutes with .3 in the symplectic case, and in the contact case T and • coninute when restricted to the contact distirubtion. References
[1]
Abbena, E.
(4]
In Mathematics, 509, Springer (1976). Blair, D.E. On the set of metrics associated to a symplectic or contact
manifold which is not Bolietino U,M.I. (6) 3—A (1984) 383-392. (2) Berger, N. Quelques fonnules de variation pour une structure riemannienne, Ann. Sci. Ecoic Norm. Sup. 3 (1970) 286-294. (3] Blair, D.E. Contact Manifoid8 in Riemannian Geometry, Lecture Notes
(5)
[6) (7)
(8]
form, Bull. Inst. Math. Aced. Sinica 11 (1983) 297-308. Blair, D.E. Critical associated metrics on contact manifolds, J. Austral. Math. Soc. (Series (1984) 82-88. BlaIr, D.E. and lanus, S. Critical associated metrics on symplectic manifolds, (to appear). BlaIr, D.E. and Ledger, A.J. Critical associated metrics on contact manifolds II, (to appear). Goldberg, S.I. Integrability of almost manifolds, Proc. A.M.S. 21 (1969)
(9)
An example of an almost
96-100.
Gray, A. and Hervella, L.M. The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pure Appi. (4) 123 (1980)
35-58.
(10] Muto, Y. [11] Muto, Y.
On
Einstein metrics, J. Diff. Gecin. 9
(1974) 521—530.
Curvature and critical Riemannian metric, J. Math. Soc.
Japan
26 (1974) 686—697.
5
Nagano, T.
on the existence of an Einstein metric, .1. Math.
A
Japan 19 (1967) 30—31. (13) Olszak, 2. On contact metric manifolds, TOAhOku idath. Boo.
J.
31 (1979)
247—253.
(14]
Sekigawa, K.
(15]
manifolds (to appear). Thurston, W. Some simple examples of symplectic manifolds, Proc. A.M.S. ss (1976) 467-468.
On some 4-dimensional compact Einstein almost
David E. Blair Michigan State University East Lansing, Michigan 48824
6
FEBURSTALL •
T'wistor fibrations of flag manifolds and
harmonic maps of a 2-sphere into a Grassmannian INTRODUCTION
A twistor fibration is a Riemannian submersion ir:Z + N of an almost Hermitlan manifold Z onto a Riemannian manifold N with the property that holomorphic maps into Z project onto harmonic maps into N. Such fibratlons have been studied by Eells and Salamon (2), Salamon (5] and Rawnsley [4]. In this note, based on joint work with J.H. Rawnsley, we consider a natural class of twistor fibrations of generalized flag minifolds over Riemannian symoetric spaces. In particular we will show that any harmonic map of S2 Into a complex Grassmannian is the projection of a holomorphic S2 in a flag manifold. Further we demonstrate that such a harmonic map has a number of which are also harmonic maps' of S2 into Grassmannlans. 1.
TWISTOR FIBRATIONS OF GENERALISED FLAG MANIFOLDS
Definhiorr 1.1
Let Z be an almost Hertnltlan manifold and.N a Rieniannian
manifold. A Riernannian submersion N' is said to be a twietor fibration if, for any almost Hermitian manifold (with twietor space Z) co-closed o Kähler form and any holomorphic map 4:M is a harmonic map. Example:
Let N be an even-dimensional Ri,emannian manifold and 1T:J(N)
N
the fibre bundle of almost complex structures on N. Eells and Salamon (2] a natural non-integrable almost complex structure on J(N) for have
is a twictor fibration with respect to a suitable metric. A generalised flag manifold is a compiex homogeneous space F = Gt/P where
which
is a semi-simple complex Lie group and P is a parabolic subgroup. then 6 is transitive on F and so Let 6 be a compact real form of F
GIG A P and 6 A P is the centralizer of a torus (see Wolf [7]).
Denote
G fl P by H.
For simplicity we restrict attention to 6 simple. Denoting the Lie algebras of 6, H, P by
k, q respectively, we have
7
q = ht
(1)
+
(2)
The decomposition (2) induces an invariant complex structure, denoted J1, on G/H which is Kähler with respect to a suitable Invariant metric. This complex structure is, of course, that induced by
where it is a nhlpotent subalgebra.
the diffeomorphism between G/H and GC/P.
be a maximal torus with Lie algebra t. Then is a Cartan it*. Since + n is parabolic, there with root system A is a positive root system A4 and a subset J of the simple roots {cz1,.. .,at} so that, defining Nj(a) by Now let T c subalgebra for
Nj(a)
H
for a
=
=
we have
9a,
C
h
C
E
>0
=0
Now for m El let m
and define m
m
o
Then there is a non—integrable invariant almost +i-eigenspace at the identity coset given by + n'.
structure on G/H with Call this almost com-
plex structure J2. It can be shown that the decomposition + nt' and thus the definition = of Is independent of the choice of maximal torus and root system and so only depends on the parabolic subalgebra q c Let • be the (unique) highest root in A with respect to A'. The positive integer is called the height of Gt/P. The properties of the almost complex structures and are reflected in the height. Examples:(1) 8
if the height of Ge/P is one, then J1 = J2 and the decomposition
-
+
+
syninetric space.
gives G/H the structure of a
is an orthogonal twistor structure if the height of G1/P is two, then in the sense of Bryant [1] and the J structure arises from the 3-syninetric (2)
structure on G/H which at the identity coset has eigenvalues + ht respectively, see Salamon [6]. +
—i-—,
1
on
Now observe that if
p=gn(p+p),Iz=k+(m+m)ng +
+
then
c
[p,p) c h,
p
and (k,k] c
k
so that
is a symetric decomposition with h c space be 6/K, with H c K. The Importance of the
Let the corresponding syninetric
almost complex structure comes from the following
theorem:
The homogeneous fibration (G.K,J2) -' 6/K is a twistor fibration. This result follows from the general twistor theory of Rawnsley [4] or by using the Rlemannian submersion equations of O'Neill (see Salamon [6]). In each case the fundamental ingredient is the following formula: Theorem 1.1
+
+
[m ,p ) C p 2.
+
TWISTOR EQUIVALENT FLAG MANIFOLDS AND
HOLOMORPHIC FIBRATIONS
We have seen that any generalized flag manifold has a canonical choice of almost complex structures and J2, with J1 integrable and a a suitable symetric space.
Definition 2.1 Two generalized flag manifolds Gt/P and Ge/P' are said to be twietor equivalent if P fl G P' n G = H and the corresponding 12 almost complex structures on 6/H coincide.
The following construction will yield all twistor equivalent flag manifolds and holomorphic homogeneous fibrations between fhg manifolds for a given simple g
simple, 6 a compact real Let Gt/P be a generalized flag manifold with the Lie algebra of a maximal torus In H, Let form of G and H = 6 n P. J the positive root system and subset and the roots with respect to of the simple roots corresponding to P as in Section 1. and write Let • be the unique highest root in with respect to
•
k1cg1. =
J such thatK1
Suppose that there is a simple root a•
10
Then
is a Weyl basis for another positive system
u
{ct1,... ,&.
= 1. 0
10
of
roots in (see Wolf [8]). We now distinguish two cases:
(1)
Ge/P has even
height. n
E
putting J' =
Then
for a=
U
let
flB.
E
and define =
=
E
E
9a +
cz:Njs(a)>0 a
parabolic subalgebra with corresponding sub-
group P'. Clearly, P' n G = P n G = H and in fact the flag manifolds G/P and G/P' are twistor equivalent, as can be seen by comparing Nj(cz) with
(ii)
Gt/P has
odd height.
Now
put J'
=
Then, defining
c h't and in fact, if H' 6 n P', then H c as above, we see that 6/H' is J2—holomorphic. homogeneous fibration G/H
hit, H'
p.
and
In fact, these procedures are the only way that twistor equivalent flag fibrations can arise.
manifolds and Theorem 2.1
Let Gt/P
Gt/PP
be generalized flag manifolds with real forms
G/H and G/H'.
(I) If H = H' and G/H Is twistor equivalent to GIN'
'O•J with K.
there exists a simple
1
even height, and P' is constructed as above. then G/H
'3
(ii)
10
If H c H' and the homogeneous fibration GIN + G/H' is J2—holomorphic,
J with
then G/H has odd height, there exists
I and P' is con— 0
0
structured as above.
in the highest root and so we have a good supply of twistor equivalent flag manifolds and J2-holoqnorphic fibrations which give rise to harmonic 'transforms' as we shall see in the following section.
In particular, if G = SLJ(n) all simple roots have coefficient
3.
1
TWISTOR FIBRATIONS OVER GRASSMANNIANS
Let P be a parabolic subgroup of = SL(n,C) then P is the isotropy subgroup of a flag in i.e. an increasing sequence of linear subspaces
0 =V0CV1C...cVk —c n is K-i.
where the height of
Since SL(n,() is transitive on the set of all such flags, we can identify with the set of flags F = {O
W0E W1 c
...
c
WK
=
:
dim
dim
for all 1}.
The complex structure on is that induced by the inclusion of F into a product of Grassmannians given by (W0 E W1 c •.. C
the flag 0 V0 c V1 E ... c VK ually orthogonal subspaces (El,...,EK) with The leg8
E. Now G
of
consist of the Set of mut-
=
= V1 n
SU(n) so that H = SU(n)
that stabilise each E1.
P is the group of special unitary matrices
So if dim
= r1, H
S(U(r1) x
x tJ(rK)).
It Is convenient to identify G/H with the legs of F, that is, with the set F(rl,...,rK) given by
F(rl,...,rK) = {(Fl,...,FK)
1. F3
i
j, dim F1 = r1}.
Then the J1 complex structure on G/H is that induced by the inclusion x Gr +r
x ... x Gn_r(Cfl)
(F1,F1 + F2,...,F1 + ... + 11
Now let no — Zr21, then the synnetrlc space G/K can be identified with the and the fibratlon Gill G/l( is just the projection
lr:F(rl,...,rK) 4
given by F21.
,
It is clear from the constructions of Section 1 that the J2 almost complex structure is obtained by flipping the orientation of the structure on ker dir.
For the c*. of clarity. let us explicitly describe these almost complex structures: Ffrl....,rK) is equipped With tautological sub—bundles F1 of K, whose fibre at is F.. As is well F(r,%...,PN)x C'1; t I known, the complexified tangent bundle of F(rI,...,rK) is naturally isomorphic to z
ij&j
L(F1IF)
a J
1)1.1
— —j
complex conjugation.
where the upper bar space with respect to
Then the (1 ,0) tangent
at (F,,...,Fx) Is given by
Z
and the kerne' of * by E
i-j€2Z whence the (1,0) tangent space with respect to +
E
i<j 1-jE2Z+1
is given by
E
'I
i-jc21
Now suppose that P' is another parabolic subgroup of SU(n) with P' n SU(n) =H.
Then P' is the isotropy subgroup of a flag n
with the same set of legs as
12
V0 E •.. c VK = £
0
SK, the permutation group on K objects, with
so that there is an element c E0(1). =
induced by the inclusion
Thus the corresponding G
r(1)
((fl)
xG
fl
+ •.. +
(F0(1))FØ(1) +
and putting
the corresponding twistor fibration G/H
the projection F(rl,...,rk)
G/K' is
given by
(FI,...,FK) In suneary, GIN
F(rl,...,rK) admits Kt J1 complex structures obtained from the first essentially by permutating the factors E1 and each has a corresponding J and twistor fibration onto a Grassmannian. Now, if K is odd, the height of G /P is even and so admits twistor equivalent J1 structures. These are obtained from J1 by cyclic permutation of the =
The homogeneous fibrations
(F(r1 ,r2,r3) ,J2)
G
r1
C's)
G
r2
(ta)
G
r3
are all twistor fibrations. In general, for K odd (F(rl,...IrK),J?) admits K
twistor
fibrations onto Grassmannians. If K Is even, the homogeneous fibration
given by 1T(Fl,...,FK) = (F1e FK,F2,...,FK1) is J2 holouorphlc and is the 13
only such fibratlon of 4.
OF S2 INTO A GRASSMANNIAN
H%RMONIC
We have seen that J2 holomorphlc maps of an almost Hermitian manifold with fonn into a flag manifold project onto harmonic maps into co—closed Grassmanflians.
In case that the domain is a 2-sphere, all harmonic maps Into Grassmann-
lans arise in this way. Let 4:S2 + G
be a harmonic map.
We identify sub-bundle of
with
on S2.
Let z be an isothermal coordinate
where I is the tautological
-.-',. where ci,8
•
are local sections of 4,1 '(see (3)). Let B* be the adjoint of B, 'then B*ct Is a nilon each fibre of I.
$tent
This follows-since the harmonicity of • implies that B*a is a local holoequipped with a suitable structure, whence aU coefficients of the characteristic polynomial of are holomorphic
morphic
on
S2 and therefore vanish.
then there Is a K such that either .
(a)
a
(b: In
is non-zero almost
and
0
is non-zero almost everywhere.
and
say • has 2K-I; in the second, that • has The. • has nil-order 1 if and only if • is antl—holooiorphic.
case, have
-
-, 6 be harmonic of nil-order K with rank Then tli.re,is a J2 holemorphic map p:S -, (F(rl,...,rK+l),J2) wtth Let
and
14
Almost ewas'3nthere, if 11,(X)
then
-
I E
(Ima(8*ci)l)i
E
Remark
n
The assumption on the rank of a does not
the 1.
applicability of the theorem since if either • or • must sat$sfy It by Propositiofl4.1. Now, if we apply the results of Section 3, we have
I
Let •:S2 G (C'') be harmonic of nil—order n—r. If K is even then there are K other harmonic maps of construct*dkfrom • essentially by differentiation. If I ts odd, there are Theorem 4.4
K—i such
Let us suuuarise our results in Theorem 4.5
Let
+
), a twistor fibration n:F
fold
Them there is a flag maniG (C") and a holI..órphlc map
be a harmonic map.
+ (F,J2) such that = or Further there are associated to
djL
1T04
a number of
of S2
Into Grassmannians all of which are harmonic.
Theorem 4.3 is a generalization and reinterpret*tiem of a theorem of Ramanathan (3) who classified harmonic maps i*to G2(C4). Ramanathan's results provided the inspiration for the above work. Remarks (I)
(ii)
Theorem 4.4 may be viewed as a kind of BScklwSd maps of 2—spheres into Grassmannians.
for harmonic
Referemces
(1] (2] (3]
Bryant,
Lie groups and twistor spaces (to lIpplPat').
Eells, 3. and Salamon, S. Twistorial constnactios of surfaces into four-manifolds (to appear). Ramenathan, J. HarmonIc maps from to G24. J. Duff.
maps of 19
207—219.
(4) Serinnar
f-structures, f-twistor Bianchi (to appear). 15
[5] Salamon, S. Bianchi
[6] [7]
Harmonic and holomorphic maps, Geometry Sentinar Luigi
(to appear).
--
Minimal surfaces and sytmietric spaces, (these proceedings). Wolf, J.A. The action of a real semi-simple Lie group on a complex flag manifold. I: Orbit structures and holomorphic arc components. Bull. Salamon, S.
Amer. Math. Soc., 75 (1969) 1121-1237.
[8] Wolf, J.A.,
Space8 of Conetant Curvature, Publish
Berkeley (1977).
F.E. Burstall School of Mathematics
Un)verslty of Bath Claverton Down Bath BA2 lAY U.K.
16
or Perish Inc.,
J L CABRERIZO & M FERNANDEZ ANDRES
CR-submanifokls of a locally conformal Kähler manifold 1.
INTRODUCTION
Let (V2m,J,9) be an
Hermitian manifold with almost complex structure 3 and Hermitian metric-g and let N be a Riemannian submanifold isometrically inmiersed in V. Recently, in (1), A. 8ejancu the notion of a CRsubmanifold of V. Precisely, N is a CI—submanifold of the almost Hennitian
manifold V if there exists on N a C" holomorptiic distribution D, i.e.: c for any x N, such that its orthogonal complement & in N is totally real in V, i.e., c for any x N, being the normal space at x. If dim 0 = 0, N is called a totally real submanifold and if dim N is called a holomorphic submanifold. Already, many papers on these submanifolds have been published (2, 4, 5, 11]. On the other hand, I. Vaisman and other authors have studied [6, 7, 9]
locally conformal (almost) Kihier (l.c.(a).K) manifolds, which are (almost) Hermitlan manifolds such that every x V has an open neighbourhood U such that for some ci:U + R, is an (almost) kähler metric
If one can take U = V, the manifold is called globally conformal (almost) Kahler (g.c.(a).K). Examples of l.c.K. manifolds are provided by
on U.
the Hopf manifolds.
me elm of this paper Is to start the study of CR-submanifolds in I.c.k. equations of the and with respect to the local
discuss the
with respect to the metric conformal Kihier trics, and we shall establish thereby the analytical conditions that characterize important types of subinanifolds. submanlfold
2.
PRELIMINARIES
Throughout this note all the manifolds and tensor fields are assumed C".
Let (VJ,g) be am al.ost Hermitlan manifold.
((9]) that (V,J4g) closed 1-form
Then, it Is easy to see
a i.c.(a).k. manifold if and only If there is a global
cm V (the Lee form of V) such that dc2
a g.c.(a).K manifold if and only If
Is also exact. p
u,
and (V,J,g) Is
In case w
0 on V. 17
the manifold (V,J,g) is an (almost) Let (V,J,g) be an I.c.a.K. manifold.
manifold. Then, we can deftne ([9]) the vector
field B (the Lee field of V) by g(X,B)
Next, let
(2.1)
w(X)
be the Levi-Civita connection of g and put -
—
(2.2)
+
is a torsionless linear connection on V which is caflad the Weyl connection of g. It is easy to see that Then
-
•
(2.3)
((9]): The almost Hermitian manifold (V,J,g) is an l.c.K. -manifold 1ff there is a closed 1-form w on V such that the Weyl connection is almost comTheOrem 2.1
plex, i.e. (2.4) Let (V,J,g) be an I.c.K. manifold and let N be a RIe.annian manifold iso— .etrlcal!yineersed in V. We denote by g the metric tensor of V well as that Induced on N. Let V the Gauss-Weingarten formulas for N which respect to are given by
+ a(X,Y), +
(x,Y).
—AIX
=
(2.6)
+ +
(2.&)
for
vector fields X, Y tangent to N and F normal to N, where a Is second fundamental form of N with respect to and 0(b) is the normal connectton. The formulas (2.6) are the Gauss-iJeingarten in of The second fundamental teniors are to a, by
(2.7) 18
X tangent to N and
For any
normal to IN put
-
(2.8) are the tanØntla4 and normal comFor the Lee field B put
where PX and FX
ponents of JX (respectively
l
Bx
(2.9)
X ER
(resp. (BX)2) Is the tangential (resp. normal) of an almost Hermitian manifold (V,,J,g), denote J0.L Hence we have In by v the compleetsr'y orthogonal sub-bundle of JD', v.
where
3.
OF THE BASIC DISTRIBUTIONS
INTEGRABILITY
First of all, Lenina 3.1
give sØe general identities.
of an l.c.K. manifold (V,J,g).
Let I be a
Then
we have:
(1,Y)
(3.1)
- lw(Y)X + Ig(X,Y)B1
—
s(X,Y)
•
(3.2)
Ig(X,Y)B2
(3.3)
+
(3.4)
for any vector fields X,Y tangent to N and Proof
This
The following
normal to N.
follows iediately from (2.2), (2.5) and (2.6). Is well known ((2]):
The totally real distribution l.c.K. manifold is integrable. Theorem 3.2
of any CR-submanifold of an
For the holomorphic distribution 0 we have: be of an l.c.K. manifold V and let Let N be a Then the cte. Suppose dim the maximal holomorphic subspace of
Theorem 3.3
19
holomorphic distribution D is Integrable if and only if the second fundamental or equivalently, a(X,JY) - a(JX.Y) + form satisfies = c1(X,Y)B2 = 0, for all vector fields X, V D. If N Is a CR-submanifold, the integrability condition of D In Theorem 3.3 can be replaced by a weaker condition: Theorem 3.4
Let N be a CR-submanifold of an l.c.K. manifold V.
The holo-
morphic distribution D is integrable if and only If g(o(X,JY) —
for all X,Y -
=0
+
D.
Theorems 3.3 and 3.4 follow easily from similar theorems In the
case ((2]), from (3.2), and the fact that locally V is endowed with metr¼s
LeQ1-Clvita connection is
regard to integral submanifolds of Di and I) (provided D is integrable), we have:
For a CR-submanifold N of an l.c.K. manifold V, the leaf
Theorem 3.5
N
if and only if = 0,
+
I.e. g(a(Z,X), JW) = for any XE D, Z,W Proof
Di.
From (2.2), (2.4) and (2.5), for any X
D, Z,W
Di, we obtain
-'
g(JvzW,X) + Ig(Z,W)g(JB,X)
But Ni is totally geodesic In N if and only If
(3.5)
Di for all Z, W. Di
and then (3.5) gives the theorem. Let N be a CR-submanlfold of an l.c.K. manifold V. If the holomorphic distribution D Is integrable and NT is an integral submanifold
Theorem 3.6
of 0, then NT Is totally geodesic in N if aisd only if g(Jo(X,Y) +
to
—
0
for any X, Y
D.
From (2.2), (2.4) and (2.5), for any X, YE D, Z €01, we have
Proof
= g(vx(JY),Z) +
g(Ja(X,Y),Z) + But
NT
is totally geodesic in N if and only if
(3.6)
D for all X, Y
D;
then, (3.6) gives the theorem. 4.
CR-SUBMANIFOLDS
First of all we shall give some identities for later use. Let P, F, t and f be the endomorphisms and vector-valued 1—forms defined by (2.8).
The follow
ing lema can be easily obtained from (2.4), (2.6), (2.8) and (3.3).
fold V.
an isometrically imersed submanifold of an l.c.K. mani
Let N
Lenina 4.1
Then:
t(X,Y)
— AFYX =
+
-
=
+
(4.2)
+
=
(4.3)
+
=
(4.1)
+
'4.4) (4.5)
for
any vector fields X,Y tangent to N and normal to N. Now we shall study totally umbilical and totally geodesic
Let.N be a totally umbilical CR-subcnanfiold of an l.c.K. manifold V. Then we have: (a) either dim = I or the component HJ(TN) of the mean curvature vector H In J(TN) is given by Theorem 4.2
HJ(TN) =
If dim > I and N i.s proper (neither holomorphic nor totally real), such that B Is tangent to N, then N is totally geodesic. (b)
21
Proof (a) Since N is totally umbilical, tangent to N, and hence, for any W
= g(X,Y)H for
X, V
(4.6)
g(c(X,X),JW) = g(X,X)g(H,JW).
From (2.4), (2.6) and (2.7) it is easy to see that (4.7)
= AjwZ'
and then, taking an unit vector field X = Z E (4.6: and (4.7) give
perpendicular to W, (3.3),
g(H,JW) = g(AjwZ,Z)
1W(JW)
g(—4B2,JW)
so that part (a) holds. (bl
Since dim D1 > 1, from (2.8) and (a) we haverth = 0. _töuH = AfflU - PAHtJ,
for any U tangent to II.-
Therefore, for any V tangent to N, =
Now, (4.3) gives
-
(3.2) and (3.3) we
= —g(U,lt)gftI,i}.
(4.8)
Putting V = Pu in (4.8), we obtain
equation holds because Pt
1INIlIt4an manifold (11). the" (4.9) gives
(4.9)
0.
—g(L,P2U)g(H,FI)
any CN-subinanifold of an almost
Moreover, It is easy to see1 ((11)) that P2 = -I-tF
g(U,U)g(H,H) — g(FU,FU)g(H,H) = 0.
SIIØ fro
Is proper, we can choose a unit vector field .X irr D. (4.10) we havkH 0.
(4.10) Thus, FX
0
Let N be a totally geodesic CR-sub.anifold of an1.c.K. manifor all x N, then 0 is integrable and any fold V. Then (a) If integra' submanifold NT of D is totally geodesic In V. (b) If B Is normal submanifold N1 of to N, then any is totally geodesic In V. Theoren4.3
for any x £ N, and in this if and only jf Furthenisor., D Is NT of 0 Is totally geodesic in V. case any integral submanifold Proof
(a)
is tangent to N, from Theorem 3.4, the distribution D Is an Integral submanifold of D. For any X tangent to N, from (2.4), (2.6) and (2.7) we get
Since
integrable.
Let N
YED, Z€
= —g((X,JY),JZ).
By using (3.1) and (3.2), we find g(vxZ,Y) —
+
a -g(a(X,JY),JZ)
(4.11)
0.
0, (4.11) gives g(V ZY) = 0, or equivalently g(V Y,Z) 0. There0. Thus N Is totally geodesic in N and hence in V. (b) If B fore is normal to N, from Theorem 3.5 any integral submanifold NL of 0.L is totally
If X
geodesic in V. and 3.6.
The second statement follows Imedlately from Theorems 3.4
Corollary 4.4 Let N be a totally geodesic proper CR-subinanlfold of an l.c.K. manifold V such that for any x N. Then N is locally the Riemannian product of a submanifold and a totally reel submanifold of V.
From Theorem 4.3 N is locally the product of a. holomorphic submanifold and a totally real submanifold N1 of V. Since w 0 on N, we have induced structure on NT. Moreover, it can easily be seen that the pro-
Proof a
jection map p (or q) onto D CD1) is parallel with respect to local product is actually a local Rieniannian product. Next, we consider the particular case when N is
that this
holomorphic or
'totally real. LenIua 4.5
Let N be a holomorphicsubmanifold of an l.c.K. manifold V.
Then
the sub-bundles TN and T1N of the bundle TV1N are holomorphic. have
(JX,Y) =
(4.12)
=
-
23
.
(4.13) (4.14)
=
(4.15)
=
for any vector fields X,Y tangent to N and
normal to N.
As V is locally endowed with Kähler metrics whose LevI-Civita these fomulas follow from similar formulas in the connection is Proof
case.
Let N be a holomorphic submanifold of an l.c.K. manifold V. (a) The mean curvature vector H of N is given by H =-1B2; (b) N Is totally umbilical If and only if Weingarten's endomorphisms are coninutative. Theorem 4.6
Proof
(a)
If dim N = 2k, let {e1,.. .,ek,Jel,...,Jek} be an orthonormal
basis for TEN, x E N.
Then k
(tr
z
+
k z
(4.16)
from (3.2) and (4.12), (4.16) gives 2kHz be a vector field normal to N. From (4.5) and (4.13) we have (b) Let 0
Hence
=
0, and therefore from (3.3) we have
=
defined in (2.8) induces an The endomorphism f of the normal bundle f—structure in 11N ((10]). For any X tangent to N and normal to N, put =
When
Leema 4.7
24
-
4.16
-
(4.17)
= 0, the f—structure f is said to be parallel ([4]). Let
be a totally real submanifold of an l.c.K. manifold V2m.
Then
for any X,Y tangent to N.
(a) (b)
If n = m we have
=
= v'f.
Cc)
(d)
(JY) =
If the f-structure f is parallel, then for any
=
(4.18)
v.
If Weingarten's endomorphisms are coninutative, then there is an orthonormal local basis in N such that, with respect to this basis, (e)
0. A
0
Je1
= 1,...,n.
I
,
(4.19)
0
b
(a) and (b) follow ininediately from similar formulas in manifolds (see the remark after Theorem 3.4). From (3.4) we easily obtain (c). In order to prove Cd), take X tangent to N. Then (c) gives: = 0. Therefore 0. By using (4.4) this yields 0, and hence from (3.3) we get (d). Finally, to prove Ce), from (4.5) we have 0 for any normal to N. Then we can find an orthonornal local basis in N (with respect to the local Kahlerlan metric = such that with respect to this basis: Proof
—
0
AJe
1
11i0
=
0
Starting from
we can get an orthonormal local basis in N (with respect to the metric g) {e1,... such that with respect this basis (e) holds.
be a totally real and minimal submanifold of an Lc.K.
Theorem 4.8
Let
manifold
such that the Weingarten endomorphisms are coninutative and the
f-structure f is parallel. Then (a) if n
2, N is totally geodesic If and 25
only If B is tangent to N;"ib) If n a
and B Is orthogonal to v, -
I
a geodesic (curve).
N is
then
Since Weingarten's endomorphisms are coninutative, let {e1,... be an orthonormal local basis as in Lenina 47 (e). From (3.2) we have Proof
(a)
0a
a
=
i
i
=
1,...,n.
Thus, from (3.3) and (4.20),
Ajeej =
I
—
=
(4.21)
l,...,n.
Hence, if n 2 and B is tangent to (4.21) gives Aje a 0, j a a a 0 for an3 Moreover, from (d) In Lema 4.7, v. Therefore a 0 for any vector field n normal to N. On the other hand, if there Is x E N such, that (82)X 0, then from (4.21) and (d) in Lenina 4.7, we can take a vector field normal to N such that 0. This gives (a) in the theorem. Let X be a unit vector field tangent to N. Then 0 = g(H,JX) (b) a
g(AjxX.X),
Lenra 4.7,
and hence Ajx = =
= 0
0.
But if B is orthogonal to v,
for any
v.
Therefore
= 0
from
(d) in
for any vector
field n normal to N.
2) be a totally real and totally umbilical submanifold of an l.c.K. manifold such that the f-structure f is parallel. Then N" is totally geodesic if and only if B is tangent to N. Theorem 4.9
Proof
Let
(n
be an orthonormal local basis in N.
Let
Since
is
totally umbilical, by using (3.2) and (3.3), for any X tangent to N we find =
But from (3.3) and (d) in lema 4.7 we also have 26
(4.22)
for any
0,
(4.23)
v.
= 0 for any vector fields ri, t normal to • Therefore, from (e) in Lenma 4.7 there Is an orthonormal local basis
On the other hand, N.
in N such that with respect to this basis (4.19) is verified. But from
we have
ii = 1,...,n..
=
(4.24)
2, we can take j #1 and then (4.19) and (4.24) give A1 = 0, and together with (4.23) we Therefore, A,je = 0, I =
Since n
= 1,...,n.
0 for any normal to N. No' if B is tangent to N, from (3.3), N is totally geodesic. Conversely, if N is totally geodesic, from (4.18) we have 0 = for any 'v. Thus, B is normal to 'v. Moreover, from (e) in Lema 4.7 we can get an orthonormal local basis {ei,...,en} in N such that have A
0 A1
=
l,...,n.
I
=
'I
2, this me ns w(Je1) = 0, i = 1,...,n, so that B is normal to J(TN), and therefore it must be tangent to N. Since n
5.
CR-PRODUCTS
Let P,F,t,f be the endomorphisms and vector valued 1—forms defined by (2.8). Put
vu(PV) -
for all U,V tangent to N. Then P Is called parallel if (2.5) and (2.6) is easy to see that =
(5.1)
= vu(PV) -
+
—
+
= 0.
From (2.2),
(5.2)
But from (4.1) and (5.1) we obtain =
+ AFVU,
(5.3) 27
Definition 5.1
A CR-submanifold of an l.c.K. manifold V is called a CRa holomorphic submanifold
product T N
and a totally real submanifold m of V.
Let N be a CR-submanifold of an l.c.K. manifold V such that the Lee field B is normal to N. Then N is a CR—product if and only if P is
Theorem 5.2
parallel.
Srie
Proof
is normal to (3.3)
llel,
from (5.2) we have
= V'P.
If P is para-
') we find (5.4)
(FV)U.
Fnr any X
0,
Fr.
,
ond then (5.4) gives
+
Q
or, equivalently, g(c(U,X),JV) +
= 0,
for any V tangent to N.
Therefore
u(U,X) =
(5.5)
0, the last equality gives u(X,JY) — cj(JX,Y) = and by Theorem 3.3, D is integrable. Let NT be an integral submanifold of D. For any W D1, (5.5) yields g(c,(U,X),JW) = and hence by Theorem 3.6, NT is totally goedesic in N. Now, let N1 be an integral submanifold of From (5.5), if U Dl, then a(U,X) = 0 and by Theorem 3.5, N1 is totally geodesic. Taking U
Conversely, suppose that N is a CR-product. that B for any X 0 and U tangent to N.
First of all we shall show
Since N is locally a Riemannian product of NT(holomorphic submanifold) and N1 (totally real subinanifold), it 01. In fact, from is sufficient to prove that D for any X c 0 and Z (2.4) and (2.6) we have =
+
(Z,JX) — J(Z,X).
Since N1 is totally geodesic Thus, if W 0, for any W in N, from (3.2) and Thoerem 3.5 we obtain g(vzX,W) Then, from (2.4) we D for any U tangent to N. E V and hence Thus = so that £(U,X) (U,JX). + find JVuX + J(U,X) Then,from (5.1) we get =
for any X
=
—
(5.6)
0
—
0 and U tangent to N.
Similarly we shall prove that and U tangent to N. In fact, Since N is a CR—product it suffices to show this for U *X D. from (2.4) and (2.6), for any V D we find — g(J(X,Z),Y)
=0
where the last equation holds from (3.2) and Theorem 3.6. we have orthogonal to 0. on the other hand, if W
Thus
is
= —g(o(X,W),JZ) +
= 0, i.e.,
But from Theorem 3.5 we have Therefore then
Is normal to N, and
(5.7)
- PVuZ = 0.
Now, from (5.2), (5.6) and (5.7) we obtain Theorem 5.3
= 0.
CR-submanifold of an 1.c.K. manifold V such that
Let N be
for each x N. If P is parallel then N is a CR-product. verse does not hold unless dim D = 2 or B = 0 on N. Proof
The con-
Since P is parallel, from (5.2) and (5.3) we have =
+
-
+
-
(5.8)
IfXEDthenFX =Oand (5.8)gives =
+
-
—
(5.9) 29
for any vector fie1d V tangent to N. —Jci(X,tJ) =
For any Z
From (3.2), (5.9) yields
+
—
(5.10) gives
=
(5.10)
— ig(JB,X)U.
for any U tangent to
N, and hence
(5.11)
AjzX =
Next, for V
D, from (5.11) we have
g(o(X,V),JZ) = 0, for X, V Therefore
-
any
(5.12)
D, Z
Theorem 3.4, D is integrable. ie ul U is totally goedesic in N because d
of (5.12) and iheorem 3.6. Now, let N1be an integral submanifold of D1 .
For any WE
from
(5.11) we have =0
g(Ajz +
and this means that N1 is totally geodesic In N (Theorem 3.5).
Thus N is a
CR-product.
In order to prove the converse, we first give: If N is a CR-product in an 1.c.K. manifold V such that
Lemaa 5.4
N, then
for ,any x
(5.13)
0
(5.14)
£
(5.15)
foranyXEbandZED1. Proof
If X JVZX
30
D, Z c
then from (2.4), (3.1) and (3.2) we have - Ja(Z,X) + Vz(JX)
+ o(Z,JX).
(5.16)
For any W
D1, (5.16) yields: = 0.
= g(V2X,W) = g(AjwZ +
is totally geodesic in N D and this is (5.13). From (2.2), (2.4) and (2.5) we obtain
The last equation holds because any leaf N1 of (Theorem 3.5).
Thus
Now, take X, V
D, Z
g(JVxZ,Y) = _g(AjzX,Y) = -g(o(X,Y),JZ) = 0. The last equation holds because of Theorem 3.6.
(5.17)
If XE D, Z,W
D1, (2.2)
(2.4) and (2.5) give g(JVxZ,W) = 9(AjwZ,X) But from Theorem 3.5 we have
g(AjwZ,X) - g(AjzW,X) =
=0
+
and hence
g(JvxZ,W) = 0.
(5.18)
Now (5.14) follows from (5.17) and (5.18).
Finally, (5.13) and (5.16) give (5.lc). Proof of the converse of Theorem 5.3: From (5.1) and (5.13) we have for any X C and U tangent to N. Put U = V + Z, Vu(JX) where V 0, Z Then, from (5.15) we obtain =
(5.19)
—
But (2.2), (2.4), (2.5) and (2.6) —
=
-
—
+
(5.2(
Then we have
(a)
if dim 0
4 and
0 for
right—hand side of (5.20) does not (b)
N, there are .'a
ED such that
ish at x. Therefore P is not parallel.
if dim D = 2, then the right—lar. side'of (5.20) vanishes, and hence
31
(5.21)
= 0 for any X E 0 and U tangent to N.
and U tangent to N, and therefore
for any Z
But (5.14) implies
= 0.
= Vu(PZ) - P(VuZ) =
(5.22)
Then, from (5.21) and (5.22) we have that P is parallel. References
[1)
Bejancu, A.
manifold, I, Proc.
CR—submanifolds of a
69 (1978) 135-142. Blair, D.E. and Chen, B.Y.
Amer. Math.
Soc.
[2]
Israel (3]
.1. Math.
Cartan, E.
On CR-submanifolds of Hermitian manifolds,
34 (1979) 353-363.
Leqone
our la
doe
Espaces do Riemann, Gauthier-
Villars, Paris .(1946). manifold, I, J. Differential [4) Chen, B.Y. CR-submanifolds of a Geometry 16 (1981) 305-322. manifold, II, J. Differential [5] Chen, B.Y. CR—submanifolds of a Geometry 16 (1981) 493-509. manifold, (6] Kashiwada, T. Some properties of locally conformal Hokkai4o
(7] conformal
[8)
Math. J. 8 (1979)
191-198.
T. and Sato, S. On harmonic forms in a compact locally manifold with parallel Lee from, Kodai Math. J. 3
(1980) 70—82. Kobayashl, S. and Nomizu, K.
-
Foundations
of Differential Geometry,
I and II., Wiley, New York (1963/1969). I. On locally conformal almost KThler manifolds, Israel J. [9] Math. 24 (1976) 338—351. [10) Yano, K. On a structure defined by a tensor field f of type (1,1) Vol
satisfying f3 + f = 0, Tensor [11]
N.S. 14 (1963) 99-109.
Yano, K. and (on, 14. CR—eubmanifolde Boston (1983). folds,
J.L. Cabrerizo -and M. Fernandez Andres Departamento de Geometria y Topologia Facultad de
C/ Tarfia s/n
41012 Sevilla Spain 32
of
and Sasakian
Mani-
R CADDEO
The Almansi property of the distance in the Riemannian context 1.
INTRODUCTION
Let A be the Laplacian of a Euclidean space or, more generally, the BeltramiLaplace operator of a Riemannian manifold. We shall call s-harmonic or poiyharinonic of order s a function f satisfying the equation A5f = 0, where is defined by A(AS_lf)
ASf
for
s
2.
In the theory of polyhannonic functions in an important role is played by the following property. be a system of Cartesian orthogonal coordinates and r2 = E Let i=1
Then one has
Property 1.1 [1] 0
= 0.
In other words, if we multiply an s-harmonic function by the square of the distance from the origin of the coordinates, we obtain a polyharmonic function of order s+1. The interest in this property and its consequences is mainly due to the fact that they provide a very effective method for simplifying the solution of the equation = 0 and, in. particular, that of the equation A2 = 0, which is well known to be of special importance in elasticity theory. Emilio Almansi noted this ,property in 1898, in a paper (1) which can be considered the origin of the theory of polyharmonic functions - a subject th8t has interested many mathematicians, for example:
T. Boggio, E. Goursat, G. Lauricella, E:E. Levi, T. Levi— Civita, R. Marcolongo, P. Montel, P. Pizzetti, F. Sbrana,
1898
L. Tonelli, 0. Venske, G. Vitali, V. Volterra; 1982:
N. Aronszajn, V. Avanissian, S. Bergman, A. Charnes, R. Fosdick, P. Frank, M. Krakowski, R. von Mises, M. Nicolescu, 33
N. Picone, N. Schiffer. The purpose of this coninunication is to discuss the Riemannian analogue of the Almansi property. We briefly describe here some of the results which
are contained in [4], (5] and (7], where the reader can find detailed proofs and complete references. 2.
RIEMANNIAN MANIFOLDS SATISFYING
=
The research of a Riemannian version of the Almansi property leads at once the behaviour of the distance function with respect to the to the stud operator. In fact, the existence of an analogous iterated space, possibly with rk, for some k ER, instead property on a
of r2, imposes rather strong restrictions on the manifold and its dimension, as well as on the class of the allowable functions.
First of all,
rk must be biharmonic.
Thus we have to study n-dimensional
Riemannian manifolds (M,g) that satisfy the condition A2rk
= 0, for some k ER,
(2.1)
at all points m E N and all sufficiently small r, where r(p) = d(m,p) denotes the distance from p to in and is the Beltrami-Laplace operator of N. This problem is related to the problem of determining all harmonic spaces (2), (12]. A hai,wnic apace can be characterized as a Riemannian manifold r alone (that is, for which at all points m one has that does not have any dependence on angular coordinates).
The conjecture
that harmonic spaces are locally isometric either to Euclidean spaces or to —.a rank one syimietric space is still open. It Is easy to find n-dimensional Riemannian manifolds (M,g) with the property that Ark
0,
(k it 0)
holds at all points and for
sufficiently small r. It is not difficult to
show that such manifolds must be flat and k = 2-n. It Is somewhat of a surprise, then, to find that the problem of determining
manifolds that satisfy A2rk = 0 at all points and for all sufficiently small r is considerably more difficult and interesting. It turns out that k must be equal to 0, 2, 2-n or 4-n, and that most
frequently condition (2.1) implies that M must be flat (it is easy to check for these values of k). But, for example, one that rk Is biharmonic on has that (E1)
= 0 for a three—dimensional space M3(A) of constant sectional curvature A, and
(E2)
t.2r2 = 0 for M = M3(Al)x...xM3(Ak) where A1 +
This is easily seen taking into account that the
radial
+ Ak — 0.
part of the Beltrami-
Laplace operator on 113(A) in a system of polar geodesic coordinates has the expression (see, for example [11]) .2 R
2
.
s3(A)
:
a
+
r
+
r
(2.2)
2
Then the proof of CE1) is just a straightforward computation of a2r, and in Theorem 2.1 it will be shown that the converse of CE1) Is true. To prove (E2) we recall that on any product Riemannian Manifold H
Mlx...xMk one
always has
and
On the other hand it can be easily checked that If constant curvature
Ii
-8A
A1
Is a space of
then
I
Hence for H = M3(Al)x...xM3(Ak) we obtain =
+ ••
=
+
+
•.. +
+ •.. + = 0.
Now we state the exact results we obtained studying the kiemannian unanifolds that satisfy condition (2.1). They are Interesting In themselves and recently have been considered in [8) and [13]. We thafl only enunciate the 35
two theorems in which they are mentioned referring to [5] and (7] for the proofs. Let (M,g) be a Riemannian manifo1d with Riemannian connection Vand
curvature operator given by Rxy = V[xy) - [Vx.Vy]. The components of the curvature tensor R and of the Ricci tensor p with respect to an orthonormal respectively, and the scalar curvature basis will be denoted by Rjjkl and will be denoted by 't. With this notation we have: Theorem 2.1
(5]
Let (M,g) be an analytic, n—dimensional Rieninanian mani-
fold such that = 0, k 0, for all m E M and all sufficiently small r. Then there are three possibilities:
(i)
k = 2 and M satisfies
(ii)
k = 2-n, n 3(n—4)1R12
(iii) k
4-n, n
= 0 and
p12
=
1R12;
2: M is an Einstein space that satisfies =
(Sn2
2, n
—
2n-12)1p12;
4: then there are two subcases:
n = 3, and N has constant sectional curvature; n
Remark 2.2
3, and N is locally flat. As we have seen, example (E2) is a non—fkit manifold satisfying
= 0, but a complete classification is not available for this case. 2-n seems to be very difficult to handle, but the calculations (5], [6] gave some support for: Remark 2.3
The case k
Conjecture
Let (M,g) be an n-dimensional, analytic Riemannian manifold such
0, n > 2, for all m (14,g) is locally flat.
that
N and all sufficiently 'small r.
Then
The second theorem seems to reinforce this conjecture, and at the same time reveals a connection between this problem and the theory of harmonic spaces.
[7] Let (M,g) be a Riemannian manifold satisfying the hypothesis of the conjecture. Then (M,g) is Theorem 2.4
36
Ci)
a harmonic space for n odd;
(ii) locally flat if it is reducible; (iii) locally flat if it is three-dimensional;
(iv) locally flat if it 's symmetric. The basic tools used in the proofs of Theorems 2.1 and 2.4 are the power of the distance function and the decomseries expansion of the Laplacian position of the Beltrami-Laplace operato' into its homogeneous components, which are given in the following two lemmas. Let {x1} be a system of normal coordinates at m M. If = ra1, where = 1, then Lenina 2.5
[10)
=
+
:
cz.r1, where
is a symmetric covariant
tensor of type (o,i+1) is determined by the Riemann curvature tensor and its covariant derivativts. [9) Let m M. Then near m the Beltrami—Laplace operator of M (operating on fuflctions) can be decomposed as where, in + E Lemma 2.6
=
a system of normal coordinates {x1} centred at m,
k=0
is given by
-.2 =1
a second order differential operator that maps degree polynomiaTs in normal coordinates into (p+k)th degree polynomials. Using the formula
k
+
(2.3)
and these two lemmas, one finds that the condition o is equivalent to an infinite series of conditions involvin functions of the curvature of M. The theorems follow then from these equations, by a straightforward computation and by integration on the unit sphere of the tangent space TmM (Theorem 2.1), and by a procedure of iterated polarization (Theorem 2.4).
37
3.
THE ALMANSI PROPERTY FOR
I
rk
Now suppose that the function rk is biharmonic on a Riemannian manifold (M,g). to show that in this case rk has the Almansi property with respect to We
some class of functions. In a system of geodesic polar coordinates (r,62,...
at m
N the
Beltrami—Laplace operator has the expression 2
'
1
+
—
(3.1)
Here e is the change of volume factor induced by the exponential map, the prime denotes radial derivative and is the Laplacian of the geodesic sphere Gm(r) of centre in and radius r. In this system of coordinates we have g11 =
= 0 for j > 1,
and
1
where the
are
the inverse components of the
Now, If f is any
function on this chart, we can write grad f = of
•
a
+ grad f,
where
f
.
•
,:
i,j>1
Finally, let < , > denote the scalar multiplication with respect to the metrIc g. Then we have: Theorem 3.1
for all m
Let (M,g) be a Riemannian manifold satisfying A2rk = 0, k # 0,
N and all sufficiently small r. rkA5f
+
+
Then we have
2skrk
(3.2)
+
if and only if =
38
—
(j-21)
2(j—1)(k-1)(k-2) +
2(k;1)
(3.3)
—
j-2
f>,
for every j = 2,...,s. From this theorem there follows iat once:
Corollary 3.2 Let (M,g) be a Riemannian manifold satisfying the hypothesis a (s-1)-hannonic function, s 2, that verifies of Theorem 3.1
condition (3.3).
= 0.
Then
of Theorem 3.1
On any Rietnannian manifold one has
A(f.g) = f.frg +
+ 2
.
(3.4)
Taking into account (2.2), (3.1) and (3.4) we obtain, by induction on s, that
(3,2) is true If and only if (3.3) holds. 4.
AN EXAMPLE
On M3(x) the distance function r is biharnionic.
In this case we have k
and (3.3) becomes =
•
(4.1)
for every j = 2,...,s. This condition implies that f is a radial function. On S3(A) we have then, for example, that Af,Otv'Lr + B
is harmonic,
Arcotv'Lr + Br + C
is biharmonic,
+
Br2 + Cr + D
+
Brk
+
is 3-harmonic
+
...
is (k+1)-harmonic,
for any k, where A, B,C, D,... are constants. References
[1]
Almansi, E. Sull'integrazione dell'equazione differanziale Ann. di Matenvtica, November (1898) , 1-51. 39
Geodesic8 are Closed, Ergebnisse der Mathematik 93, Berlin, Springer, (1978). (3] Caddeo, R. Riemannian manifolds on which the distance function is
[2]
Besse, A.L.
[4]
biharmonic, Rend. Caddeo, R. Sulla
Manifolds
All of Whose
Sem. Mat. Univ. e Politeon. l'orino, 40 (1982),
93-101.
[5]
di Almansi della funzione distanza su una Riemanniana, Conferense delia Uni.v. di Ban, Ban (1984). P (to Caddeo, R., I4atzeu, P. Riemannian manifolds satisfying
(6]
appear). Caddeo, R., Matzeu, P.
[7)
distanza biharmoniche, Rend. Seni. Faa. Sci. Univ. Cagliari, LII (1982). = 0 on a Riemannian manifold" Caddeo, R., Vanhecke, L. Does
[8]
imply flatness? (to appear). Elchhorn, J., Schinining, R. Rlemannlsche Manigfaltigkeiten, bei denen
Riemanniane con potenze
funzione
elne Potenz des Radius k—harmonisch lst(to appear).
[9] Gray, A., Pinsky, M.
The mean
exit time from a small geodesic ball in
345—370. a Riemannian manifold, Bull. Sci. Math., 10? (4) [10) Gray, A., Vanhecke, L. Riemannian geometry as determined by the volumes
of small geodesic balls, Acta Math., 142 (1979),.157198. [11] Gray, A., Willmore, T.J. Mean value theorems for Riemannian manifolds, Proc. Roy. Soc. Edinburgh, 92A, (1982), 343-364. 112] Ruse, H., Walker, A.G., Wilimore, T.J. Harmonic Spaces, Cremonese, Roma (1961).
Riemannian manifolds for which a power of the radius is k-harmonic (to appear).
(13] Schinuning, R.
R. Caddeo
Dipartimento di Matematica Via Ospedale 72
09100 Cagliari
Italy
40
C CURRAS-BOSCH
Infinitesimal transformations with bOunded
norm In
1955 (see [5] or [6])) Kostant proved that, given a Killing field X on a
compact Riemannian manifold, its Ax_operator, defined by Ax Lx =lies in the holonomy algebra at each point. In this paper we consider complete noncompact Riemannian manifolds and we
prove that, for each Killing field X with bounded norm, its Ax_operator also lies in the holonomy algebra at each point. The same result for Killing fields with finite global norm was proved in
[1]. With respect to vector fields with bounded norm it is interesting to remember that each infinitesimal affine transformation with bounded norm on a complete noncompact Riemannian manifold is a Killing field (see [4] p. 126
or (5] p. 244).
-
Preliminaries Let (M,g) be a complete Riemannian n-manifold. At any x of N, 6(x) represents the holonoiny algebra at x, and E(x) the Lie algebra of skew-symetric endo—
If we consider in E(x) the standard Inner product, we have E(x) = 6(x) • Lx represents the Lie derivation with respect to X, and V the covariant derivation with respect to the Levi—Civita connection. morphisms of
Given two forms of the same order • and 4,, we denote by <$,4,> its local is agiven coordinate system then scalar product;
if
dx ld¼...Aclxk,
• 1
i
k
"
4,
= 4,.
.
a given point x
1k
= (1/kl)q, Cx) of M, If M is oriented we denote by <<4,,4,>> the global scalar product, when • or has compact support, defined by =J
d vol.
N
According to this we define the local and global norm as
41
and
2
=
d denotes the exterior derivative and o acting on A5(M) is given by 6
1)flS+fl+1 =
(
* d*.
Let (M,g) be a complete noncompact Riemannian n-manifold and let X
Theorem
be a Killing field whose norm is bounded; i.e., there exists a positive coneverywhere. stant k0 such that IXI < lies in the holonomy algebra at each point of M. Then Ax = Lx We consider the universal covering of M, (M,ir) wi the induced Riemannian metric. The vector field x on M,ir related to ' is also Killing. x t44 x •.. x Mk, where Because De Rham's Decomposition Theorem = is a Euclidean space and each Mf 1 it is well j k, is locally known (see for instance [5]) that can be decomposed the form x = x0 + x1 + ... + Xk* where each X, 0 a k, is a iilling field on each Proof
N. a
-
XI = lxi at corresponding points, the norm of x i also bounded and the same hold5 for each X. As is a Euclidean space, X0 must be parallel Since Ax = A + ... + at corresponding points, our original SO Ax = 0. x1 As
problem can be reduced to a locally irreducible Riemannian manifold; so from now on we have to study if on a locally irreducible manifold it is possible to have a Killing field X, with bounded norm, such that Ax does not lie in the holonomy algebra at any point.
In order to simplify notation we represent this locally irreducible Riemannian manifold by (M,g). G(x) and
where
For each x of M we have
=
+
6 G(x)1.
It is well known that Bx is parallel (see [5]) or [6]) and, as the manifold is locally irreducible, Bx must be q01 where is a given constant and I is a parallel complex structure. Without loss of generality we can suppose q0 = and, as I is orthogonal to G(x) at each x of M, the Ricci tensor must vanish, so the hol.onomy algebra of the manifold (M,g,I) must be either su(m) or sp(k), where 2m = n or 4k = n respectively. Let be the 1-form associated to X via g. Let o be a fixed point of N; p(x) denotes the geodesic distance from o to x, and we define B(r) = r). 1
42
We know that, given two positive numbers r1, r2, 0 < r1 < r2, there exists a Lipschitz continuous functfon f, almost everywhere differentiable, such that, for some constantc > 0 not depending on r1 end r2, we have Idf( 0
f(x)
x
Such a function can be defined by p(x) + r2 — 2r 1
r2 - r1 where • is a smooth function defined on K, with Q
and •(t) * 0 for t
1, q1(t) =
I for t
2 (see, for instance, (3] or (7]).
Now we consider
dlv(f2AxX)
- f25(x,x) + ,
where $ Is the Ricci tensor. Since Stokes' theorem holds for Llpschltz differential forms, taking r2 = 2r, as S = 0, we have J B(2r)
dvol
+
I
JB(2r)
fA
12 X
dvol = 0.
So
=
As Ax Is skew-syninetric, we have =
<<2df A
fAX>>B(2)l.
Schwartz inequality yields I<<2fdftAxx>>B(2)I
2df
IlfAxIl
8(2r)'
(H) 43
but as
From (!) and (U) we obtain IlfAxil B(2r) IdfI < (c/r), we have IlfAxil B(2r)
+ I, we obtain
As Ax =
= n volB(r), so n volB(r)
B(2r)
4(c/r)2
volB(2r), because
<
everywhere.
According to ([2] p. 42) as S = 0 we have volB(r9) ,
if r2 > r
volB(r1)
which is wrong, so Bx must be null.
So (n/4(ck0)2)r2 Examples
Finally we give some examples of complete, noncompact and locally irreducible Riemannian manifolds admitting Killing fields with bounded norm. Example 1
g = dx2
+
In R3 = R2 x dy2
+
R
let us consider a Riemannian metric of the form
f(x,y)2 dt2, where f is a positive, smooth and bounded
function on R2.
is a nonparallel Killing field, whose norm is bounded. If we suppose that grad f has not a constant direction one can prove that
this manifold is locally irreducible. More generally, we could take Fl = N x R, where (N,gN) is a complete + f2.dt2, where f has the same Riemannian manifold, then we define = is a Killing field under the same conditions properties as above, then as above.
let us consider a Riemannian metric of the form g = (ct(t))2(dx2 + dy2) + dt2, with cz(t) a nonconstant, differentiable, positive,bounded function on R. is a nonparallel Killing field with bounded norm. It is not difficult Example 2
In R3 = R2 x
R
to see that this manifold is locally irreducible. 44
More generally, we could take M = N x R, where (N,gN) is a complete Riemannian manifold with some Killing field X with bounded norm (for instance one can take N compact), then define 9M = (cz(t))2gN + dt2, with ci(t) as in the above conditions; then X provides in M a Killing field with bounded norm. References
(1]
C.
Ar-operator on complete Riemannian manifolds (to
appear).
Finite' propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete
(2]
Cheeger-'Gromov-Taylor.
(3]
Riemann manifolds. J. Diff. Geom. 1? (1) (1982) 15-55. Dodziuk, J. Vanishing theorems for square integrable harmonic forms. Geometry
[4] (5)
Vol.
(6)
[7]
and AnalyBi8.
Papers Dedicated to the Memory of V.K. Patodi,
Springer—Verlag (1981) 21-27. Kobayashi, S. Transformations Groups in Differential Geometry. Ergebnisse der Mathematik. Springer-Verlag (1972). Kobayashi, S., Nomizu, K. Foundations of Differential Geometry,
I, mt. Publ. (1963).
Kostant, B. Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold. Trans. A.M.S. 80 (1955) 528-542. Yau, S.T. Some function theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25
(1) (1976) 659—670.
Carlos Departament de Geometria I Topologia
Facultat de Universitat de Barcelona
45
J EELLS
Certain variational principles in R.iemannian geometry 1.
INTRODUCTION
Consider the following two loosely formulated problems: (P1)
LWhen
can a given Riemannian metric be deformed into
a beat metric?
instance, we can ask when a given manifold admits an Einstein metric; or one 'of prescribed sectional or Ricci curvature. Or when can It be embedded in another Riemannian manifold wi,th prescribed properties (as in the problems of Mlnkowski and Weyl). Often such problems lead to a equation of the form For
= p, for a given function ii,.
det (P2)
£Gi van a map $:M
deformed into
N between twO Riemannian manifolds, when can
be
a beat map?
instance, harmonic maps are reasonable candidates for being best. There are others, however. In both problems, "best" means an extremal of a geometrically significant variational problem. Our objective is to exhibit large interrelated classes of such. For
If
Ø:M +
is a map between two Riemannian manifolds (I't,g) and (N,h), we
N
define:
(I) its first
fundamental
(ii) its second fundamental
form
4*h,
a syninetric 2-covariant tensor field on
a syninetric 2-covariant tensor field on II with values in the induced tangent bundle p1TN. form
=
In local charts, * =
ij "ij
(B(A))Y
+
c*B'
subscripts on q denoting covariant derivatives. 46
We are interested in various variational principles associated with inteintegration with respect to the grals of the form =
J
volume element Vg over compact domains of N.
Following a program initiated in [16, §1], we now look at special constructions with substantial geometric content. Given a synrietric function (m = dim N), we define the function : N -*R which at every point of N assigns the a-function of the elgenvalues of with respect to g.
Then
(1.1)
J
is a first order functional of the maps order functionals of the form =
-* N; similarly for the second
(1.2)
J
once we have clarified the meaning of the integrand. Remark
1
of a map
Of course, we can define the kth fundamental form (k 3) N as the k order symmetrized iterated covariant differential
where
(vki)(vl,...,vk)
=
z
and develop the variational theory of In much of what follows we could allow nondegenerate metrics of any signatures. Remark 2
Remark 3
We can formulate our variational principles for sections
of
Rieniannian fibre bundles (38]. We shall not proceed in any of those directions now. 2.
ENDOMORPHISM INVARIANTS
Let V be an rn-dimensional real Euclidean vector space and A:V V a selfadjoint endomorphism. denotes the spectrum (with multiplicities) of A and -' P a syninetric function, then 47
(2.1)
a(A) =
is a well defined real number depending only on A. A well-known theorem of Newton (for polynomials) and Glaeser (18, 24] states that any reasonable syninetric function is a function of the elementpm ary syniiletric functions : P (0 p m), with the convention that = 1.
The characteristic polynomial of A is
det (A - xl) k=O
=
ii
=
(x.1 —
i=1
A Induces seif-adjoint endomorphims :
p
0
A1'V
with A0A = the identify transformation of A°V = P.
sists of the
numbers (Aj ...A1 = Trace
The spectrum of
con-
Thus
(2.2)
=
In particular, = det A
Trace AmA = A1...A and
(m)
(det ADA) (det Am
A) p
(2.3)
.
In terms of an orthonormal base (ej)i<j<m of V we have the matrix representation of A; then — —
(,j)
a
(i)
(J)
sumed over all strictly increasing p-tuples (j) with Its trace I
(Yp
48
j1
<
...
<
is that minor of A determined by the (i)-rows and (j)-columns
Here
(8].
1
(A)—'
-
pa. ...a. 1
i
p
setting
It is convenient to normalize the pQlynomials we have Newton's inequalitIes (20, No. 51) p—1
p+i
p2'
Furthermore, if A is non-negative, then
with equality 1ff A1 = ••• = Am• —1/2
—1/rn
(2.5)
m1
1
Then
—
2.6)
a1.
We define the adjugate of A to be the endomorphism A : V -+ V of the oriented Euclidean space V. given by the composition
Am-ly
>
Am_i
where * denotes the Hodge duality Isomorphism.
Relative to an orthonormal
base of V, the matrix of A has entries which are the cofactors of those of A.
A is self adjoint, and Ok(A)
am_k
1
=
More generally, we have
k
(2.7)
.
m
(Am_PA) * as an endomorphism of
A
det
= (det A)':p_l)
so that =
ak(APA
) °m' 1
k
(m)
Apparently, general formulas for ak(APA) (in terms of the elementary symmetric functions) are not known explicitly. qth The V -, V czeaociated to A is defined by Newton endomorphiem Xq(A)
xq(A) =
j=0
q—j
(2.8) 49
Equivalently, Xq(A) can be defined inductively by
x0(A) = I, Xq(A) = oq(A)1 - Xq_t(A)•A• Then
q aq(A)
o1(A.Xq_1(A));
c1(Xq(A)
-
Cayley-Hamilton theorem).
Xm(A)
The spectrum of Xq(A) consists of the m nUmbers
Other Symmetric Functions
Next, define the symmetric functions lTk(k
0) by the formula
A
=
E
det(A — Al)
1=1
A—
to obtain EA1
,
of non-negative integers such that
suuueed over all m-tuples E
14 uk.
-
Then
i—I
-
-
+
= 0.
(2.10)
Setting —. k
we hove
inequalities (20, No. 220] k
50
1
(2.11)
with equality iff A1
= Am; and if A is non—negative, then
=
—i/k
—1
(2.12)
For k a non-negative integer, polarization gives m
=
=
01(Ak).
1=1
Then (9]
— 0k-1°1 + •.. + (_1)k_101,k_l
+
(_i)kkar = 0,
1
k
(2.13) k > m. —
Wrltlng u3 = (—1)
,j+1
j
1
m, we obtain
k(r +...+r -1)! m
1
r11
...
r
...
rm, m
k
(2.14)
1,
sunuied over all m-tuples (ri,...,rm) of non-negative Integers with m E
1
r = k.
1=1
We have liapounoff's inequalities (20, No. 18], equality 1ff A1 ... = Am•
10k+1'
with
Bilinear Maps
If g denotes the inner product of V viewed as a bilinear form on V, its nondegeneracy provides an inverse linear map
:
V.
Now any symmetric
bilinear form B
:
VxV
has a canonical interpretation as a linear map V V*; consequently the o B is a seif-adjoint endomorphism of V, with composition
o B(u),v> = B(u,v) for all u,v C V. Wc shall write systematically 51
a(B) =
cl(g_1
o
(2.15)
B).
Now let W be another Euclidean vector space, and B V x defined by bilinear map. Then for each w W we have Bw(u,v)
= for all u,v
÷ W
a synliletric
(2.16)
V.
polynomial function a: Rm
For any
V
÷ R we have a(B)
Q*W 4efined
by
(2.17)
a(B)(w)
using the canonical identifications (the use of which
QkW
suggested by J.
= Pk(W)
=
of elements of ?w with polynomial functions of W. In particular, that, that polynomial function identification assigns to v ® ... ® v
... ®
e w>.
for all w
given by w +
=
W
Consequently, if a is k-homogeneous then (2.18)
W.
Alternative definition: a(B) is determined by the ak(B), 0
k
m, and
these can be evaluated as follows: = Trace B(k) +
where B(k):AkV
B
B(u,v) =
(2.19)
characterized.by means of
is
det
A uk, V1 A ... A Vk)
A
The adjugate
ekw,
:
Vx
V
+
of 8
:
Vx
for all u,v
B(u1,v3).
(2.20)
W is.given by
V
V.
(2.21)
composition with a function
f
:
Q*W +
(2.22) 0
gives the real number f(ci(B)). 52
There are many choices available, such as
f(a(B)) =
(2.23)
the norm2 on Q*W induced from the given Euclidean structure of W; or Io(Bw)Idw,
f(a(B))
(2.24)
= J siw
where S1W denotes the unit (n—t)-sphere of
or
f(o(B)) =
(2.25)
In case o is a 2k—homogeneous polynomial, where
zation of 3.
o2kw
is the
the induced inner product of ekw*.
Qkh,
CERTAIN VARIATIONAL INTEGRALS
In General
Let V -t N a Riemannian vector bundle of fibre dimension m over a RiemanaRm -+ nian manifold N and be a syimietric function. If +N
denotes the bundle of seif-adjoint endomorphisms of V and A denotes a section, then we have the density function a(A) N and its integral J0(A)
(3.1)
J
Its Euler—Lagrange operator can be derived easily, using the basic formula (28, Leoiaa A) °
Trace
(3.2)
0
where (At)
Is a 1-parameter variation of A = A0 and Xr_i(A) Is the (r_l)th
Newton endoorphism.
integral of
The adjugate
=
J0
is
(33)
J
If W + N Is another Riemannian vector bundle, we the bundle f:9*W of sp'mnetrlc bilinear maps V x V -' W. Given -sP and a synretric poly-
we can define the variational Integral
nomial
=
J
(3.4) 53
of the sections B of J$(V,W).
Its adjugate integral Is (35)
J
B denotes the adjugate of B, a section of See t17J for varieties of examples different from those considered below.
where
First Order Integrals Example
•:Pl + N between Riemannian manifolds we form the den-
For a map
1
lity a($*h) : M -sR and its Integral as in the Introduction. ai/2 we have the energy integral of •: o =
Jai
Taking
(3.6)
J
Its EuleV'—Lagrange operator is ai(8(4})i known as the ten8iofl
fiel4
of
,; -It Is a section of The adjugate of E is
(,*h)
(3.7)
=
J
Its tuler—Lagrange operator is
(3.8) a
we
obtain the voiwne
integral of •: (3.9)
Its adjugate is =!(4)• We
observe
that if
m
2, then both variational principles (3.6) and
(3.9) are asif—adjuga-te. Example
3
Take a =
54
1
=
k > 0. so j IId$II
(3.10)
k
(Hubert-Schmidt norm as in (3.6)) - not to be confused with I
.
In part-
m and applying Newton's inequalities (2.5), we obtain
icular, taking k
1(4) with equality 1ff • is weakly conformal. widely studied; see [32], for instance. Example 4
For
(,) is
m the Euler-Lagrange operator of I
p
1
That borderline case has been
Op
(3.11)
= —2 Trace v(d4
T0
I,
see [383 for a study of that variational principle. is a confoiinai invariant of (M,g).
If dim N = m = 2p, then
p
Example 5
Let 1 —2
0
in and
p
ni/4p
—
—
- °p-1 °p+13
is
Then
a
coafonnal
invariant of (Meg).
has inte-
ad,jugate
corresponding to the symmetric function
(.2 0.-p
——
.
(p-1)m/2p °m
°m—p+1
Second Order integrals Example 6
If fls the
the tension field of •. we find Taking a = = kth power of the norm (k EP(> 0)), the total kth tenBion inte—
grai of a
N is (3.12)
f studied by Chen [11, 133 and Wilimore [35] in case • is an ininersion.
Its
adjugate integral is J
lkJm_i(B(4)'JtI
If •:H
kvg•
N is an Inmiersion, Its a-mean curvature function in the brackets below. For instance, Example 7
is
the real Is the 55
Llpschltz—Killing curvature of H In N in the normal directions. The total a-mean curvature of$ is
JiJ51i.,1 a(BV(,))dv]v9.
(3.13)
The integrals with a = have been studied by Chern-Lashof (15) for k= and p = m; and by Chen (11] for arbitrary k,p. 1
Example 8
For any map
N then
-
Ji
(3.14)
defined using (2.25). And if 4, is an inrmraion then it is a con— invariant of (N,h). See [14) and [25]. The Euler-Lagrange operator of (3.14) has been computed (for ininerslons) by Karcher-Voss (21], and recorded in [37]. The variational theory of this example goes back to the early 1920s; see Thonisen [31], and in particular the reference there to W. Schadow; then Blaschke (7], Chen (12). For ininersions •:M
of hyper-surfaces In space forms (with constant sectional curvature c) Rund (30] and Reilly (26, 28] have derived the Euler—Lagrange operators for integrals for = f rather general a. For instance, if a then the Newton tensor is divergence free; and the Euler-Lagrange operator Is Example 9
-(p +
+
. pm+1
the functions a2k(B($)). 0 2k m, are (up to constant factors) the intrinsic curvatures of Weyl [34). Similarly, X2k are the Einstein of Lovelock (22]. For
Take for (N,h) the real line. Then is the Hessian of the function $:H -, P. Reilly (27, 29] has calculated/transformed the integrals of based on his formula Trace 1
(P 56
P
,a;
p
a1 b
b
p-i
c
R
He makes several good applications, using p = 1. 4.
OPERATORS
Functions
Given a function •:t4 the function
R, we define its
()*h
density (real form) as
(4.1)
where dot denotes the contraction of the adjugate of the Hessian of • with its first fundamental form. The associated Euler—Lagrange operator is the determinant
(4.2)
Let us compare that variational principle with its standard complex analogue on a manifold H (of complex m) (4-6, 2]. IPt •:M + £ be a complex function. As integrand we have the 2m—form (ddc,)m_l
where d = d" +
A d4 A
dS
and dC = 1(d" - d').
(43) Its associated Euler-Lagrange operator
is (ddc,)m.
(44)
K. Grosse—Srauckmann (19] has derIved (4.2) from a version of (4.1) by taking real slices in the complex form. 1. Aubln (13 and I.J. Bakelman (3) have used a different Integrand in case H • producIng (4.2). As an rn-form their Integrand Is cohomologous to that of (4.1). We prefer (4.1) because of what comes next.
If •:M N is a map, then the contraction is a section of UsIng (2.22), we can then define the density f(B(,).,*h) for any function f L The of -. N can reasonably be defined as
57
(4.5)"
N is a path, then .*h = is'
Example 11
12,
and B(s) = B(s) =
so
Is'!2D5'/dt. Example 12
Suppose that •:M .. N is an ininersion, so that ,*h
g.
Then
=
the adjugate tension field of •. of construction whenever •:14
t
More generally, we can make the same sort is a map with 5*h non-degenerate, forming
any
The case m 2 is especially interesting. Then B(5).,*h is a variation of q. In local charts it has the form (subscripts on denoting differentiation)
- 2B12<52,s1> + BT1<52,52>, (1
If 5:11 -. N is weakly conformal (i.e., 5*h = jig for some function
then its
density becomes
0)),
which is the total
tension density (Example 6) for isometries. Suitable parameter normalization N produces the form (4.2) of its Euler-Lagrange operator, as described In (17, Example 6). of a map •:ii2
-
References
(I]
Aubln, 1.
Equations de
i4elles. J. ?un. Analysis 41 (1981)
354—377.
equations. (2] Aubin, 1. Nonlinear Analysis on Manifol.da. Springer (1982). equat(3] Bekelman, I.J. Variational problems and elliptic ions. .1. Diff. Ceo. 18 (1983) 669-699. (4] Bedford, E. and Taylor, B. The Dirichlet problem for a complex Mongeequation. mv. 3? (1976) 1-44. E. and Taylor, B. Variational properties of the complex Monge(5] equation. I. Dirlchiet principle. Duke Math. J. 45 (1978) 375403. 58
Variational properties of the complex MongeIntrinsic forms. Amer. .7. Math. 10 (1979) 1131-
(6) Bedford, E. and Taylor, B.
equation. II. 1166.
(7] Blaschke. W.
(8] (9]
Vorleaungen
(1929). Bourbakl, N. Bourbaki, N.
fiber Differentialgeometrie III. Springer Hermann (1948).
aeaquiiin&ziree at for,nae quacfrotiquae.
Hermann
(1959).
(10) then, B-V. On theorem of Fenchel-Borsuk-Willmore—Chern—Lashof. Math. Ann. 194 (1971) 19—26. (11) then, B-Y. On the total curvature of imoersed manifolds I-Ill. Amer. .7.
Math. 93 (1971) 148—162; 94 (1972) 799-809; 95 (1973) 636—642.
An Invariant of conformal mappings.
(12] Chen, B-V.
Proc. Amer. Math. Soc.
40 (1973) 563—564.
On a variational problem on hypevsuvfaces.
(13] Chen, B-V.
.7. London Math.
Soc. 6 (1973) 321-325.
(14] Chen, B—V.
and their appli-
of
Some conformal
cations.
Dci. U.14.I. 10 (1974) 380.385. (15] Chern, S—S. and Lashof, R.K. On the total cwrvatur. of limnersed manifolds I, II. Amer. J. Math. 79 (1957) 306'318. moh, J. s 5—12.
(16] tells, J. and Sampson, J.H.
of Riemannian manifolds.
Harmonic
Amer. .7. Math. 86 (1964)109-160.
(17] Fells, J. Variational theory in fibre bundlest Gao. Math. Thys.
Math.
P2'oc.
3.
Diff.
(1983)
Reidel
148-158.
[18]
6.
Fonctions composdes dlffdr.ntlables.
Ann. M1th.
(1963)
193—209.
(19]
K.
Variationai.prcp.rtiea of
equationa.
M.Sc. Thesis, Warwick (1984).
(20] Hardy, G.H., Littlewood, J.E. and Polya, 6.
InequàUtiea,
(1934).
(21] Karcher, H. and Voss, K. J H2dA und
Oberwolfach.
(1972).
(22] Lovejock, D. 12
The Einstein tensor and its
.7. Math.
PP4JB.
(1971) 498—501. 59
Combinatory analyeia I,
(23]
MacMahôn, P.A.
[24)
(1960). Malgrange, B. Ideal8 c'f
II.
Cambridge (1915, t916),
Functiona.
Tata Inst. Bombay
(1966).
(25]
Extrinsic rigidity theorems for compact submanifolds of
Reilly, R.C.
J. Diff. Ceo. 4 (1970) 487-4W.
the sphere. [26]
Variational properties of mean curvature. Proc. 13th
Reilly, R.C.
Canad. Math. Congrea8.
[28)
(1971)
114.
Reilly, R.C.
On the Hessian of a function and the curvatures of Its
graph.
Math. J. 20 (1973) 373-383.
Mich.
Reilly, R.C Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Diff. Ceo. 8 (1973) 465477.
(29]
ReIlly, R.C.
(30]
Rund, H.
Applications of the Hessian operator in a Riemannian manifold. md. Math. J. 26 (1979) 459-472.
Invariant theory of variational probi.me on eubepaoee of a nt3nifold. Harrb. Math. Einzelschriften No. 5 (1971).
[31] Thomsen, G. Grundlagen der konformen FlSchentheorie. Ab. Math. Sem. Rastburg 3 (1924) 31—56. (32) Uhienbeck, K.K. Minimal spheres and other conformal variational problems.
Ann. Math.
Studiee 103 (1983) 169-176.
Einlge differentialgeometrlsche Kongruenzsatze fdr geschlossene Flkhen und Hyperflkhen. Math. Ann. 3 (t956) 180-218. [34) Weyl, H. On the volume of tubes. Amer. Math. J. 61 (1939) 461-472. (35] Willmore, T.J. Tight Ininersions and total absolute curvature. Bull. London Math. Soc. 3 (1971) 129—151. (36) Willmore, 1.,]. Mean curvature of Riemannian iu,uaersions. J. London (33]
Voss, K.
Math. Soc. 3 (1971) 307—310. (37) Wilimore, T.3. Total curvature in Riemannian geometry.
E. Horwood
Series (1982). [38]
energy—related functional., and their vertical variational theory. Thesis, University of Warwick (1983). Wood, C.M.
SGne
Eells Mathematics Institute University of Warwick Coventry CV4 7AL U.K. 60
F GOMEZ & .M MUNOZ
The topological dimension of the cut locus I NIRODUCT ION
We study in this paper the possible numbers appearing as topological dimensions of cut loci relative to any Riemannian metric on a given compact connected smooth manifold.
Let N be a compact connected smooth manifold of dimension n and let C(g,x) denote the cut locus of x E P1 relative to a Riesnannian metric g on N. Thus
C(g,x) is the set of cut points for all geodesics emanating from x (a cut point for a geodesic starting at x being the last point to which this geodesic minimizes distance). The point x in N may be left fixed since the above set of numbers coincide for any two points x and y in P1 (one simply consiiers a diffeomorphismfof P1 sending x toy and observes that C(g,y) a
f(C(f*(g),x)). The main results we prove are contained in the following two theorems: Let M be a compact connected smooth manifold of dimension n and let m be the least positive integer such that the mth homotopy group of M does not vanish. Then dim C(g,x) n-rn, for any Riemannian metric g on N. As a consequence of Theorem A one deduces, using duality and the Hurewicz isomorphism theorem, that either dim C(g,x) n/2 for all Riemannian metrics g on M or N must be a homotopy sphere. Theorem A:
There exist Riemannian metrics on the n-sphere, with its usual smooth structure, whose cut loci have any prescribed dimension from 0 to n—I. Theorem B:
1.
PROOF OF THEOREM A
We begin with two lemmas. Let B be a smooth real vector bundle of rank r and let C be a subset of Ewhose topological dimension is less than r. There exists then a smooth cr6ss-section s of such that (Im s) n C = Lemma 1:
Proof:
Assume first that
is trivial; E = B x
Rr.
There
then v £ 6t
such that (x,v)
C for all x
Rr would
with Rr, contradicting the hypothesis on the dimension of
B, because if not the projection of C onto
C.
In the general case, choose another vector bundle
such that
S
trivializes and define C' as the set of points z1 • 22 in S such that E C. It is clear that dim C' = dim C+ rank rank • c'). Thus, we can choose a smooth cross-section s 0 s' of • such that (Im(s • s')) n C' = 0.
Therefore s is a cross—section of
and (In s) n C
0.
Lema I implies that a vector bundle whose rank is greater than the dimepslon of its base, admits a cross-section without zeros. One simply takes as C the image of the zero-section. Remark:
Lennia 2:
Let M be a connected smooth manifold of dimension n and let C be a subset of N. Assume that dim C < n-rn, n > 2m and the mth homotopy group of H-C equals zero. Then the mth homotopy group of N is zero. Proof:
Let
i!m(H,xo) with x0
(N,x0) be a continuous map representing an element of C.
5m
Since n > 2m and
is compact we may assume that
Is a submanifold of N and f is the Inclusion. neighbourhood of Sm in N and use Lenina
such that Urn s) n C = 0. (Sm,X0)
1
Let U
5m
be a tubular
to choose a cross-section s:STh
U
The composite map
e-4(N,s(x0))
is homotopic to the constant map Sm -, s(x0), since we have by hypothesis = 0. But F:Smx(O,1] U given by F(x,t) = ts(x) shows that the
Inclusion f is w-homotopic to s with w:(O,I] Therefore f is c&-homotopic to the constant product of w and the constant path at s(x0).
U given by w(t) ts(x0). 5rn -. s(x0), where cz is the Thus f is 8—homotopic to the
Is the loop at x0 product of the path and the inverse of w. Finally, using the fact that the loop B is homotoplc to the constant loop at x0 (x0 fixed) and that all points in Sm are nondegenerate (see (3] Ch. 7 §3), we conclude that f is homotopic to the constant map Sm fixed), constant map Sm
Proof of Theorem A:
where
If n > 2m, Leimna 2 implies dim C(g,x) n-rn. If n = 2m, applying Lemea 2 we deduce dim (C(g,x)xR) in+1 so we have dim C(g,x) 62
together with the Hurewicz isomorphism theorem implies that N Is a hocnotopy n-sphere. Therefore m n and Theorem A also
If n < 2m, holds.
Observe that the proof of Theorem A really shows n—rn, where m is the least integer such that the mth homotopy that a(14) is defined as the minimum of the topological group of N is not zero and dimensions for all closed subsets C of M such that M-C is contractible. n-i for all Riesiiannian metrics g We clearly have 0 dim C(g,x) cg(M) onM. n-i if 6 is not (b) For a compact connected Lie group G we have a(G) simply connected and n-3 n—i if it is, because 1t2(G) 0 and a(G) Conmients on Theorem A:
2.
(a)
PROOF OF THEOREM B
Cluck and Singer (see [2]) construct a Riemannian metric g on (n 2) whose cut locus from x0 (O,O,-i,O,...,O) is an arc contained in the great = 0} and consequently of topological dimension 1. circle Sn fl {x2 ... Furthermore the metric g is obtained by deforming the round metric in a neighbourhood of x3 • 0) so as to connect E and 6' geodesic fields G and G', where 6 joins the points of E(S'1) to joins the points of C(g,x0) to E(S"). Moreover 6 and G' are Integrable In
the sense of being transverse to a foliation. Assume Inductively that a Riemannian metric g is already given on
such that the following conditions are satisfied: (a)
dim C(g,x0) = k, I
(b)
C(g,x0) c
(c)
The metric g has being obtained by deforming the round metric in fl {x3 = 0), so as to connect fields a neighbourhood of E(S") = and of geodesics 6 and 6', where 6 joIns the points of to
k
= (O,O,—1,O,...,O).
n-I, where
n {x2 = ...
0).
G' joins the points of C(g,x0) to (d)
6 and 6' are integrable in the sense of being transverse to a
foliation. n+i
satisfying We are going to construct, then, a Riemannian metric g on S the above conditions substituting k by k+I, thus closing the induction and proving Theorem B. 63
be the inclusion of Sn as the equator of
Let l:S11
i.e.
Consider the rotations about the n—plane
x1•
0 of = (x1cos
a).
generates Under these rotations 1(S") generates = transEach fl 0), and C(g,x0) generates a subset denoted by forms the geodesics of S into geodesics of connecting the point i(x ) Similarly the geodesics of 5' transforms into geodesics of to ). connecting the points of to the points of Denote by
the fields of geodesics obtained by rotating Sand 5'
about the n-plane x1 = 0. The fields and are both integrables: because It consists of geodesics of starting at the point 1(x0) and 5, because its orthogonal hypersurfaces are those of 5$ rotated In 5n+1• Thus we may apply Theorem 2 of (2] and deform the metric of in a neighso as to connect and G' in the new metric g'. It bourhood of
remains to check that C is the cut locus of g' from the poInt 1(x0), since conditions (a) through (d) are clear. We show first that c C(g',i(x0)). Let = with x C(g,x0). Either x is the first conjugate point of x0 along a minimizing geodesic y or there are at least two minimizing geodesics joining x0 to x. In the first case = $a(X) is the first conjugate point of i(x0) along the minimizing geodesic and so C(g',i(x0)); In the second case the images under of two minimizing geodesics from x0 to x are two minimizing geodesics from 1(x0) to and so x C(g',i(x0)). Finally, to prove that C(g',i(x0)) c let y be a geodesic in starting at 1(x0) and l.t be the cut point of 1(x0) along y. Suppose Is reached before C. If y is any point in y between and let y' be a minimizing geodesic from 1(x0) to y. Both y and y' belong to the new geodesic field which connects and G', and since the geodesics In G" never intersect before we conclude that y rivst be an arc of Therefore y minimizes after in contradiction with being the cut point. Therefore
64
3.
FINAL REMARKS
Is possible to show Following a construction of Gluck and Singer in (2] the existence of a Riemannian metric on any smooth manifold N, which is to be also compact, whose cut locus from a point has dimension n-i. be interesting to know, as for the case of the sphere, whether at not It it is possible to find Riemannian metrics on N whose cut loci have any prescribed dimension from the minimum value up to n—i. In Section 1 of this paper, this mininum value?
Is cz(M), as defined
References
[1] Cheeger, J. and Ebin, D.G.
Theorema in Geometry. North Holland, (1975). [2] Gluck, H. and Singer, D. Scattering of geodesic fields, I. Ann. Math. 108 (1978) 347-372. [3] Spanier, E.H. Algebraic Topology. McGraw Hill (1978). Francisco Goinez
Department of Mathematics
Faculty of Sciences
University of Spa in
Miguel
Department of Mathematics
Faculty of Sciences University of Palma de Mallorca Spain
65
A GRAY
Erstaz Chern polynomial and Weyl's tube formula 1.
INTRODUCTION
One of the main reasons why the study of Kähler manifolds is richer than that of general Riemannian manifolds is the existence of Chern forms. Although of a manifold P of dimension 2P coincides with the top Chern form 1p the Euler form x (which is defined for general orientable Riemannian mani-
folds), it is not in general possible to define the other Chern forms Y1,...,Yp—I.
Nevertheless there is an interesting curvature polynomial k(P,t) which is defined for all compact Riemannian manifolds, orientable or not. This polynomial, which we shall call the ersata Chern polynomial, reflects many of the properties of Chern forms, the most important of which is the Whitney sum formula. The polynomial k(P,t) arises naturally from the study of Weyl's tube formula and also is relevant [12] to the study of Chern's kinematic formula (1]. The polynomial k(P,t) is interesting because of the following result: Let P and Q be Riemannian manifolds for which the ersatz Chern polynomial is defined. Then Theorem 1.1
k(P x Q,t)
k(P,t)k(Q,t).
k(P,t) = pk(P,t)
(1.1) (1.2)
for a p-fold covering P. Formula (1.1) is proved in (9] but a more transparent proof is given in Section 2. Formula (1.2) is an easy consequence of the definition of k(P,t). To explain k(P,t) let us first look at Weyl's tube formula for a complex submanifold P c C".
When P has compact closure (and even more somewhat.
generally) one has [6, 7]
(r) =
66
r j
'P
y
(irr' + F)'1,
(1.3)
is the total Chern form of P and F is the form of P). Furtherform of C" (which when restricted to P becomes the where
=
+ ... +
I +
more the volume of P
vol(P) =
by
eF.
F" J
=
(1.4)
J
(In formulas (1.3), (1.4) and subsequent formulas we assume that all relevant integrals converge. Furthermore the integrands, if non-homogeneous are to b? expanded in power series and all terms not of degree P are to discarded.) It is important to observe that whether or not P is actually a submanifold of C", both (1.3) and (1.4) are meaningful; in particular (1.3) can be regarded as a definition for any n and any compact Kähler manifold. In fact if we sum (1.3) over all n we obtain n
2
(r) =
y A eF.
(1.5)
JP
n=O
This leads us to consider y(2rrt) AeF
k(P,t) where y(t) =
k
=
c=D
(1.6) is the Chern polynomial.
c=O
Even though (1.6) is not available in general, It is still possible to define k(P,t) for a large class of Riemannian manifolds including all compact Riemannian manifolds.
Definition Let P be a Riemannian manifold of dimension p. polynomial of P is
The ersatz Chern
C
where
'P t' —— cU2clJ 1
1
C2C(RC dP
'
and R is thi. curvature operator of P. Here c2C(RC) denotes the complete contraction of the power of R. (See for example E2, 3].) S4e show in Corollary 2.2 that, for Kähler manifolds, (1.5) reduces to (1.4). Thus the ersatz Chern polynomial is defined for all manifolds for 67
which the relevant integrals converge, and generalizes the Chern polynomial Of course the main difference is that the y(t) = I + ty1 + ... + ersatz Chern polynomial does not give rise to the cohomology classes Nevertheless it appears to be an important function of t11] [In_I:l. curvature associated with any Riemannian manifold. Furthermore when P is compact and even dimensional the top coefficient in the ersatz Chern polynomial k(P,t) is, by the Gauss-Bonnet theorem, equal to where X(P) is the Euler characteristic of P. So the ersatz Chern polynomial can also be thought of as an extension of the Euler characteristic which depends. on curvature. The ersatz Chern polynomials for low dimensions are as follows: dim p
k(P,t) = vol (P).
0,1:
dim P = 2, P compact:
k(P,t) =
dim P = 3:
k(P,t) = vol (P) +
dini P = 4.
Jt dP.
vol(P) +
P
TdP
+
4ir2t2X(P).
Jp
dim P = 5:
k(P,t) = vol(P) +
Jp
tdP
Zr +
)p
{11R112—411p112 +
t2}dP.
It is it. eresting to compare the ersatz Chern polynomial with Weyl's tube formula, which can be rewritten as n
(hp) — c=O
k
2c
(P)
,
2
(2)C
where now p is the real dimension of the Riemannian manifold P.
(18 Again it is
to observe that if P is not given as a submanifold of Rn then (1.8) can be regarded as a definition. Furthermore k(P,t) determines Vr(r) for n. Conversely If is different from zero for 0 c then (r) determines k(P',t). It should be pointed out that there Is a nonintegrated version of the ersatz Chern polynomial.
Definition Let R be a tensor on a vector space V that has all the symetries 68
of a curvature tensor.
Put 4C
t(R,t) =
c c=O
2
cU2c)!
Clearly
(1.9)
k(P,t)
= J
is the curvature operator of P. In this paper we shall establish some of the properties of k(P,t) includ-
where
ing the relation between the ersatz Chern polynomial and Weyl 's tube formula. We compute k(P,t) for certain spaces such as spheres and complex projective spaces. For example, when P is a sphere, k(P,t) is a variant of a Hermite polynomial. Finally we make some remarks on the ersatz polynomial of symmetric spaces. 2.
PRODUCT FORMULAS
In this section we sumarize and sharpen some results of ( 9]. In particular we give a more transparent proof of Theorem 1.1. Let R be tensor on a vector space V with inner product < , > that has all the usual properties of the curvature tensor field of a Riemannian manifold. We say that R is curvature—like. R:A2(V)
-.
Thus
we may regard R as a symmetric linear mapping
A2(V) that satisfies the (first) Bianchi identity. It is possible power RC of R (for example, see t3).) This is a symetric
to define the
Let C2C(RC) be the complete contraction;
linear mapping thus C2c(Rc)
=
The polynomial 9.(R,t) has a much simpler description than that given in
Section 1. For this we first need to extend R to a map (also to be denoted by R) from MV) + MV). We put E
c R.
C =0
Then an easy calculation shows Lemma 2.1
We have
= tr (etR)
(2.1) 69
and so
k(P,t) =
r I
Jp
tr
(etR
P
(2.2)
For I = 1,2 let R1 be curvature like tensors defined on vector spaces V1. Leimia 2.2
Proof
is multiplicative.
The polynomial
We have
etR2)
• R2,t) =
tr(etRl)tr(etR2) = t(R1,t)&(R2,t).
=
Now Theorem 1.1 follows from Leimna 2.2 and (1.9). 3.
GENERALIZED VOLUME FUNCTIONS
It will be convenient to study a function slightly more general than that given by (1.8). We put
VP (r,t) =
[hp)
k
c=O
(3.1)
=
When P Is a complex submanifold of real dimension 2p this formula becomes yA Jp =
1
+
(3.2)
y(t) A (1rr2 + F)'1.
There is a simple relation between the generalized volume functions and the ersatz Chern polynomial. Leimna 3.1
We have 2
k(P,t) =
n E
n—p even
70
(r,t)
(3.3)
Proof
This is a simple calculation using (3.1) and the Taylor expansion for
If P is a complex manifold then
Corollary 3.2
L
=
=
-co
=
E
JP
n=O
y(t)J.
(3.4)
manifold we see that k(P,t), as defined by (1.7),
Thus when P is a
reduces to (1.6). It is worth remarking that just as the exponential function gives rise to the ersatz Chern polynomial, it is possible to write the right—hand side of WeXl's tube formula as the integral of a Bessel function. In fact:
tr =
J
For details see [8). 4.
THE ERSATZ CHERN POLYNOMIALS OF SPHERES AND COMPLEX PRO1JECTIVE SPACES
denote a sphere of radius R of dimension p.
Let
Theorem 4.1
The ersatz Chern polynomial of =
2
(t)2P
is given by (4.1)
H
/21 DP2C1 r\
=
c!(p-2c)!
c=O
•$enotes the Hermite polynomial of degree p.
where
cuefficients
(i).
fo
V'
(r)
can be 'umputed from Weyl's formula
On the one hand we have
[hp) c=O
2C4ic!k 2c
-
—
(4.2)
(2c+1)!
But this volume can also be written as the difference of the volumes of spnees: 71
4
4
(r) =
V",,
S"(R)
—
'
(R_r)P+t}
(4.3) EIP]
—
p+1
p—2c 2c+1
c=O
Upon equating like powers of r in (4.2) and (4.3) we find that k2
cI(p2c)I
C
Thus
R)P2C
=
c=O
Accorldng to RainvIlle [13, p. 187] the [hp] H Cx)
Hermite polynomial is given by
,,,
(45)
E
cUp—2c)!
c=O
Now (4.1) follows from (4.4) and (4.5).
The first few of the ersatz polynomials of spheres are: k(S0(R),t) = 2,
k(S1(R),t)
k(S3(R),t)
=
k(S5(R),t)
= 1T3(R5 +
k(S6(R),t)
2ir2(R3+3Rt), lOR3t
Ji!L (R6
+
k(S2(R),t)
2i11,
k(54(R),t) +
15R4t
=
+
=
4it(R2+t), 6R2t +
3t2),
15t2), +
9OR2t2
+
15t3).
The 3rsatz Chern polynomial of complex projective space sectional curvature is pinched between x and 4x) is given by
(where the
Theorem 4.2
(2tA)C = A
Proof
The Chern polynomial of CP"(X) is y(t) = (1 +
4.2, follows easily from this fact and (1.6). 72
Now Theorem
is a Riemannian fibre bundle with E, F and M compact.
Suppose that E
Is it the case that (4.6)
k(F,t)k(M,t)?
k(E,t)
This would be a nice generalization of the product formula (1.1). It is well vol(F)vol(M) and that xlE) = x(14)X(F). However In genera known that vol(E) there is no product rule for the Poincaré polynomial x(E,t), even though
In fact it Is easy to construct a
x(P x Q,t) = x(P,t)x(Q,t) [11, p. 277).
For this we use the Hopf fibratlon. The metrics are chosen so that the fibres are totally geodesic and the fibrarion is a counterexample to (4.6). Riemannian submersion. Theorem 4.3
The generalized product formula fails for the Hopf fibration (even though
s3(R)
vol (s3(R)) = vol(S1 (R))vol Proof
We have
= (2,rR)( R2+4irt) but k(S3(R),t)
21T2(R3+3Rt).
To describe the ersatz Chern polynomial of a synmetric space, it is
necessary to extend a little the definition of the bracket of vector fields. M)2C
For a differentiable manifold 14 let Pf C
2 ci
GE$2c
a
(X
ci
÷ x(M)C be defined by )N. . .#(X
,X 01
-
02c-1
3,
,X
°2c
Clearly Pf1(X,Y) CX,Y). Let II be a symetric space and write 14 = G/K. Then each tangent space of 14 can be identified with m where g = k • m is the decomposition of the Lie algebra of G. where # denotes the
4.3
For a locally syninetric space M of compact type with curvature
tensor R we have
Rc(Xl...X2c)(yl...y2c)
(Cl)2
(4.7)
for Xi•••X2c Em; 73
c2C(RC)
(4.8)
;
2
c=O
k(P,t)
C
(2c)!
(4.9)
II
vol
It is well
that the curvature of a sywetric space is given by <[W,X], [Y,fl> for W,X,Y,Z m. So for XI...X2C,Yi...Y2C m we have
Proof
known
RC(Xi...X2c)(Yi. =
'I
I 0
I
2-4c
a, =
n C
a
C
a
II
i,j=i
R(X
, X
021_i
021
)(Y
,Y
02j-i °2j
)
n
a
a
II
i,,j=1
<X
021_i
,X 021
], (Y
°2j-1
, Y
02j
(cD2,
proving (4.7). The proofs of (4.8)-(4.1O) now follow easily. (The proof of (4.7) follows (iOh in this paper a similar calculation is used to show. that the cth sectional curvature of a synmietric space of nonnegative sectional curvature Is also nonnegative.) References
Chern, S.S.
On the kinematic formula in integral geometry, J. Math
16 (1966) 101-118.
(2] Gray, A.
A generalization of the theorem of F. Schur, J. Math.
Soc.
Japan 21 (1969) 454-457.
Some relations between curvature and characteristic classes, Math. Ann. 184 (1970) 257-267. (4] Gray, A. Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula, Topology 21 (1982) 201-228. (5] Gray, A. Volumes of tubes about submanifolds expressed in terms of them Classes, J. Mzth. Soc. Japan 36 (1984) 23-35.
(3] Gray, A.
74
A product formula for the volumes of tubes about manifolds of complex Euclidean space, Riv. Mat. Univ. Pared (4)
sub-
[6] Gray, A.
10
(1984).
(7] Gray, A. Volumes of tubes about complex submanifolds of complex proJective space, Tran8. Amer. Math. Soc. Gray, A. Weyl's tube formula as the integral of a Bessel function (to (8] appear).
(9) Gray, A. and Lee, S.Y.
Product formulas for tubes, P'roceedinga
Conference on Differential Geometry (1983) 7.7—85. [10) Greub, W. and Tondeur, P.
of the
and ite Applicatione, Nove Meato
On the sectional curvatures and characteristic
of homogeneous spaces. Proc. Amer. Math. Soc. 1? (1966) 444-448. (11) Hu, S.T. flamotopy Theory, Academic Press (1959).
On Chern's kinematic formula In integral geometry, j. Differential Geometry 9 (1974) 475-482. E.D. Special F'unctione, Chelsea Publishing Co. (1971). (13] (14) Weyl, H. On the volume of tubes, Amer. J. Math. 61 (1939) 461-472.
(12] Nljenhuis, A.
Alfred Gray Department of Mathematics
Unl"ersity of Maryland College Park, MD 20742 U.S.A.
75
JFTLOPERA
Conformal vector fields on a certain class of nilpotent groups 1.
INTRODUCTION
Y. Haraguchi studied a set of nhlpotent groups of dimension pq+p+q, = H(1,p), we prove: denoted there (H,p,q), p,q N. Writing In
Proposition 1
If a differentiable vector field on the group
zi
'p =
{
o o
I
oij
;
x, z
pP,
the p x p identity (1)
is conformal with respect to any of the Riemannian metrics + dy
0 (dzi_Axldy),
dy +
(2) zT
X,t €R—{O}, XT =
=
V=
isometry for that metric.
then It is necessarily an
tare left invariant if and only if A = 1. Therefore, The metrics (for every discrete subgroup F of the quotient R$emannian manifolds being the Riemarinlan metric induced by acting on the left
it,
on
have no other conformal vector fields than infinitesimal isometries.
We remark that (H0,g2t) is not a locally symetric Riemannian space. Let us note that Proposition 1 cannot be deduced from a well-known theorem by N. Tanaka (7, Th. 2] dealing with Riemannian spaces of type In the sense of Sasaki-Yano; indeed, one has:
Proposition 2 The Riemannian spaces (Hpi9xt) are not of type L
has no other projective vector fields than infinitesimal isometries, through an explicit computation of T. Hangan (4] has proved that (H1
76
those fields. A Bouyakoub has extended that result for We study the projective and Killing vector fields of
in another
examples to In [1], Cordero-Fernandez-de Leon use manifolds. of compact The group H1 is the well-known three-dimensional Heisenberg group. Howis not isomorphic with the (2p + 1)-dimensional Heisenberg ever, for p > 1, group. To appreciate the differences between both groups in terms of the contact geometry, compare the papers of Lutz (6] and Goze-Haraguchi [3).
paper.
2.
BACKGROUND AND NOTATION
Let us recall that on a Riemannian manifold (M,9) a vector field
formal if and only if the Lie derivative of g in the direction of
is con-
satisfies (3)
=
for some real function
The relation between
and
is .
(4)
is the divergence of
and m the dimension of M. be a moving frame on an open set U of M, {81,•••,Øm} the Let dual moving coframe. Let v be the Levi-Civita connection of (N,g) and write,
where
for every vector field =
1
on M,
90i —
'k —
8
=
=
1,
—
=
,
i
a, where a = aike •
=
8k
(*)
k
and so on for any other tensor field on M. We also recall that for any
vector field
on M one has
=
(5)
+
(*) Sums must be read when repeated indices appear up and down on different factors. Exceptions to this rule will be specified. In the remaining
situations, E is employed.
In this section each object is
77
If Is a conformal vector field on (N,g) and the comRonents of the are and {e1,..., curvature tensor of V with respect to then, from (3) and (5), (6)
+ 9ik*,j -
+
of this system of equations in Any solution Integrablitty conditions, partial derivatives must satisfy the following obtained from (6) by means of the Ricci identities:
where lpk
(i,j,k,l); l,j,k,l = 1,...,m;
k < 1,
(7)
lii +
-
+
+ (l/2)(9ik*jl - 9il*,Jk + 9j1*,lk - 9jk*,il) the components of the covarlant derivative of
=
the curvature tensor). 3.
For details and more information see (2].
COMPUTATION OF COEFFICIENTS OF EQUATION (7)
From now on denote (p) = {1,...,p}; y = I
I'
p + 1, x'
(p)). Let us choose, as global moving coframe on
= z1 (for every {e°,01,•..,02P)
given as follows: e0
dx0
= dy;
= dx1, i
01
(p);
i
=
I
t(dx1_AxldxO),
R {0).
(p); A ,t
defined by
The global moving frame
a/ax'
=
(p); X1.
(1/t):/axl' = ai'' i
X0=a0+AE k=l
(p);
XakI
is left—Invariant if and only if A = 1, dual of {0a. a = O,1,...,2p), orthonormal with to the metric 9A,t'. (2), and their Lie are
78
= 0, i,j,k
=
(p)
(X1,X0] = AtX1, I E (p)
In the sequel we will write s = At for short. Let V be the Levi-Civita connection associated to
{O,l,...,2p}.
a,b,c
VXXb =
and write (9)
It is imeediately seen that jS —
I
0=
1'
0 =
=
0
= s/2, I £ (p), (10)
0 In the remaining cases. Up to the Identities RabCd
_Rbacd
_RabdC
Rbadc
RCdabI
the only nontrivial components of the curvature moving frame
are —s2/4
R1jJ.1
i,j R0101
(p)
(11)
—3s2/4
By virtue of (10) and (11) it follows that the components Rabcde of the covariant derivative of the curvature tensor are nufl when the indices a,b,c,d,e (0,1,...,2p} verify one of these two conditions *: (1)
Two or three indices belong to (p) (respectively: to (pu) {lu,...,ps)) and at least two of the others belong
to (Pu) (respectively: to (p)). (Ii) Two or three indices belong to (0} and at least two of the remaining ones belong to (p) (respectively: to (p')). In particular, we have, for every a *
However, VR
(0,1,...,2p},
0. 79
a
0
R
j'kP,m = RaSIk,l,ffi
= Ra jk'l,m' = R R
a
=R
j'oi',o
a
4.
= R OOj,1
=
=
=
"
a
jOl,0
,
OjO,1
PROOF OF
From
a
j,k,l,m
(p)
(12)
PROPOSITION I
(3) and (5) we deduce p,
We
(13)
a=
of the integrability conditIons (7) In order to prove notice that in the following ones the is zero by virtue 0f (12), a - O,l....,2p.
come
that the function ip
coefficient of (0,1,1,0)
must be
(l',i,i,i')
,ll
+
(i',O,O,i')
(O,i',i,O)
,l'j'
+
+
+
+
(14)
(p)
+
=
=
0, i
(p)
(15)
=
0.
(r)
16)
1
= 0.
+
1
(p).
(17)
From (13) and (16) we obtain, respectively, =
=
=
i,j
(p)
(18)
i,j
(p).
(19)
Bearing in mind (18), (15) and (16), we arrive at =
80
(P).
•
(20)
Using (13), (14) and (20), we have
(3s2/2)*=
(21)
(p)
I
(p) together with (16) on'y once, we Suming (15) and (17) for all i find, by virtue of (19), (20), (21) and (4), +
O
I fixed in (p),
+ 2iP
o = (4s2/4)g'
(p),
i
+
(22)
By (10),
a = O,1,...,2p, therefore equations (20), (21) and (22) can be written (p)
I
+
=
(23)
k=l A2 =
—
+
82)*
0, j
(p),
where, for any real function
(24)
82
5s2/4
(25)
defined on
From (24) we deduce +
*=
(p)
I
(26)
meaning the non-dependence of the two functions f,
on the variab'e
Let us Introduce (26) into (25): +
+
+
B2g)(21), i,j
therefore, once i,j are fixed, we see -
+
B2f = 0,
+ B2g
= 0,
j
(p).
(28)
On the other hand,
81
kI)(e
x
xkfOk,
+
=
f+ (xk)2fk.k.)
+
e (p).
+
From this equality and (24), (27), (29) we obtain o=
- A2f
xkfOks -
+
X2
(xk)2B2g),
(29)
this Identity being true for every i (p). If we fix the values of all the variables except x1, for an arbitrary index I given a priori, then
(29) can be read in the form P(x1) +
o=
Q(x1),
x'
(30)
with P and Q polynomials in the variable x, both of degree two. Since the exponential function is not algebraic, it follows that P(xi) = 0, Q(x')
0, x'
and thus
f=0, g=0, that is to say, (26),
= 0; this ends the proof of Proposition 1.
of any of defined by the left action For all quotient manifolds by the induced on its discrete subgroups, 1, the Riemannian metric locally and (r-.H , left—invariant metric (2), makes (H exist no isometric. Therefore, we derive from that conformal vector fields on other than infinitesimal isometries. A direct computation shows that on the left—invari,ant vector fields 1
(p), satisfy
=
ea
82
0, a = O,l,...,2p
and therefore they induce p independent infinitesimal isometries on any of those quotient Riemannian manifolds. For examples of uniform discrete subsee (3]. groups of 5.
PROOF OF PROPOSITION 2
Let us recall that an rn-dimensional Rfernannlan space (M,g) is of type
the sense of Sasaki-Yano, k E 14 u (o'}, if its Ricci tensor R is zero, (type or If there exists a (1,1)-tensor field H on N of constant rank k veryifylng: (I) H Is parallel with respect to the Riemannian connection, (ii) H2 = H, (lii) g(H(X),Y) g(X,H(Y)) for all tangent vectors X, Y on N, and (iv) given a moving frame {X =
(k
on an open U of H, the components
and
=
— (m — k —
—
satisfy
=
1)gjh(6hj —
(31)
(type
of Its
it is ininediate that the
Coming back to
Ricci tensor, with respect to the global orthonormal moving frame
given in SectIon 3 are, by virtue of (11), Rab
0 if a $ b, a,b = O,1,...,2p; 2
= s2 /2,
R11 = —s /2,
1
R00 = —ps2/2,
(32)
(p),
is not of type We will show that, for every k 7, a parallel (1,1)-tensor field H verifying condition (31) cannot exist, independently of the nature of its rank and of (ii). In fact, from (31) we therefore
defined in Section
derive, using the orthonormal moving frame 3,
+ (2p — k)óab)/(2P - 1), a,b = O,1,...,2p,
Hab =
(33)
and thus, some components of the covariant derivative of H are different from zero; for example, we have, because of (33), (32) and (10), H
II
•1'
H)
(H
— H'
(no gum)
0 =
s3/2p—1,
I
(p), 83
which Is not zero, (s types for k not
A, t
is
R.{O1, see (2)). Therefore,
14.
Acknowledgement
We are indebted to 1. Hangan and R. Lutz of Mulhouse for encouragement and aid, to L.A. Cordero and the Departamento de Geometria y Topologia of
Santiago for broad support, and to A. Bouyakoub,
11. Goze and
14. de Leon
for fruitful conversations on this subject. References
(1]
[2]
Cordero, L.A., Fernandez, N. and De Leon, N. Examples of compact non— almost Kahier manifolds, Proc. Amer. Math. Soc. (to appear). Elsenhart, L.P. Geometry, 5th edn, Princeton U.P. (1964)
(first published 1927). [3]
Goze, M. and Haraguchi, Y. Pari8, 294,
[4]
Hangan, 1.
(1982)
Sur les r-systèmes de contact, C.R.A.S.
95-97.
Sur les transformations
Heisenberg, Rend,
Sam.
de l'espace de (1984). des structures de contact,
Fac. Sci. Univ. Cagliari
(5]
Haraguchi, V.
(6]
Lutz, R. Quelques remarques sur la geometrie metrique des structures de contact, in Geometrie Synrpiectjque at de Contact, Coil. "Travaux en cour8", Hermann, Paris (1984) 75-113. Tanaka, N. Conformal connections and conformal transformations, Tran8. Amer. Math. Soc., 92 (1959) 168-190.
Sur une
Muihouse (1981).
[7]
J.F.T. Lopera Institut des Sciences Exactes et Appliquées tjniversitg de Haute Alsace.
4, rue F4res
68093 14.alhouse
France
NOTE ADDED IN PROOF: These results have been extended to a larger class of
generalized Heisenberg groups, which includes A. Kaplan's groups of type H (Geometria Dedicata, 1982) and Haraguchi's groups H(p,q), proving that there is no conformal global transformation on them when endowed with suitable metrics. Our technique here also works at the infinitesimal level in this more general situation.
84
A MACHADO & I SALAVESSA
Grassman manifolds as subsets of Eucidean spaces 1.
INTRODUCTION
Let E be a Euclidean space. Following Palais, we identify each vector sub— space F of E with the orthogonal projection nF:E + F. In this way, the Grassman manifold G(E) of all vector subspaces of E appears as a submanifold
of the Euclidean space L(E;E) of all linear maps from E into E (with the Hubert—Schmidt inner product). The aim of this paper is to present some explicit formulas concerning the differential geometry of G(E) as a submanifold of L(E;E). Most of these formulas extend naturally to the case where E space, although in this case there is no Is an Infinite dimensional natural inner product in L(E;E). 2.
NOTATION AND PRELIMINARIES
LetE and F be finite or infinite dimensional Hilbert spaces. We will denote by L(E;F) the vector space of all continuous linear maps from E into F. If C L(E;F), we will denote by L(F;E) its adjoint linear map, the one defined by the identity =
The following identities will be used quite often: -
idE* = idE.
(2.1)
= The map L(E;F) -' L(F;E), L(E;E) is eelf—adjoint if + is a real linear map (even if E and F are complex spaces, it is not a complex linear map) and the set Lsa(E;E) of seif—adjoint linear maps is a real vector subspace of L(E;F). In case E or F is infinite dimensional, we will look on L(E;F) merely as a Banach space (with the sup In case E and F are finite dimensional, we take in the finite dimensional vector space L(E;F) the Hubert—Schmidt
A linear map
inner
prothwt, defined
by
85
(2.2) 1
xl,...,xk is an arbitrary orthonormal basis of E.
We will use the
following Identities concerning these Inner 4A°X,fl>
=
oA*>.
(2.3)
The word "manifold" will always mean an embedded submanifold of some
finite dimensional or Banach vector space B and the tangent vector spaces will be considered as vector subspaces of the ambient vector space B. In
fact, one can even define, for each point a of an arbitrary subset N of B, a notion of tangent vector subspace
differentiability (see, for example, (2]).
which behaves well with respect to
In the same spirit, by vector
bundle we will mean a vector sub-bundle of constant one. A vector bundle E where each is a vector subspace with basis M will be a family a fixed finite dimensional or Banach vector space E, verifying the usual properties, and we will use the same symbol E to denote the corresponding subset of Mx E. It will be useful to allow a vector bundle to have as basis an arbitrary subset M of a finite dimensional or Banach vector space B. If = (EX)XEM Is a vector bundle with c E, we identify a connection N, which is a bilinear in £ by its second fundamental form at each point x x E such that map (2.4)
is
each smooth section W = (Wx)XEM of E, the covariant derivative given by the formula For
-
=
(2.5)
If E is a Hubert space, the metric connection of E is the one defined by the condition that if is the is orthogonal to the fibre is a smooth map from N into L(E;E) and we orthogonal projection, then x nX have the following formula for this connection: =
We will use also the following characterization of a connection e in the vector bundle E = (Ex)XEM, where M 86
(2.6)
tensor of a manifold
and
E :
assuming that x
is a smooth map from H into the space
L(E,B;E) of bllineIr maps, such that each curvature tensor is the trllinear map x
is a restriction of
the
x
defined by =
(2.7) 3.
THE GRASSMAN MANIFOLDS
Let E be a finite or infinite dimensional real Hubert space.
For each closed
vector subspace F E, we will denote irE the orthogonal projection from E onto F. We have hence a natural bijective map between the set of closed vector subspaces of E and the set of orthogonal projections. We will denote
by G(E) the subset of L(E;E) whose elements are the orthogonal projections onto closed subspaces, and we will call G(E) the manifold of E.
The fact that G(E) is indeed a manifold is proved in Akin (1], who attributes this result to Palais (unpublished preprint), but we will sketch here an independent proof.
The following characterization of the elements 0f G(E) is well known: 3.1.
belongs to G(E) if and only If it is self—
A linear map
adjoint and verifies
o
=
We can consider a morphism from the constant vector bundle
with
basis G(E) and fib1e E, into itself, associating to each
G(E) the linear map + E. The fact that the image of an idempotent morphism is a vector bundle allows us to state: 3.2. There exists a tautologioal vector bundle with basis G(E), whose fibre in each 1TF is F. metric connection, we deduce: Using formula (2.6) for 3.3
The metric connection of the tautological vector bundle is defined by =
8?
G(E), w
each
and
As a corollary of the local constancy of the dimension of the fibres of a of G(E), whose I vector bundle, we see that, for each n, the subset elements are the 1TF such that F Is n—dimensional, Is open in G(E). Let F c E be a fixed closed vector subspace. It is a well known simple
linear algebra result that, for each closed vector subspace G c E, the following two properties are equivalent: (a)
(b)
E
eG
(direct sum);
is an isonorphism from G onto F;
anu that, if they are verified, the projection E
G associated to the direct To each a L(F;F1) we associate its graphic sum is = {x + which is a closed vector subspace of E verIfying the conditions above. Inversely, for each closed vector subspace G c E verifying the conditions above, there exists one and only one a whose ° graphic is namely a "F1 We well-known considerations in the proof of the
I be a H,lbcrt space and let F cE be a closed vector subc be the set of the orthogonal projections space. Let G(E) such and that E = F1 e *(E). Then is an open subset in G(E), containing defined by there exists a diffeomorphism that verifies (I)F(1TF) = 0.
The considerations before the statement show that is a bijective associates to each map from onto L(F;F1), whose Inverse *F :L(F;F'1) + o the orthogonal projection onto the closed vector subspace {x + cz(x)IXEF. is open In 6(E) and that both and All we have to show Is that are smooth mips. For that, we consider the morphism from the tautointo the constant vector bundle FG(E) logical vector bundle whose value at G(E) is • th. fact that e if and only if the flbre of the morphism at is an Isomorphism implies that Proof
the Is open In 6(E); takIng the restrictions of ths vector bundles to fact that the inverse of a (smooth) Isomorphism is smooth Implies that the L(F;E), is also Is smooth, hence map UF 88
-
Now, we have an Injective morphism from the constant vector bundle FL(F.F•i•) Into the constant vector bundle EL(F;PL)I whose fibre at a €L(F;F1) smooth.
Is the linear map F + E, x -' x + (1(x), hence the image of this morphism is and this implies that the map a vector bundle with basis + L(E;E) is smooth. As a corollary, we have: 3.5.
If E is a real Hilbert space, then G(E) is a manifold in L(E;E).
If E
E is n-dimensional, then the dimension of G(E) at
is N—dimensional and F is n(N—n).
3.6.
let E be a real Hilbert space, F c E be a closed vector subspace and + be the diffeomorphism defined in 3.4. For each and we have
n
=
-
In particular, Proof (See
=
+ L(F;E) be the smooth map defined by
Let
the proof of 3.4). = w,
let w
F arbitrary.
=
Differentiating the identity
we obtain =
0,
FL.
On the other hand, we have a smooth section of the tautological vector bundle associating to each Its hence
covariant derivative with respect to the metric connection, which, by (2.5) and 3.3, is equal to must hence belong to
projection of The fact that
onto F4 associated to the direct sum E shows that hence
=
is also the projection of
associated to the direct sum E = F4 made before 3.4, this projection is equal to onto
is the
We can now conclude that
and, by the considerations
89
—1
To
show that
maps F into obtain complete.
—1
(w)) -
/F
=
—
(w)))).
it will be enough to know that each
I
(G(E)) 1TF
To see this, we differentiate the identity F o = and ° hence n ° ° n and the proof is = = -
+
We present now several equivalent characterizations of the tangent vector spaces to G(E).
Let E be a real HUbert space and let F E be a closed vector sub— space. The tangent vector space (G(E)) is then contained in the vector
3.7.
F
space Lsa(E;E) of self adjoint maps and, for each n conditions are equivalent: (a)
n
(b)
0(F) c
n
F;
WF = (Id_hF)
(Id-iTt)
(e)
(f)
o
n
Lsa(E;E), the following
;
° =
(2hF1d)o n.
=
The fact that each
(G(E)) is contained in Lsa(E;E) is a consequence of the fact that G(E) c The equivalence between the four last conditions is trivial. Assuming (a), we obtain (c) simply by differentiating the identity F in the direction of n. It is readily seen that at Proof
condition Cd) Implies that 0(F) c that condition (e) implies that c F (Id_wr Let us prove now that condition (b) implies condiis a diffeomorphism from the open set tion (a). The fact that In G(E) onto implies that + is an isomorphism. We can hence take n' c
(G(E)) such that F =
Then n' is
and verifies condition (b), hence
+ F is
the adjoint map to fl',F:F .+ We deduce now that
= n/ri
hence rì
=
F Is the adjoint map to and the proof is complete. S
To feel what is happening, assume that £ is finite dimensional and take an orthonormal basis Xl,...,XN of E, whose first n vectors constitute a basis for F. •Then the matrices of are respectively Id_IF and Remark.
[Id 10
01
10
0
[Id
1
0
-Id
10
and condition (b) says that the elements of I (G(E)) are the linear maps F
whose matrix has the form [0 LA
4.
0
THE DIFFERENTIAL GEOMETRY OF GRASSMAN MANIFOLUS
4.1.
Let E be a real Hubert space and let F c E be a closed vector subspace. Lsa(E;E) the following conditions are then equivalent:
For each
(a)
ri(F) c F and n(ri) c
(b) (c)
n.
0 (Id—irF) = (Id_1TF)
We will denote by I
(G(E))1 the set of seif-adjoint linear maps nEL sa (E;E)
verifying the preceding conditions.
The fact that (b) and (c) are equivalent is trivial. It is readily
Proof
seen that (b) implies one sees that
for x
c ri.
F and that (c) implies
0 r(x) = n(x)
Assuming (a),
F
=
for arbitrary x and (b) is
0
proved *
4.2
Let E be a real Hilbert space and let F c E be a closed vector subspace. (E;E) is the sum of the closed vector subspaces I (G(E)) and sa
Then I
T(G(E))1 and iT
1
:L5 (E;t ) -• T(G(E)) and
"F
Lsa(E;E)
associated to this direct sum are defined by
91
(ri) = (Id—lrF)
'
(ri)
°
o(Id_flF)
+
°
+
=
Conditions (a) of 4.1 and (b) of 3.7 show that the intersection (6(E))1 is {O}. It is readily seen that, for each (6(E)) n
Proof
F_
F
n E L sa (E;E),
it
(n) applies F into F1 and F1 into F and (ii)
into F and F1 into F1, hence F
-
(ri) applies F (G(E))'.
(G(E)) and F
All we have to note now is that, for each ri,
(ri)
+
F
F =
ii.
F
If E is a finite dimensional real Hubert space and if we consider in
4.3
Lsa(E;E) the Hubert-Schmidt inner product, then, for each vector subspace F c E, the subspaces (6(E))1 of Lsa(E;E) are mutually (G(E)) and orthogonal, hence each is the orthogonal complement of the other. Proof
TitF(G(E)) and n'
Assume n
Choose an orthonormal
TitF(G(E))'.
basis Xl,....XN of E such that the first n vectors constitute a basis of F and the last N-n vectors constitute a basis of F1. Conditions (b) of 3.7 and (a) of 4.1 assure that, for each 1 k N, = 0, hence 0 (2.2)). The preceding result explains why we employ the notation
(6(E))1 and
F.
If E is a finite or infinite dimensional real Hilbert space we will define connection in the manifold 6(E) as the one that verifies the (6(E))' of the linear map condition that 0 (ri,c*) belongs to the kernel •ITF F ii :1. (E;E) I (G(E)), for each ii and in (6(E)). In an analogous way
the canonical
F
to that used in the case of a metric connection, it is easily seen that this connection is synluetric and also is defined by the formula (4.1) 'tF
=
This is the connection that we will always consider in the Grassman mani-
fold 6(E). Of, course, in case E is finite dimensional, this connection is the metric connection with respect to the Hilbert-Schmidt inner product. We can obtain a more explicit formula for the connection on 6(E) by calculating the derivative in (4.1), using the formula in 4.2, 92
0
= (Id-c) 0
o (Id—c).
o
This gives °
-
+
=
Ii
(4.2) =
0
+
us now obtain, using (2.7), two formulas for the curvature, the first for the metric connection of the tautological vector bundle, and the second Let
for the canonical connection of the Grassman manifold. In the first case, = for each w we take E and (cf. 3.3) obtaining B(a(w)) -
=
ct(B(w)),
in
and w
=
(Id-2E) °
°
o
=
for each a and
a Id
In the second case, we take
and obtain, noting that
and that, by 3.7(f),
conrutes with the composite
of any two elements of 0 a
=
o
B-rI
° B ° a +B ° a °
—
a°
(4.4)
° n.
Assuming that E is finite dimensional, we obtain, for the sectional curvatures: = 2 -
=
2.
(4.5)
prove it, all we have to do is to apply the formulas in (2.3), remembering that a and are seif-adjoint. The fact that (aoB)* = B oa implies that To
ao
B
and B o a have the same norm and we can hence apply Cauchy-Schwartz to
conclude that One
0 and
=
0 if
and only if aoB
= Boa
can also establish easily the following formula for the Ricci curvat-
ure:
Ricci
=
±..?.
(4.6)
where N Is the dimension of E. Grassman manifolds (or, more precisely, their connected components) are sometimes represented as homogeneous spaces of the orthogonal group.
The
following considerations will compare this approach with the one we are using.
Let E be a real Hubert space and let 0(E)c L(E;E) be the orthogonal 93
= + E such that group, I.e. the set of the toplinear isomorphisms It is well known that 0(E) is a manifold (a Lie group) and that, for each
L(E;E), we have
(E) and a
if and only if ct*
a
o
+
a = 0.
(4.7)
In the case where E is finite dimensional, the Riemann structure in 0(E) induced by the Hubert—Schmidt inner product is readily seen to be bi-invarare defined by iant. The orthogonal projections -' (4.8)
-
Even in the case E is infinite dimensional, we define projection maps by formula (4.8) and we have an associated symetric x connection In 0(E) defIned by the bilinear maps -
=
(4.9)
Of course, in the finite dimensional case, this will be the metric connection. Now assiine that E is a finite or infinite dimensional real Hubert space is a' fixed closed vector subspace. We can define a smooth map 0(E) the orthogonal projection onto G(E) associating to each •:O(E) denoting by ir:E H the orthogonal projection, it is easy to see that and H c
E
we have
(4.10)
Although • Is not a totally geodesic map, we can nevertheless state:
G(E) has totally geodesic fibres and, in case E is finite 4.4. •:O(E) dimensional, is a Riemannian submersion. Proof
a
Given a
3.7(c) and (2.3),
94
is defined by
The derivative linear map +
and B
° it
C
arbitrary, we obtain, using (4.7),
=
+
=
+
+
=
+
=
=
where, using (4.7), we can see that adjoint linear map
Hence, the
c
is defined by
ciT _80
= 280
It is not difficult to verify
that
= 8,
which means precisely that
is a Riemannian submersion.
To prove that 0(E) and let 00(E) be the fibre of over 00(E)isatotally geodesic subnianifold of 0(E), all we have to see is that, for each a and 8 in we have where we see that the is the connection of 0(E). Using the formula for fact that a and 8 are in is equivalent to Let
and, using the same formula, oneconcludes easily that, if (B,A) then (8,A)
T(
0
CE))) if and only if
X*0
a0iT08*+(30iToa*+AoiTo
and the characterization of
Now, using formula (4.9) for given in 4.7, we obtain Uo
iT
0 8* + 8 ° a ° it
-
-
=aono3* + 8olroa*+ +
+ 30 iT 0
+ 8*0a0 +
=ao,ToB* + Bolroa* —
+ —
1fioifoB*_ 95
and the proof is complete. We are going now to present a formula for the geodesics in G(E) with arbitrary initia' conditions. Let E be a real Hubert space. 4.5. For each E G(E) and f:R + G(E) defined by
f(t) =
there exists a smooth map
(Id +
+
and f is a geodesic of G(E) that verifies f(O) = Proof
and f'(O)
We note first that, from 3.7(f), we conclude that
wrth
coninutes
It is now trivial that f(t)
and anti-coninutes with
Is seif—adjoint and, noting that
= Id and -
= Id,
+
we obtain
f(t)of(t) =
(Id +
+
+
+
+
sin
+
+
+
+
whence we conclude that f(t)
f'(t)
G(E).
f(t),
+
Next, we see that 2
= (cos
in particular f'(O) =
—
Next, we obtain
fM(t) = (—2sin(2tri) On the other side, remembering 3.7(f), we have 96
+
2
-
f'(t)0f'(t) =
cos(2tri) ° n
— -
o
+
°
°
+
=
2
°
+
+
=
sln(2tr,)2) °
=
and, using (4.2), we have now
+ f'(t)0f'(t)o(Id—2f(t))
=
+
2
a
—
-
=
—
whence we conclude that f is indeed a geodesic.
4.6. Let E be a real Hubert space. G(E) is then a syninetric space and, for each it G(E), the symetry Sym:G(E) G(E) with respect to it is by
= (Id-2it)
°
° (Id-2g).
It is trivial that
Proof
(Id-2n) o
(Ld—2it)
SymU) E G(E). =
D
is a seif-adjoint map and.the fact that
= Id shows that
o
is trivial to see that Sym(ir)
hence
ir and that
We have
(Id—2n) ° a ° (Id-Zit), 97
(4.2),
hence, vD
=
=
(Id—2n)o(Id—2it)
—
)
=
-
+
(Id—21T)octo$o(Id-2w) —
-
+
- (Id_2rr)oczoBo(Id_21r) + 2(Id_2i)oct
=
0,
that is to say, Sym is a totally geodesic diffeomorphism. Now, if -* 6(E) is a geodesic with f(O) = iT and f'(O) = we have
f(t) =
(Id +
(2iT1d)ocos(2tfl) +
hence
Sym(f(t)) =
+
(Id+(2iT—Id)°cos(2tfl) —
-
and
the proof is complete.
5.
THE COMPLEX GRASSMAN MANIFOLDS
= f(—t)
Assume that E Is a complex HUbert space, whose inner product will always be denoted by < product
,
Then E is also a real Hubert space,wlth the inner
Re
and the following two facts are trivial: 98
(5.1)
If F c
Is a complex vector subspace, then the orthogonal projection ir:E .* F Is the same when we consider in E either the complex or the real Inner product. 5.1
5.2.
E
If
+ E is a complex linear map, then the adjoint map + E is the same when we consider E to be either a complex or a real HUbert space. We will denote by the vector space of all continuous real linear maps and by its vector subspace whose elements are the complex linear maps. In the case where E is finite dimensional the Hubert-Schmidt inner
product that we will consider in L(E;E) will be the one associated to the real structure of E and we will consider in the closed subspace Lc(E;E) the induced inner product.
If E isa complex Hilbert space, we will denote by
the set of the
orthonormal projections Onto closed complex vector subspaces, and we call G1(E) the complex Graeeman nvinifold of E. G(E) will denote the real Grass-
man manifold of E, i.e. the Grassman manifold of E, when consldere€I Hilbert space. It is trivial to conclude that
(5.2)
G(E) A
All that has been said in SectIon 3 applies mutatie rautandie to the com-
is only a real plex Grassman manifolds, but one must be aware that the complex vector space Lc(E;E). The essential reason for
manifold within
this is the fact that the map + is not (—linear, but it was natural to anticipate this because, in case E Is finite dimensional, (like G(E)) is compact (because It is closed and bounded) and it is well known that there exists no compact nontrivial complex submanifold of a complex vector space. For each closed complex vector subsapce F c morphism *F:UF +
where
Is open in
E,
we still have a dlffeoand contains (cf.
3.4), hence: 5.3. If E has complex dimension N and F c E has complex dimension n, then the real manifold G1(E) has dimension 2n(N-n) in The tangent vector space T
is contained in the real vector space
Lcsa(E;E), whose elements are the seif-adjoint complex linear maps and, for Lgsa(E;E), the fact that E is equivalent to each of the each n
conditions (b) to (f) of 3.7; in other words: 99
I
(5.3)
(G (E)) = I (G(E)) n L1(E;E).
is only a real submanifold of Lt(E;E), it admits a complex
Although
structure: 5.4. Let E be a complex Hilbert space. Then the real manifold G1(E) admits a complex structure defined by the linear maps
= In
e
are in fact
For this structure the real diffeomorphisms IpF:UF + holomorphic. Proof
into itself we use 3.7(f), rememberId and noting that
aoplies
To see that
ing that =
in
=
It is also trivial that -n. The fact that this almost complex structure is indeed a complex one comes froiii fact that the real diffeomorphisms *F:UF + L(F;PL) are holomorphic; this is a simple
of
the formula in 3.6, °
the formula
being a simple consequence of the
=
to is the identity. Note that, in case E is finite dimensional, If we choose a complex ortho—
fact that the restriction of
is a basis of F(E), then, if
normal basis xI,...,xN of E such that nE
has matrix tiA0
has matrix
_iA* o
The considerations in 4.1-4.3 and (4.1)-(4.5) apply ,m4tatie rautandi to the complex Grassman manifolds and we have in particular a canonioai connection in Gt(E) defined also by =
°n °
This implies in particular that:
100
°n °
(5.4)
is a totally geodesic submanifold of G(E).
5.5.
We see now that:
from the
5.6. If E is a complex Hilbert space, then the morphism J =
vector bundle T(Gc(E)) into itself, is parallel. Proof
From
=i
we obtain
(ct)(B) = 2iBo a + ie (B,a) °
— e(1 8° c
= 218° a +
o
+
i
o
°
$°a+
=21 Boa -18 °a—
8=0.
In the case where the complex Hubert space E is finite dimensional, we
note that the Hilbert-Schmidt inner product in L(E;E) is the real part of a complex inner product (the one defined by (2.2) with <
,
instead of <
,
>)
and we see that =
=
Bo(2F_Id)> = =
hence:
5.7.
If E is a finite dimensional complex Hilbert space, then
is a
manifold.
We end with the remark that (4.7)-(4.10) and 4.4 work equally well in the complex case, the usual notation for 0t(E) being U(E) (the unitary group).
References
[1] Akin, E. The Metric Theory of Banach Manifol4e, Lecture Notes in Mathematics, 662, Springer (1978).
101
£2]
Machado, A.
A locally closed set with a smooth group structure is a
Lie group.
Amer. Math. Soc. 84 (2) (1982).
Armando Machado a.nd Isabel Salavessa
Faculty of Sciences of Lisbon "Centro de Matemftica e Aplicaçoes Fundamentais",
Av. Prof. Gama Pinto 2, 1699 Lisboa—Codex
Portugal
102
S M SALAMON
Minimal surfaces and symmetric spaces INTRODUCTION
If N admits an Invariant orthogonal almost complex structure, that structure is automatically integrable and makes N a Hermitian synmietrlc space. The metric Is then and It follows that any holomorphic curve in N is a minimal surface, corresponding in fact to a minimum of the area functional. The above situation may be generalized as follows. A Hermitian symetric space is an example of a 3-syninetric space, that is, a homogeneous manifold G/K for which the Lie algebra g admits an autornorphism P of order 3 with ker(P—1) = k. The point is that many even-dimensional Riernannian symetric Let N be an even-dimensional Riemannian symetric space.
spaces possess a homogeneous bundle whose fibre Is Hermitian syninetric, and
whose total space is 3-syninetric. Over the fixed syninetric space N, each one of these bundles arises from a twistor structure on the involutive Lie algebra of N, as defined and classified by Bryant in [ 2]. On the other hand, any 3—syimietric space has a canonical almost complex structure, equal essen-
tially to (2P +
Actually if the fibre of E is more than one point, then the total space admits at least four invariant almost complex structures which we denote by ±J1, ±J2; J1 is Integrable whereas is not. Nevertheless it is the latter that Is associated to the 3—syninetric structure, and we prove that holomorphic curves in relative to J2 project to minimal surfaces
inN. The advantage of working with 3-syninetric spaces is that the almost complex structure is unique, and J2 holomorphic curves are minimal surfaces. However we consider other twistor fibrations in which the total space may admit many non-integrable almost complex structures with respect to which holomorphic curves project to minimal surfaces in N. This provides a considerable extension of previously known composition laws involving horizontality, which we illustrate by considering minimal 2-spheres in the corn-. plex Grassmannian G24, as classified by Rarnanathan [16).
This note is based upon [18] and joint discussions with F.E. Burstall, J.H. Rawnsley and J.C. Wood. 103
AND 3-$YN*TRIC IPACES
1.
Let j • Pi • m be an orthogonal Involutive Lie algebra, so that
(II,h] c h, (h,m] c m, [rn,m] c h, and the group of transformations of g generated by adgh is compact. Furtherwhich amounts to assuming that the more we shall suppose that g Is
Lie algebra is effective, Irreducible, and not of the form g
g' +
gS,
h
In particular any admh invariant inner product on in is unique up If G Is a connected, simply-connected Lie group with Lie to a algebra g, and H the connected subgroup with algebra h, then G/H is a Rie— diagonal.
manniart synunetrlc space.
The synuuetric space G/H Is Herinitian symoetric 1ff there exists an ad h invariant orthogonal transformation J rn m with J -1. Extending J to for A E h, gives g by setting .JA J(A,X]
(JA,X) +
(A,JX]
(A,JX], A
h,
X £ m.
It follows from this that J Is actually a derivation of g, and that J = ad(j) for some element S In the centre of h (see (11, Ch. XI, Th. 9.63); thus
JX • (J,X],
m,
X
With the above motivation, and following Bryant (2], we define g to be an element 5 Ii such that etruoture on the Involutive Lie (admi)2 • -1. Since in general restriction of 5 to h will not be zero,
it Is not necessary that h have a non-trivial centre.
The element 5 defines
an orthogonal almost complex structure on m, so its complexification admits the usual decomposition + rn°"
Into the elgenspaces of admj. kC
• (m1'°,m0'1),
n1'0
Define subspacEof =
(m1'°,m''0), fl0'1
hC
=
by
(rn0'1, rn°'1].
Because (m,m] + m Is an Ideal of g, necessarily [m,m] = h, and so 9c
104
=
+
+
+ rn''0 + rn°''
(1)
in which the subspaces on the right-hand side are the eigenspaces of ad9J with elgenvalues 0, 21, —2i, 1, —1 respectIvely.
If fl1'0 = 0, then j is admh invariant, and we are back to the situation In which N is Henultian symmetric. Accordingly we shall say in this case that
the twistor structure j is trivial.
general the behaviour of the elgenspaces in (1) under Lie bracket is clear from an inspection of their elgen— values. For example let w denote a primitive cube root of unity, and define a linear transformation P of g to have eigenvalues 1,w,w2 with respective In
el gens paces
kC, n°'1
+ rn1'°,
n1'0
+
rn°'1.
Then P3 = 1, and P[X,Y] = (PX,PY] for all X,Y £ g. In particular, k is a Lie subalgebra of h, and if K is the corresponding connected subgroup of G,. the coset space G/K together with the automorphism P is called 3-symmetric (8). In fact P can be integrated to give a 3-fold symmetry about each point of G/K. Any 3-symmetric space G/K has a natural Invariant almost complex structure defined by decreeing that the w-elgenspace of P consist of vectors of type (1,0). If we call this almost complex structure J2, then on the tangent space
- 1)/2.
we have P =
In our case the 3—symmetric space G/K also
admits an invariant complex structure which we call J1, for which the space
of (1,0) vectors is n1'° +
rn1'°.
To summarize:
Proposition 1 A nontrivial twistor structure on the involutive Lie algebra g = h + m gives rise to a 3-symmetric space G/K which has at least four invariant almost complex structures ±J1, ±J2. Since + rn1'°,
fl1'0
is integrable (Cf.
+ rn1'°)
c n1'0
Th. 4.5)).
+ rn1'°,
(2)
In contrast, unless the twistor
structure is trivial, J2 will not be integrable. The difference between J. and J21s best understood geometrically with reference to the fibration G/K --> G/H. Regarding this as an associated bundle, It has a natural connection arising from Riemannian structure of G/H. The horizontal distribution in T(G/K) defining this connection consists of the translates by G of in, regarded as a subspace of the tangent space to G/I( at the identity 105
coset, whereas the translates of n are vertical. Consequently the almost and complex structures coincide on the horizontal spaces, but differ by a sign on the verticals. The possible twistor structures on a given orthogonal involutive Lie algebra g = h + m were classified directly by Bryant in [2]. Up to an obvious notion of equuivalence, the possible triples (g,h,k) with g of compact type and simple and h
k are:
(so(ni+2n), so(m)+so(2n), so(m)+u(n)), m
(so(2n), u(n), so(2)+u(n—1)). n
1, n
2
3
(su(m+n), s(u(m)+u(n)), s(u(m)+u(k)+u(n-k)), in 1, n (sp(m+n), sp(m)÷sp(n), u(m)+sp(n)), m
1, n
2, 0
k
n
1
(g2, so(4), u(2)) (f4, sp(3)+sp(1), sp(3)+u(1))
(f4, so(9), so(7)+so(2)) Ce6, su(6)+sp(1), su(6)+u(1))
(e6, su(6)+sp(1), s(u(5)+u(1))+sp(1)) (e6, so(1O)+so(2), so(8)+so(2)+so(2)) Ce7, so(12)+sp(1), so(12)+u(1)) Ce7, so(12)÷sp(1), so(10)+so(2)+sp(1))
(e7, su(8), s(u(7)+u(1))) (e8, e7..sp(1), e7+u(1))
(e8, so(16), so(14)+so(2)).
This list may also be obtained from the classification of 3-symmetric spaces (20, Th. 7.t], [10, Ch. X, exercises E). It is an easy matter to pick out those 3-symmetric spaces G/K such that g = h + m is an orthogonal involutive Lie algebra admitting a twistor structure. For G compact, they are 106
precisely those for which ICis the centralizer of a torus listed in [20, In (1), [n''°, n1'0] —0, so the fibre H/K of Section 6, table 1). Note the fibratlon G/K --> Gil Is necessarily a Hermitlan syninetric space. By dropping the assumption that g
h + in be orthogonal, one obtains other
twistor fibrations [20, tables 7.11, 7.12]. preserves the fibres, it induces at each x C G/K an almost comSince plex structure JxOfl the tangent space Tit(x)GIH below (J1 Induces the same almost complex structure). In this way we obtain an embedding of G/K into the associated bundle over Gill with fibre ,(t(x) SO(2n)/U(n) parametrizing all oriented orthogonal almost complex structures on Tit(x)G/H. In fact the manifold also admits almost complex structures and J2, and the above embedding is doubly holomorphic (18]. The first study of twistor fibrations of the above type was made by Wolf in (19]. He showed that corresponding to each simple Lie algebra g of compact type, there is a Riemannian syninetric space G/H which is quaternionic i.e. whose holononiy lies in Sp(n)Sp(1) S0(4n). The twistor space of G/H is a complex contact manifold with fibre Sp(1)/U(1) S2, and the induced almost complex structures behave like the 2-sphere of imaginary quaternions. 2.
MINIMAL $URFACES
By a minimal
surface in
a Riemannian manifold
we shall mean a branched
This is then equivalent to asserting that • is conformal and harmonic; for definitions and explanation, see [5;9]. with G/H Riemannian Let ,r:G/K --> Gill be a non-trivial twistor invariant inner syimietric, and G/K 3-symetric. Firstly extend the product on m to an invariant inner product on n.m; the corresponding Riemannian metric (.,.) on G/K Is then almost Herniltian relative to J2 (and minimal Ininersion $:M --> N, where N is a Riemann surface.
Such a metric is not unique since there is at least a one parameter family corresponding to a and makes the projection it a Rienannian submersion.
horizontal versus vertical scaling factor. However all the reeuite of this the corresponding section be independent of the choice. Let levi-Civita connection on G/K, and v the Levi-Civita connection on G/ll. The next two propositions synthesize results of (12, Section 163, (20, Section 8) and [17, Section 10). 107
If
PropQsitipfl
are vEtor fields on 6/K of type (1,0) relatIve to
then 0. a
Proof First note that the left-hand side is tensorial in in the sense that the value of at x E G/K is independent of the (1,0) vector field used to extend the value of at x. To prove the proposition, we may there-
fore suppose that ct,8,y are vector fields belonging to the subspace fl0'1 + rn1'° of the space gc of complexified infinitesimal isometries. Taking x to be the identity coset of 6/K, we have at that point
I
2(V_B,y) = —(&,{8,y]) + (8,Ey,&]) +
(using for instance [11, Ch. X, Th. 3.3]). But k, so the right-hand side vanishes [y,&),
/
fl1'0 + required.
and
If we introduce the fundamental 2-form X,Y
=
T(G/K),
-
then the property expressed by Proposition 2/s equivalent to the assertion that the 3-form dw2 has no component of ty/ (1,2) relative to J2. An almost
Hermitian manifold satisfying this condl/n Is called eyn,piectio (20;18], and the class of Hermitian ones.
manifolds
or (1,2)—
the class of
2 holds for any almost Hermiproperties are obtained when the metric comes bi—invariant one on Th. 8.13]. A mapping between almost comp/ex manifolds is said to be holomorphic if Its coøunutes with,Ahe respective almost complex structures. kcordlngly we say that a from a Rlemsnn surface II to 6/K is We emphasize that
tian G-invariant metric; more
:!
1ff J2(** a/az) • I
for any complex z on N. A local •xlstemce theorem for such maps has been given by Nl4mnhuls and Woolf (143.
/
Proposition 3 Any J2 holomorphic map 108
•-> G/K Is a minimal surface.
Proof
The trans1ates of the subspacesn0't, m1'° define sub-bundles
of (T(G/K))C, which are vertical and horizontal respectively for the submersion ¶. Since * Is J2 holomorphic,
rnTPU
= x + V1
At points where X + V is not + V) is well—defined and represents the trace of zero, the expression the second fundamental form 0f the mapping 1p; moreover It is real. Putting = X + V in Proposition 2 therefore gives a= where V
rn''° at each point of
X
+ V) = 0. The proof is completed by observing that ha/az) = 0 whIch Implies that 'i, is conformal. Is harmonic.
This means that
We shall now apply the Riemannian submersion equations of O'Neill [15]. Let !p:M --> G/K be J2-holomorphic again, and In the notation of the previous proof, fix a point x c *(M) where X is not zero. In a neighbourhood of x,
extend X to a baoic vector field and consider it also as a vector field on C/H.
The horizontal component =
+ AxV
(3)
does not depend on the extension of X, so, by Proposition 2, it Is a (1,0) vector at x relative to If Y is any other basic vector field, the tensor A Is defined by the equation —
Now for any horizontal fields X,Y, the value of at x depends only on the values of V,X,Y at x. In our case, belongs to rn1'0 at x, so choosing V £ at'° and using the relation
[rn1'0, rn1'°] =n1'°
(4)
is a (1,0) vector at x. From (3), the same is true of VRX, is but this is real so it must vanish. This means that the projection shows that
harmonic at 11(x); it is also conformal since (X,X) = 0. the following composition law:
Therefore we obtain
109
--> C/K is a J2-holonxrphic map, then its projection 14 --> GIN is a minimal surface.
If
Theorem I
--> G/H, It makes sense to study the set of Given a minimal surface --> G/K such that minimal surfaces = 4. Proposition 2 and TheorOm show that in favourable circumstances this set has distinguished representatives. I
3.
A GENERAL
.At4D. EXAMPLE
coincide on horizontal directions, the minimal surfaces produced by Theorem 1 include those introduced by Bryant consisting of projections of J1-holomorphic curves in the 3-syninetric space E — C/K that are horithy exist zontal. One could call the latter minimal because the horizontal distribution Is holomorphic with the complex structure J1 [2]. Furthermore it is known that any mInimal surface or is i-isotropic, where E equals respectively s2 ——> --> Since J1 and
U(n+1)
SO(2n+fl
is the ldt' homogeneous
.
r. .,oine k, 0 •'g
1w any
not
-->
NPrI
In contrast there exists a into quaternionlc projective space which Is space (18]. k
n
-
course, not all minimal surfaces in a Riemannian be accounted for by projections of J2-holomorphlc curves in
It is
space wlfl spaces
true that any compact minimal surface in an even—thmensional
manifold can be realized as the projection of a curve in the manifold g mentioned at the end of Section 1 (17;1$]. However, in a homogeneous context, our results can be extended to a wider class of fibratlons as follows. Let (5)
with G/H Riemanniansyamietric, and suppose only that for which the fibres of 11 are complex 6/K has en invariant structia'e (1) with components Th. :1. alg.b,'e g of G now has the (2). Let J2 denote the almost complex structure obtained from by reversing sign on the vertical space n''°. Then with these hypotheses, be
1lzeor6m I i'eiraina valid. Indeed, Proposition 2 stIll applies provided that B,y are both horizontal, and the theorem follows from the fact that or [mt'°,m1'0] and [m1'01n1'°] have no components in Important examples of the above situation are provided by taking G/H to be an inner Riemannian synnietric space, and K = 1 a maximal torus of G lying in H, as explained by Bryant in the last part of (2]. The Set of Invariant complex structures on G/T is then in bijective correspondence with the set of positive root systems of g [1]. We conclude by relating these facts to work of Ramanathan (16]. From now on N will denote the complex Grassmannian G2
of the fibratlon
which may be identified with the base
subspaces of
U(4)
>
-
(1(lIxU(1)XIJ(l)xU(1)
of two—dimenslortel
U(4) U(2)xU(Z)
In Explicitly a typical element of F can be labelled (L1,12,L3,L4) corresponding to the Hermitian
Let F denote the total space of "full" sum
flags
• L2 0 L3 • 14 and complex flag
=
cl1 • 12 c
• L2 • L3 c
as tautologous we set *(L1,L2,L3,L4) = 12 0 L4. Interpreting the on F has holocomplex line bundles over F, a standard complex structure and
morphic tangent bundle
T1'0(F,J1) I
• <j
(juxtaposition denotes tensor product). T1'0(F,J2)
Note that
T1'°N
0
L1L3
The corresponding J2 has
E1L4 0 12L3 0 12(4
[3L4.
is not holomorphic, as the standard complex structure on N is
Horn (V,V1)
where V corresponds V L2 W
L4.
narmonic, it is convenient to choose a local complex where coordinate z on the Riemann surface H, and Write ahz = a + Then can be interpreted as a global holomorphic T1'0N. u,i3 If
•:M -—> N
111
differential on N with values in the pullback
Moreover, as
must be explained in (16], when N has genus zero, the nilpotent, and at least one of ci,B must have rank less than 2 everywhere
(for all choices of z).
Ignoring the case in which • itself Is holomorphic we may assume'without loss of generality that rank ci = or point now is that this nilpotency condiexcept at Isolated points. The tion is also satisfied by the projection of a J2-holomorphlc map in F; 1
indeed Theorem 2 $:N ——> N
There is a bijective correspondence between minimal surfaces with rank a 1 and non-vertical J2-holomorpHc maps tp:M --> F.
Given • with rank ci generically equal to 1, one obtains a lift VM --> F by putting 1.4 = ker ci cV, L3 = ima cV1• (and using standard Proof
arguments at isolated zeros). The conformality equation = 0 implies that the horizontal component of the differential has J2-type (1,0), and
p is thó only lift with this property. It is easy to verify that the vertihas J2-type (1,0) 1ff Va,,ajci = 0, which is precisely the condition that + is harmonic. Bijectivity follows from the generalization
cal component of of Theorem 1.
describe all J2-holomorphic maps
In
the projection ir'(L1,12,L3,14) from F
onto
the 3—symmetric space
—-> F,
(L2,L3,L1 • 14) is a = U(4)/U(1)xU(1)xU(2).
observe that map
almost
and I can be represented by orienting the edges of a tetrahedron and triangle respectively. The resulting digraphs are examples of since they consist of a collection 0f nodes each two of which are joined by a unique directed edge indicating the outcome of a game (13]. The existence of closed cycles corresponds to non-integrability, and for tte 132 structures and projection ii' we obtain:
complex structures on F
L4
L3
L3
112
These diagrams can also be used to represent J2-holomorphic maps into F
and E respectively, with the arrows indicating the derivative a/az acting on It now follows the appropriate sub-bundles of the trivial bundle (P1 x and 13 define from Theorem 1 that (the puilbacks by q, of) the line bundles minimal surfaces in (P3, and so are classified by [7, Th. 6.9). In the and I = h are adjacent legs of the Frenet notation of [16], 1 = h —
frame of some holomorphic map h
(P1 ——> (P
,
1
m
3.
(The latter may
also be deduced directly by identifying the closed triangle in the second diagram with a vanishing cubic holomorphic differential on (P1; cf. [6, SectIon 12]). Now from the first diagram, L4 is an antiholomorphic and this data completely determines p. We have therefore bundle of • proved Ramanathan's result that up to complex oonjugation, mininv.zi 2—epheree in G24 are either hoicmiorphio or of th, form hm_i 1, where L is an anti— hoiomozphic line sub—bundl, of (hm.i • h11,)1.
I
The above methods have been developed by Burstall (3) any minimal 2-sphere in a complex Grassmannlan is the
has shown that
of a holomorphic curve in a complex flag manifold with Pespect ta asultable almost complex structure. Starting from the fibrarion 13(5)
IR1)xU(1JxU(l)xUu)xU(1J
>G 2,5
'
the reader may like to construct a rotationally synuietrtc tournament with five anodes which is needed to describe minimal surfaces S2 ——> G25. References
[1)
Borel, A., Hirzebruch, F.' spaces I, Am.
J.
and homogeneous
Math. SO (1958) 458-538.
[2) Bryant, R.L. Lie groups and twistor spaces, (to appear). [3] BurstalI, F.E. Twistor fibratlons of flag manifolds and harmonic maps
[4]
of a 2—sphere into a Grassmannlan, (this volume). Calabi, E. Quelques applications de l'analyse coinpiexe aux surfaces
d'aire minima, in Topics in Complex Manifolds1 Universityof
*ntreal,
1967.
[5)
Eells, J., Lemaire, 1. A report on harmonic maps, msl.L. Lond. Soc. 10 (1978) 1-68.
Math.
113
S.
(63
Twlstorlal construction of harmonic maps of
Piea. (to appear). Ann. So. Norm. .,urfa$s iMo (7] tells, J.,.Wood,J.C. Harmonic maps from surfaces to complex projectlye.
C8] ,Gray, A.
Math. 49 (1983) Riemannian manifolds with geodesic syrmnetries of order 3, 7
.1.
(1972) 343-369.
GullIver, R.D., Ossennan, R., invnersions of surfaces, Am. J. (10] Uelgason, S. Differential Academic Press, 1978. (11] Kobayashi, S., Nosnizu, K.
2,
of branched A (1973) 750-812.
H.1.
(9]
.
Groupe, and
of Differential
Volume
1969.
[12] Llchnerowlcz, A.
Applications harmoniques et Syinpoeia ?dathmwxt'ioa 3 (1980) 341-402. [13] Holt, Rinehart and Winston, 1968. TIpioB in A., Woolf, W.B. Some Integration problems in almost—complex (14] and
[15]
Ann.
Math. 77 (1963) 424-489.
O'Neill, 8. The fundamental equations of a submersion, Mieh. Math. J. 10 (1963) 336-339.
[16)
Ramanathan, J.
Harmonic maps of S2 Into G2
J. Diff.
19
(1984), [17] Rawnsle$, J,H.
207—219.
f—structures, f—twistor spaces and harmonic maps, in (to appear). Nei'mofflc and holomorphic maps, in Saninar Luigi
(18]
(19]
homogeneous contact manifolds and quaternionic
Wolf-, J.A.
J. Math. Ma&z. ii (1965) 1033-1047. (20]
A.
mrpbis1,11, J. 5.1.. $Miosi Oxford OXI
114
Homogeneous spaces defined by Lie group autoDiff. Qeom. 2 (1968) 77-159.
U SIMON
Eigenvalues of the Laplacian and minimal immersions into spheres In this paper we discuss some results on
subuanifolds of spheres and their background in the spectral theory of Riemannian manifolds. Let (14,g) be a closed, connected two-dimensional Riemanmian manifold. We + SN(l) into the N-dimensional unit consider isometric minimal ininersions sphere of the real Euclidean space RN41. In case (N,g) is a sphere S2(K) of
constant curvature K, the following existence and uniqueness results are known [2,3):
Let s E14, K(s) 2[s(s+1):11. For each s there exists an iso.+ SN(l) is an metric minimal ininersion If arbitrary Isometric minimal iumnersion such that (K)) Is not contained In a hyperplane of then, modulo a rigid motion, for some s; in particular, K K(s) and N = 2s. Because of CalabVs result we expect the following to be true: Theorem A
Let (M,g) be a closed, connected 2-manifold with curvature K. + 5N(1) be a minimal isometric lemersion such that K(s+1). Then either K = K(s) or K = K(s+l) on (M,g) and
Conjecture I
Let s eti and let K(s)
or x
K
resp.
This conjecture is true for s = 1 and s 2 (Cf. (81 fer a — I and N — 4; (7] for s = 2). To Illustrate the background in [1] for s = 1, N Riemannian geometry we consider the canonical
j:SNUJ
such
that the centre of j(SN(I)) is the origin of a Cartesian coordinate system and we also denote the position vector of the in the Euclidean space isometric Inunersion x: (a
j.
by x.
Then each coordinat, function x(Q)
= l,...,N+1) fulfils (12] + 2x(a) a 0,
(1)
denotes the Laplacian. Calabl [2] constructed in Theorem A using the spherical hernamics of ordur s *5:S2(K(s))
where
as coordinate functions of
The bousids
K
115
correspond to the values s = 2,3 in Theorem A, resp., and to the second and x3= 12K(3) (counted without multiplicity) = 6K(2), 2 third eigenvalue 2 = of the Laplacian on S2(K(s)). It Is well known that the corresponding eigenfunctions on S2(K(s)), i.e. the spherical harmonics of order s, are character-
ized by systems of partial differential equations and that the nontrivial solvability of these systems characterizes again the sphere; we recall this for s = 1,2,3: Proposition B Let (M,g) be a complete, connected n—dimensional Riemannian manifold, n 2, and f Each of the following conditions (1), (ii),
(iii) characterizes the sphere S2(K): (9] There exists a nontrivial solution of the system
(1)
:=
(ii)
0.
+
[13] N is simply connected and there exists a nontrivial solution of
the system
B2(f)jjk
ijk + K(2fkgjJ +
(iii) (4) N is
+
connected and there exists a nontrivial solution of
the system 3
B
+ K(3fklgl,f +
:
+
+
+
+
+
+
+
Thus, the eigenfunctions of order s define a tensor field B5(f) of order (s+1) on (N,g), iddch is totally symmetric because of the Ricci identities. The situaLlon on the sphere S2(K(s)) motivates corresponding constructions
Riennian manifolds; to illustrate the method we first sketch the proof of the above conjecture for s = 2, and point out afterwards how to
on
proceed on two—dimensional Riemannian manifolds.
Proof of the conjecture (s
I, 116
2 (7]):
Construction of a vector—valued (0.4) tensor field U from the coordinate
functions x(ci) which are elgenfunctions (Cf. :=
+ Xjl9jk) + 4 + K)xklgjj
+
+ X1k9J1 + xjk9il)
+
+
(1)).
-
+
+
-
then U corresponds to the covariant If the curvature is constant K = derivative of the tensor field B (f) above, while U corresponds to B (f) for K =
II.
Using the inner product of >+
= <xlJkl ,x
III.
and the Ricci identlUes, we get
2(1OI(3—19K2
14K—7) -
To calculate the first right-hand term, we need the following Weitzen—
boeck formula;
<xljkl ,x IV.
>*
Kfl2+4(14K3-23k2+12K—4).
Steps (Ii) and (III) give the simple identity below; the zeros of the
polynomial in K on the right—hand side are the values K(s) for s + 8
-
1,2,3.
+ 2(K-1)(3K-1)(6K—1).
K there are only For a closed manifold we Integrate (IV); as two possibilities: K md U(x) — B (x) — 0 or K = and 11(x) (x) 0. Tanno's and Gallot's results above imply that (N,g) is a sphere and x is one of Calabi's standard iitmiersions *3 in Theorem A, resp. Rigoli (5, p. 41] announced that for fixed dimension N 2s of the space one can omit the assumption K K(s) in Conjecture 1. Examples of E. Tjaden (Cf. (7]) illustrate that one cannot omit this assumption for arbitrary N. In (I) we could show that the proof of Conjecture 1 for s follows Inmediately from a result in spectral geometry:
V.
1
iheorem C
Let (M,g) be a closed, connected two-dimensional Riemannian maniThen there Is no elgenvalue of the Laplacian 0.
fold with curvature K >
117
If an elgenvalue x fulfils x = 6 mm K diffeoK, then K = const, I.e. (M,g) is
In the interval (2 max K, 6 mm
or x
2 max K
6 mm
K).
to a sphere S2(K). The question whether the proof of Conjecture 1 for s 2 follows from a corresponding result in Riemannian geometry is still open: Let (t4,g) be a closed, connected two—dimensional Riemannian
Conjecture 2
manifold with curvature K > 0. Then there are no elgenvalues in the interval (6 max K, t2'Rin K). If an eigenvalue A fulfils x a 12 mm K or A a 6 max 12 sIn K, then K const, i.e. (M,g) is isometrically diffeomorphic to a K sphere S2(K).
(11] tried to prove Conjecture 2 under additional but his proof Is not correct (Cf. [6]). There are partial results in (6) and (103, and 1. Pavlista [10] gives a systematic study of our constructions above. With regard to Conjecture 2, he constructs a totally utricelessu (I.e. each contraction gives zero) (0.4)—tensor field A(f) from an elgenfunction 0 and proves the integral foraila below. + xf For an (0,r)—tensor D denote by D* the corresponding totally symeetrized (0,r)—tensor. Define (0.4)-tensors by A.
ljkl
Jk1
:-
+
4fSjSi+fsjjS+fsjIS+fijsS])gkl
+
St +
+
+
and A by
Ajjkl
a
A(2)*
(AWe +
ljkl
ijkl +
ijkl
)
Furthermere define $ by sSl9kJ)*
:— —
Proposition D (10] Let f C°'(M) be an eigenfun.ction on the closed, 4 Af a 0. Then conflècted two—dimensional manifold (14,g),
118
J
- 12k)
hAil2 do
- 12K)(A — 6K)(a1 - a2)2do
+
J +
J
hl9rad
fit2 do,
•
a2 denote the elgenvalue of the sytmietric (0,2)-tensor Hess (f).
where
This integral gives a new proof for the case s 2 In Conjecture 1 and at the same time a partial result with respect to Conjecture c-f. (10]. References
(1]
Benko, K., Kothe, H.,
k.-D., Simon, U.
Elgenvalues of the Colt. natli. 42 (1979) 19—31. of surfaces in Euclidean spheres. J.
Laplaclan and
(2] (3)
Gean. 1
do Carmo, H.P., spheres.
(4)
Min11
CalabI, E. Diff.
H.P.
Ann. MTtII. (2) 93
(1971) 43—62.
Gallot, S. Equations dlfferEntielles charactdrlstiques de la sphere. Ann. Sci. teoi.
(5]
Minimal ininerstons of spheres into
9iç.
12 (1979) 235-267.
Jensen, G.R., Rigoli, N. Minimal surfaces in spheres by the method of Preprint Univ. Nancy I (1983). Kozlowski, N. Gecmetrische *bschftzungen f& kielne Eigenwerte des Laplaceoperators auf fast—sph&rlschen RotatlonselHpsoiden. Dissertation FB 3 III Berlin (1983). Kozlowski, N., Simon, U. Minimal Imnersions of 2-manifolds into
moviflg frames.
(6)
(7]
spheres. (81
Local rigidity theorems for minimal hypersurfaces.
Lawson, NJ. Mzth.
(9]
Z. 186 (1984) 377-382.
(2) 89
Ann.
(1969) 187—197.
conditions for a Riemannian manifold to be
Obata, N.
with a sphere.
J.
(10] Pavllsta, T.
Soc.
Japan 14
(1962)
333-340.
Absch8tzungen kleiner Elgenwerte des
Dissertation FB 3 TU Berlin (1984). (11] A. Global theory of structures on Riemennian manifolds. University (1979). Laplaceoperators.
Kuwait
119
[12)
Takahashi, T.
.iul inl2lersions of Riemannian manifolds. J. Math.
Japan 28 (1966) 380-385. Tanno, S. Some differential equations on Riemannian manifolds. Math. Soc. Japan 30 (1978) 509—531. Soc.
[13]
Udo Simon
Technische
BerUn
NA 8—3
0—1—BerlIn 12
120
J.
G-deformations and Weyl's tube theorem
L VANHECKE
1.
INTRODUCTION :
SOHE TYPICAL PROBLEMS
The determination of the volumes of a geodesic sphere, tube about a curve and a tube about a submanifold in an arbttrary Riemannian manifold has been considered by several people.- Here I will concentrate on some typical problems about the volume of tubes. For results and references about the volume
0f geodesic spheres I refer to [4). (See also [5], [6].) The determination of_the volume of tubes has been treated by H. Hotelling in (8]. More specifically, he considered the determination of the volume
Va(r) of a tube of small radius r about a curve a:[a,b]. (M,g), where (M,g) is an arbitrary Riemannian manifold. typical problem:
From this work we can deduce a first
Let Vp(r) denote 'the volume of a tube of small radius r about a topo— logically embedded, connected submanifold with closure in a Rieman(P1)
nian manifold (N,g). What Is the influence of the extrinsic and Intrinsic geometric properties of P on the volume For example, Hotelling obtained the following formula for the voli Vg(r) about a unit speed curve a:(a,b] ÷ (M,g), where (M,g) Is a gemeral Riemanniar manifold of dimension n: n-I V Cr)
(lIr)
(i + Ar2 + O(r4))(a(t))dt
Ja
where A
1
,
— 6(n+1) yr +
,
,
I
Here p denotes the RIcci tensor and t the scaler curvature of (M,g). Hence the first two terms in this power series expansion do not Involve the curvetires a. So, Hotelling explicitly states the problem of the
This lecture is based on joint work with 0. Kowaiski and T.J. WIlimore [II], (123, (15]. 121
volume or the volume element involves the various radii of curvature of a. He also proves that the answer for the volume V(r) is negative when a Is a curve in Euclidean space or a spherical space. To prove this he determines complete formulas for Va(r) in both cases. This gives rise to a second typical problem:
and, if Determine more terms in the power series expansion for for some special Riemannian possible, determine complete formulas for manifolds (M,g) or special submanifolds of (14,g). (P2)
As is well known, this has been considered in a paper by H. Weyl [16). He derived the following formula for when P is an arbitrary p—dimensional submanifold in
22
2 )
[p/2) c=O
k
Important
(R1)r2'
(n-p+2)(n-p+4)...(n—p+Zc)
RT denotes the Rlemann curvature tensor of P wIth respect to Induced are functions of the curvature Invariant*. also metric and the determined the complete formulas when (M,g) is a space of Curvature. This result implies that the volume of a tube about a curve in Em depends only on the length of the curve and the radius of the tube. More generally, Weyl can be interpreted In the following way: be a the formula space of constant curvature and let P (M,g) be a topologically enthedded submpnifold with compact closure. Then the volume of tubes about P for all
sufficiently small radii r remains invariant under isometric deformations of P.
From this we derive a third problem: when P is a special subProve similar invariance theorems for manifold or when (14,g) is a special Riemannian manifold. (P3)
F.J. Flaherty 13] and R.A. Wolf (17] gave similar theorems for Isometric submanifolds P and P' when P and P' are both in In (5], [6] A. Gray and the author determined more terms in the power series expansion for
when M is a general Riemannian manifold and also they
determined conrplete formulas for Vp(r) when P is a submanifold of a two-point homogeneous space. 122
More specifically they proved that the volume V(r) of a
tube about a curve In a two-point homogeneous space remains the same under Isometric deformations. Also It Is proved that this result cannot be extended
to surfaces. Here I will not consider these results in more detail because they will follow easily from the following problem which Is the main purpose of this lecture:
in case that (M,g) Is an (P4) Prove similar invariance theorems the Isometric deformation of arbitrary (analytic) Rleinannian manifold P c N Is replaced by another notion, namely by G-deformations of the first and second order.
In what follows 6 will be the group 1(M) of isometrics of (M,g). The method we will use Is not the determination of the complete formulas. In general this seems tQ be Impossible. Here we will use an elegant formula for the volume or the volume element which is obtained by using Jacobi vector
fields and a particular useful property of solutions of the Jacobi equation. First we treat this theory for curves and then we generalize it to arbitrary submanIfolds. 2.
THE VOLUME OF TUBES ABOUT CURVES
This section, md also the following one, will be divided Into three ports. First we derive the required formula for the volume, then we treat the notion of k-equivalence and finally we consider some results related to the fourth problem stated above. A.
A formula for the volume of a tube about a
The derivation of our formula for the volume of a tube about a curve in an arbitrary Riemannian manifold will be done in three steps. Let (M,g) be a Riemannian manifold and let a: (a,b] + (M,g) be a C" unit speed curve of finite length which is topologically eithedded in (M,g). 'Further, let N0 c TN be the total space of the normal bundle of a((a,b]) in N and let N(r) denote the open solid tube of radius r about the null-section of N0. For sufficiently small r, the map exp0 :
{t
[a,b],
is a C"dlffeoinorphism of N0(r) onto the open solid tube 110(r) of radius 123
about a In 11.
of 11(r). Let
be the corresponding denote the velum fôrm of the
We choose an
volume form of (14,g) on 11(r). Finally, let coherently oriented total space of the normal bundle a equipped with its
natural flat metric g0.
Then we have
= 00w0
The strictly positive of
exp0.
Lames 1
is called the volume
function
denaity function
We have
(5], (6]. Let V(r) denote the volume of the tube 11(r).
Then we
have
V fr)
JN(r) 0
çbjr a
e
vol
(ri) (1)
5n—2 n—2 J S0(5)(1)
0
e
a
: a'(a)1 + as(t)L denotes parallel translation in the In this lames normal bundle along a((ab]) with respect to the normal connection and is the unit sphere with centre 0(a) in the normal space of a at 0(a). In the second step we transform (1) by using Jacobi vector fields. Let be the s, y(O) a(t) and put such that fin II Further let be an orthonormal frame at 0(a) such • that o'(a) = e1, F = We consider now the Fermi frame field CE1,.. a 2,...,n, are obtained along a such that E1(t) = o'(t) and where the from the e5 by parallel translation in the normal bundle with respect to the normal connection. This gives rise to the so-called Fermi coordinate8 defined In a neighbourhood of the curve by
n
x1(exp
'
Z
t.E4) = t,
E
=
a = 2,...,n.
Now we turn to Jacobi vector fields along y.
along y If and only if 124
Y is a Jacobi vector field
Y° +
More specifically we consider the n—I Jacobi vector fields V1, Y, a = 2,...,n—1 with initial denotes the Riemannian curvature tensor of (M,g).
R
conditions
y1(0) = E1(t),
Y(0)
Va 1
(3)
Y'(O) = E(t), a = 2,...,n—1.
Y(O) = 0
These Jacobi vector fields can be expressed in terms of the vector fields
Indeed, it is easy to verify the following
Y1(s)
, a = 2,...,n—1.
=
Next we shall express these Jacobi vector fields with respect to the Fermi frame field along y obtained by parallel translation of the frame
along y.
Put
X = 1,...,n—l. Substitution of the
in the Jacobi equation (2) gives
B" + R o B • 0 for the endornorphism-valued function s phism Y
R
B(s).
Here R denotes the endomor-
Ivy'. ,
The initial conditions for B follow easily by using Lema 2: 1
0
...
f_Kg
0
o
B'
—
0
...
0
0
'
0 0
B'10' —
''—
0
where
This function $ tubes.
B(s)
plays a key role in the study of the geometry of
For example, we have 125
*
=
B)(s) where Ye(s)
-
Hence, we have for =
Wehave
v Cr) =
, I
I
I
Ja JO
(det
2
(4)
Note that S = B'B1 is the shape operator at 1(s) of the nonsolid tube considered as a hypersurface in (11,g).
In the last step we shall rewrite (4) in a particular elegant form. Therefore we use the notion of the W(A,B) of two solutions of the equation
+ R ° C = 0. This Wronskian is defined by tAss -
W(A,B)
tABS
and it Is well—known that, since R Is a symeetric endomorphism, W(A,B) is constant along the geodesic y. This property is very useful here. Indeed, let B be the solution considered above and let A be the solution such that A(s)
0,
A'(s) = -I.
(Geometrically, this solution is related to the geometry of the geodesic sphere Gq(S)s q= exPO(t); with centre q and radius s. (See, for example,
(1],.[2), (14), (15).) W(A,B)(s)
Then, consider
W(A,B)(O).
Using the Initial values and taking the determinant, we obtain (det B)(s) = — {g(S where S 126
exp(t)n
exp(t)n
a'(t),cj'(t)) + K!')} dot A(0),
is the shape operator of the geodesic sphere G
(s) at
a(t).
Further we have
det A(O) = s e
-
n—i
(a(t))
e
is the volume density function of the exponential map
exPo(t)n
of (M,g) at exPa(t)n evaluated at 0(t).
(See, for example, (1], (14].)
Finally we have (see (1], [14])
(a(t))
o
= 80(t) exp(t)n and hence we obtain 4
We have
= - {9(Sexp
o'(t),a'(t)) + (5)
This formula has some interesting features. It leads to an answer to the question stated by Hotelling. Indeed, it shows that on1y the curvature is involved and this in a linear way. All the other quantities are completely determined by the geodesic spheres.
The following result will also be useful In what follows. Let (M,g) be a Riemannian manifold such that all geode8io Byninetries are voliune—prenerving; then the curvature K disappears completely in the formula for V(r). Indeed, such spaces are characterized by the property
8(t)(exP(t)n) = o(t)(exp(t) and hence, we obtain
,r V Cr) = —
I
5n—l
I
a JO
9(5
exPa(t)n
a' (t) on such a space. B.
(6)
(See [10], [15] for more details abqut such spaces.)
k—equivalent curves
Now we shall give a geometrical meaning to the obtained formulas using the classical concept of deformation in the sense of E. Cartan. (We refer to 127
[7], [9] for more details about deformations.)
Definition.
Let (M,g) be a
Two parametrized curves lent
We start with the following:
0 an integer. (M,g) are said to be locally k—equiva—
Riernannian manifold and k :
(a,b]
(or k—equivalent, respectively)
if for each t
(or global, respectively) isometry
ta,b] there is a local
of M such that
°
=
is the k-jet of a map [a,bJ -* M at the value t.
Here
Note that a reparametrization changes (locally) k-equivalent curves into (locally) k—equivalent curves. In what follows we will only consider 1- and 2-equivalent curves and our results will quickly follow from the following properties which can be proved easily. 1—equivalent cu1'ves have the same length. Hence, if a,B are 1—equivalent curves of finite length, we can always assume that they have unit speed on (0,L3 where I is the common length. Secondly, for 2-equivalent curves we have Lemma 5
Let (M,v) be a manifold with affine connection and let
be two curves such that cx(a) —
B(a).
Then
=
H
if and only if
,
Vd
-
VdB
-a
-a
Results
C.
8ased on these properties we obtain from (1) and (5) the following general
result: [a,b] + (M,g) be two smooth regular parametrized curves offinite length which are topologically embedded in a real analytic RiemannIan manifold (M,g). Let P(r), and P8(r) respectively, denote tubes of eufficiently small radius r about and B respectively. If and B are 2—equivalent, then P(r) and P3(r) have the same volume. Theorem 6
Let ci,B
Note that this result still holds for k—equivalence by k—equivalence.
nianifolds if we replace local -
\
128
Formula (6) leads to a stronger result for spaces with volume-preserving geodesic synunetries.
be as in Theorem 6. If (M,g) is a space with volumeJet preserving geodesic synunetries and if ct,B are 1-equivalent, then Pa(r) and PA(r) have the same volume for any sufficiently small radius r. There are many examples of spaces with volume-preserving geodesic symetries. All syrmuetric spaces have this property and, hence, also all two— point homogeneous spaces. Moreover, these last spaces are also isotropic and so any two unit speed curves of the same length are 1-equivalent. So we obtain the following corollary which contains the already mentioned result of A. Gray and the author [5]:
Theorem 7
Corollary 8 Let (M,g) be a two-point homogeneous space. Then the volume of a tube about a curve in (M,g) depends only on the length of the curve and the radius of the tube. Remark
We refer to [11] for further
about the existence of 1-and
2-equivalent but noncongruent curves in homogeneous spaces. There we show that the theorems obtained above have a nontrivial geometrical meaning. 3.
THE VOLUME OF TUBES ABOUT SUBMANIFOLDS
A.
A formula for the volume of a tube about a submanifold
In this section we derive a formula for the volume of a tube about a submanifold P man arbitrary Riemannian manifold which generalizes the formula for curves obtaincd in Section 2. Again, we do this in three steps. Let (M,g) be an n-dimensional Rieniannian manifold and let P be a p—dimensional connected topologically embedded submanifold with compact closure in M.
Further, let
denote a (nonsolid) tube of radius r about P, i.e.
Mithere exists a geodesic y with length L(Y) = r from m to P meeting P orthogonally).
= Cm
Also, we always suppose that r is smaller than the distance from P to its nearest focal point. Next, let m P and let be a local orthonormal frame field of 129
We specialize this moving frame are fields and are normal vector so that be a system of Fermi coordinatee with fields of P. Further, let defined in an open domain respect to m and (see (6]). In this domain we consider the Riemannian volume element such that
N defined along P in a neighbourhood of m.
> 0.
Now, let u be a unit vector normal to P at m and consider the geodesic y:t expmtu through m. Further, we choose the frame field such that En(m) = u. For the volume function t • we have
and:
[6). The volume V(Pr) of the nonsolid tube
Lema 9
V(P ) r
=
J
J
(1)
r'1e
u
is given by
(r)dudP.
(7)
In the next step we transform (7) using Jacobi vector fields along i. Here we consider the Jacobi fields Y1,Y8, I = 1,...,), and a
with initial conditioni Y1(0)
E1(m),
Y(D) = 1
0'
,
= Ea(m),
denotes the Riemannian connection of (M,g).
where 10
(8)
Then we ftave
The Jacobi fields Yx, A = 1,...,n—1 are expressible as follows:
v1(t)
Ya(t) = t
'
=
a = p+1,...,n—I.
Further, let
be the Fermi frame field along the geodesic y obtained by parallel translating the frame along y and put Ya(t) = 130
ct = 1,...,n-1.
This gives rise to the endomorphism-valued function t +
such that
= 0.
+Ro
Using Lenina 10 we easily obtain
(det
=
and hence: Lema 11
We have
V(Pr )
=
I I Jp
(1)
(det 0U)(r)dudP.
(9)
Finally we rewrite (9) as in Section 2 in a form which involves only To do this we first and need the initial values for Therefore we first state information about P and the geodesic spheres of (M,g). some well—known formulas
submanifold theory.
Let V denote
connection of P. Further, let X,Y be tangent vector fields and N a unit normal vector field along an open domain in P. Put =
+
= T(N)X +
denotes the connection in the normal bundle. Further lxv = T(X,Y) is the second fundamental form operator and 1(N) is the shape operator. They are related by Here
g(T(N)X,Y) = —g(T(X,Y),N).
Also, we will use the operator .1. defined by .LxN =
Using (8) we then find quickly
r'
D(0)1 U
0,
0jI,
rT(U) D'(O)I t U
0
I
where
131
=
i,j = I,...,p and a = p÷1,...,n—1. Now we use the same trick with the Wronskian as in Section 2.
We consider
the solution of the Jacobi equation and the solution A of Section 2 which is related to the geodesic sphere Gq(r) with centre q = expmru and radius r. We write =
This gives explicitly
D(r)=[
T(u) + B(ru)
U
-1(u) I
J
where
i,J = 1,...p.
=
Here Sq is the shape operator of the geodesic sphere Gq(r) at in.
From this
we obtain
= det(T(u) + B(ru))det A(O)
det
and since
det A(O) =
=
we have
= det(T(u) + So we proved:
Lema 12
volume V(Pr) of the tube
The
V(P ) r
=
J
J
—
(1)
det(T(u) +
is given by m
ru)dudp.
(10)
A remarkable feature of (10) is that it involves only the shape operator of the submanifold P and some information about the geodesic spheres of the 132
ambient space M which B.
are tangent to P.
Equivalence of submanifolds
Before we introduce the notion of k-equivalence of submanifolds, we consider the k-equivalence of maps. Let II be an open domain in pP containing the origin 0.
Definition smooth maps
and
-, N of II into a Riemannian manifold are said to be k-
:11
equivalent at
0
Two
if there is a local Isonetry D of N such that
don D
ci).
=
For 1- and 2-equivalcnce
we
need the following leninas which are easily
proved.
Let a,8 :11 + N be 1-equivalent maps at 0. near 0, then B is also an embedding near 0 and (B ° Isoinetry of the tangent subspace Lenina 13
Lemma 14
Let a,B:U + N be two embeddings such that
If a Is an embedding
a linear c TB(O)M. Then
and
a*
x
V
0
(X
0
for any two vector fields X,Y on U. (v is the Riemannian connection on N.) Further, for the second fundamental form operators TB of the embeddings we have at 0: =
u,v ERR.
Next, let P be a smooth manifold of dime'nsion p. A parametrization of P at in £ P is a diffeonorphism of a neighbourhood II of 0 in P9 Onto a neighbourhood U' of m in P such that •(0) in. Further, let P. P' be two submanifolds of N with the same dimension p.
Definition P and P' are said to be locally k—equivalent If there is a diffeomorphism + P' (called a with the following property: for any point m P. and for some parametrization • of P at m, the maps 'and o • are k-equivalent at 0. 133
Note that this definition is independent of the choice of parametrization. Now we give some results and remarks about locally 1-equivalent subnianifolds. The proofs are easy. Theorem 15
A 1—deformation between two subrnanifolds P, P5 of (M,g) is always
Moreover, it preserves the sectional curvature of N at all 2— planes tangent to P (and P', respectively).
an isometry.
As concerns the converse we have
in a space of constant curvature, two isometric submanifolds P. P' are locally 1—equivalent. Theorem 16
For a general Riemannian manifold this converte does not hold. It even does not hold in the class of rank one symetric spaces. To illustrate this we consider two surfaces P, P' embedded in where the holomorphic 4. For P we take £P1(4) embedded as a totaUy geosectional curvature desic KMiler submanifold in £P3(4). For P5 we take a geodesic sphere of radius ri6 inRP3(1) and embed RP3(1) as a totally real, totally geodesic submanifold in CP3(4). Then P' has constailt sectional curvature 4 and it is isometric to P. But P and P5 are not 1-equivalent because of the second part of Theorem 15. This means that, in general, 1—equivalence Is stronger than isoinetry.
We finish this part with a reformulation of the corollary of H. WeyPs tube fornula. We obtain this reformulation using Theorem 16 and It is convenient for the generalizations we shall treat next. -...Theorem 17
Let (M,g) be a space of constant curvature and let P, P' be two
1—equivalent submanifolds with compact closure.
Then V(Pr) =
for all
sufficiently small r. Note that V(Pr)
theorems are also valid for the volume of a solid tube. C. Results
Using (10), Lema 13 and Lema 14 we obtain some generalizations of Theorem
We state the results, which we divide into three classes, without proofs. See [12] for more details. We start with the general situation.. 16.
134
1.
Submanlfolds In general Riemannian manifolds
Let (M,g) be an analytic Riemannian manifold and P, P' two locally V(P.) for any 2-equivalent submanifolds with compact closure. Then Theorem 18
sufficiently small r. Note that this theorem still holds for 2.
conrpaot submanifolds.
Submanifolds of special Riemannian
If we consider 1—equivalence instead of 2—equivalence, we still obtain a
result if we restrict ourselves to more special manifolds.
We have:
Let (M,g) be a locally symmetric Riemannian space and let P. P' be two locally 1-equivalent hypersurfaces with compact closure. Then
Theorem 19
V(Pr) • V(P.) for any sufficiently small r. We have already seen (Theorem 7) that for curveB
this
theorem is still
true for the larger class of all Riemannian manifolds such that all geodesic symmetries are volume-preserving. For surfacee we also obtain a result when we consider a further
restriction. Let (M,g) be a two-point homogeneous space and P, P' locally 1— holds for equivalent surfaces with compact closure in M. Then Theorem 20
any sufficiently small radius r. be replaced by the Note that local 1-equivalence in Theorem 20 For example, it is proved In (6] that the tubes weaker property of isometry. about the two surfaces P and P' considered after Theorem 16 do not have the same volume.
3.
SpecIal submanifolds of special Riemannian manifolds
The problem seems to be much more difficult for higher—dimensional subinanifolds even in two-point homogeneous spaces. But we still get some results If we only consider submanifolds of a special geometrical nature in such spaces. So, let (M,g) be again a two-point homogeneous space. A tangent element L
TmM* m
N, is said to be wnhilical If the following holds: for any geo-
desic sphere Gq(r) tangent to L at m, the shape operator Sq(r) of Gq(r) at m
restricted to L is a multiple of the identity operator. A submanifold P c M is said to be of the wnbil.ical
type if
all tangent 135
spaces
m
P, are umbilical.
are of and It is not difficult to see that all submanifolds of umbilical type (13). We also refer to [13].for a proof of the fact that a submanifold of the umbilical type in
or Cay P2 respectively, is
or
the same as a complex submanifold, or a quaternionic submanifold, or a Cayley submanifold respectively. The same holds for the non—compact duals of the last three spaces. For this kind of submanifolds we have:
Let P, P' be two submanifolds with compact closure and of the umbilical type in a two-point homogeneous space. If P, P' are 1-equivalent, Theorem 21
then V(Pr) = V(P.) holds for any sufficient small radius r. Note that this also follows from the etplicit formulas in [6]. Up to now we have considered only the volume of tubes. Tubes in (M,g) are hypersurfaces and so we can consider the integrated mean curvatures.
Remark
particular we can consider the total mean curvature (the first integrated mean curvature). We note here without proof (see (12]) that Theorems 18, 19, In
20 and 21 also hold when we replace the volume by the total mean curvature. References
(1]
Besse, A.L. Manifolds All of Whose Geodesic8 are Closed, Ergebnisse der Mathematik, vol. 93, Springe?-Verlag, Berlin-Heidelberg—New York, (1978).
(2]
[3)
then, B.Y. and J. Reine Angew.
L. Math. 325
Differential geometry of geodesic spheres,
(1981)
28—67.
Flaherty, F.J.
Illinois
J.
The volume of a tube in complex projective space, Math. 16 (1972) 627-638.
[4]
Gray, A. and Vanhecke, L..
(5]
voiwse of small geodesic balls, Acta Mat/i. 142 (1979) 157—198. Gray, A. and Vanhecke, L. The volumes of tubes about curves in a
Riemannian geometry as determined by the
Riemannian manifold, Proc. London Math.
Soc.
44 (1982) 215-243.
[6] Gray, A. and Vanhecke, L. The volumes of tubes in a Riemannian manifold, Rend. Sem. Mat. Univ. e Politec. Torino 39 (1981) 1-50. [7] Green, P1.1. The moving frame, differential Invariants and rigidity theorems for curves In homogeneous spaces, Duke Math. J. 45 (1978) 735— 779. 136
[8] Hotelling, H. [9)
Tubes and spheres in n-space and a class of statistical problems, Amer. J. Math. 61 (1939) 440-460. Jensen, G.R. Deformation of submanifolds of homogeneous spaces, i.
Differential
Geometry 26 (1981) 231-246.
[10] Kowaiski, 0. and Vanhecke, L.
Op&ateurs diff&entiels invariants et le volume, C.R. Acad. Sci. Paris
I Math. 296 (1983) 1001-1003. [11) Kowaiski, 0. and Vanhecke, L. G-deformations of curves and volumes of tubes in Riemannian manifolds, Geometriae Dedicate 15 (1983) 125-135. (12] Kowalski, 0. and Vanhecke, L. G-deformations and some generalizations of H. tube theorem (to appear). (13) Tricerri, F. and Vanhecke, L. Geodesic spheres and naturally reductive homogeneous spaces, Proc. Convegno Internazionale di Geometria
Differenziale e Analisi Complesse, Rome 1983, Riv. 10
Math.
Univ. Parma,
(1984) 123-131.
[14] Vanhecke, L. Some solved and unsolved problems about harmonic and coninutative spaces, Bull. Soc. Math. Beig. B 34 (1982) 1-24. [15] Vanhecke, 1. and Wilirnore, T.J. Interaction of and spheres, Math. Ann. 263 (1983) 31-42. [16] Weyl, H. On the volume of tubes, Amer. J. Math. 61 (1939) 461-472. [17] Wolf, The volume of tubes in complex projective space, Trans. Amer.
Math.
Soc. 15? (1971) 347-371.
L. Vanhecke
Katholieke Vniversiteit Leuven Department of Mathematics
Celestijnenlaan 200 B B—3030 Leuven,
Belgium.
137
T J WLLLMORE
The Gauss map of surfaces in R3 and R4 In this survey we pay particular attention to previous papers by D.A. Hoffman and R. Osserman (1,2,3,4]. Our results are really contained in these four papers but our treatment is slightly different and possibly more geo1.
metrically significant. 2.
DVIERSIONS IN
Let S0 be a Riemann surface and let X:S0 -. be an inrersion which is conformal. The image of X is an oriented surface which we denote by S. The Gauss map G:S -' G2,n sends a point of S to the oriented parallel 2-plane
passing through the origin 0 of 2—planes through The set of all 0 forms a differentiable manifold L,fl which can be identified with a certain —1 in CP quadric . We give a brief description of this correspondence. Let a = be two non-zero vectors in b= such that
= Ibi and a.b = 0. We write Z
(2.1)
ak + ibk, and note that condition (2.1) is equivalent to = 0.
(2.2)
k=1
We observe that a change of orthogonal basis in the 2-plane It spanned by a and b leads to the same relation (2.2), which therefore depends only upon (fl_{0) lying on the 2-plane IT. Thus a point IT of G2 n determines a point in given by (2.2). However, the nature of (2.2) shows that the quadric can be regarded as living in Conversely, given a point [zk] in 1pn—1 which lies on Zk, and this determines an we may write ak + ibk From now on we orthogonal basis for the corresponding 2-plane Ii in G2 If we compose a map with the map consider maps of the form G:S we obtain a map
X:S0
g:S0 138
(2.3)
where
g=
GoX.
(2.4)
problem considered by Hoffman and Osserman is the foUowlng. Given a map (2.3), under what conditions does this factorize according to (2.4) so that X(S0) is a surface in and G is its Gauss map? Following their treatment we let z = + in be a local coordinate system on S0 and let (xi,x2,....,xn ) be coordinates in The map defining S is The
given by z
X(z) = (x1(z), x2(z),...,x(Z)).
Since X is conformal we have (axi
=
0
ax
and
=
2-plane corresponds
The tangent plane to S is spanned by i.ne quadric
to the point (.
0n2•
Thus the Gauss map G of S is
qitta— "I
(aX an
If we write af
an 1
=
-
af = —
I
(af
+
then the above map, when
composed with X, gives rise to the map g:z
F ax 1
However, for reasons associated with classical differential geometry (which really depend upon how the outward normal of a sphere becomes inward pointing
after stereographic projection) it will be more convenient for us to consider the complex conjugate
of this map given by (2.5)
whose image also lies on We note that the metric on S is given by
139
(2.6)
A2 dz
cis2
-
2
-
=
—I have
2
available the standard result = 2H,
(2.7)
of the metric (2.6) and H is the mean curvature vector. We return now to the main problem. --'s that we are given a map from to represented by o(z)
do not vanish simultaneously and
where
= 0.
(2.8)
We note that we are really concerned with the equivalence class [43 of such 0 0 maps •, where • and.o are equivalent if • = where A E The question is whether such a map • arises from the Gauss map of some surface X(S0) in This will 'be the case if x
z
(2.9)
whenever is a complex-valued function on S0. We write = pe 0 0. Then the following theorem [3) gIves necessary conditions which must be satisfied if • arises from a Gauss map.
where
Theorem A. Let 0 be a domain in the complex plane, and let S be a surface in R" given locally by a conformal map . Let arise from a Gauss map. 140
Form the quantities .
(2.10)
=
V=
(2.11)
1.
—
3Z
Then for every z
D we have
V(z) =
(2.12)
where R(z) is a real vector moreover, on the set where V(z) 0, the function is uniquely defined modulo and satisfies the differential equation 2
(2.13)
3z
These conditions result from differentiating (2.9) with respect to i and using (2.7). Evidently is a real vector. It seems convenient to normalize the map by imposing the condition 1. Then • maps S0 into c C'L{O}. If we write •- = y V for some function y where V is the component normal to S. we have imediately y
•
= n and hence
2
Z
This gives a goemetrical significance to the "quantities" and V of Theorem = A. A brief calculation shows that if • is replaced by •, then the corresponding V is given by V =
V.
An alternative derivation of these conditions arises from taking the differential of the Gauss map and relating this to the *econd fundamental form. If we take the differential of in (2.5) and let it act on we see, for example from (2.7), that V is a complex number times the (real) mean curvature vector. A simple calculation shows that, when H 0, the real—valued function cg(z)-
imist satisfy the relation
=
from which it follows that
= Im
We now turn to the.question of when the necessary tonditions of Theorem A 141
are also sufficient.
The answer is given by the
Let S0 be a simply-connected Riemann surface and let g:S0
B
be a map into the complex quadric
in the sense that = defined as previously in terms of
If V
Case 1.
surface in
Represent g locally by a map • into The scalar and the vector field V are We consider two cases.
0 on S0, then g factorizes via the Gauss map of a minimal provided that S0 is not compact.
If V never vanishes on S0, then a sufficient condition for g to
Case 2.
factorize via the Ga'uss map of a surface S given by a conformal map X:S0 Is that the necessary conditions of Theorem A are satisfied,where R(z) is a
non-zero real vector for each z. The proof of Theorem B depends upon a standard existence theorem in complex
analysis — see, for example, (5, Theorem 1.2.2). One of the most surprising results of Osserman and Hoffman is that a is essentially determined by its Gauss map by surface in come with a one-parameter general we mean non-minimal. Minimal surfaces in family of associate surfaces all having the same Gauss map. However, except
for minimal surfaces, the following result holds: Let S be a surface defined by a conformal ininersion X:S0 of a Riemann surface S0. If the mean curvature of S is different from zero at some point, then S is determined up to a similarity transformation of by Theorem C
its Gauss map G. 3.
SURFACES IN R3
In this particular case it is more convenient to deal with the classical Gauss map than with the generalized Gauss map into the quadric Q1.
These two maps are related as follows. The map +
(1w2, i(1+w2),2w)
3
satisfies
-
= 0 and therefore defines a map k=t
142
=
-
(3.1)
c (P2.
C+
Now the identification of Q with the Grassmannian of oriented 2.planes in P yields an identification of that Grassmannian with (P where to each C corresponds the plane with oriented basis w However, the map 1
N(w)
2
(2Re(w), 21m(u,),
— 1),
(3.2)
is the inverse of stereographic projection from the North Pole. We see that 0 so that N(w) is ±1 times the oriented normal to the oriented 2-plane in Q1. Consideration of the 2-plane with basis = (1,0,0), e2 = (0,1,0) shows that N(w) is in fact minua the oriented unit normal to Thus, in order to find a surface whose Gauss map is g:S0 Q1 we may express g in the form
ti—f2, i(14f2), 2fJ
=
=
for some map f;S0 + £ U
(3.3)
This is equivalent to finding a surface whose
Gauss normal map Is
N•
.
C2Re(f), 21m(f),
I
We now apply the results of Section 2 to the map f which we write locally as w = f(z).
Using
•(z)
=
(i-f2,
i(1+f2), 2f)
(3.5)
,
(3.6)
we find that =
v=
—
(2Re f, 21m f,
(3.7)
1f12 — 1).
From (3.4), (3.7) we see bmediately that V =
-
N and, since N Is a real
vector, we see that the first condition of Theorem B is automatically satisfied. We introduce functions F, 1, S and I associated with each sufficiently 143
= f(z) as follows:
differentiable complex function 2
+
L
= 1(f) =
S = S(f) =
1(f)
F(f),
— 2?
provided that
# 0.
Then, Hoffman and Ossennan prove:
Let S be a surface in P3 and let w = f(z) represent the Gauss map. of S into the unit sphere, where z is a local conformal parameter on S0. Then it is necessary that one of the following two conditions at every point of Theorem 0
holds:
(I)
F(f) = 0, or
(ii)
F(f)
0 and Im (1(f)) = 0.
Calculation shows that condition is equivalent to (2.13) for surfaces in P3. It follows that if we aregiven an arbitrary C3-surface S in P3 and we express the classical Gauss map of S by a complex function w f(z) where z is a local isothermal parameter on S and is obtained by stereographic
projection from the unit sphere, then at each point of S, f satisfies either the firstorder equation 0, or else the third order equation 0 and Im
}
=
= This second equation is the form taken by when n = 3. f(z) be a local representation of g in a neighbourhood V of some Let w point q where z is a local complex parameter on S0 and w is obtained by stereographic projection from some point p not lying in f(IJ). Then we may prove that I dz di are smooth globally defined forms on S0 which depend only on the map g and not upon the choice of loca' paroneter. Then the sufficiency conditions are given by:
Let S0 be a simply-connected Riemann surface and let g:S0 + be a smooth map of S0 into the unit sphere in P3. Then Theorem E
144
S2(1)
Case 1.
If IF!
Case 2.
If
IdzI = 0 on S0, and if S0 is not compact, then there exists a minimal Inunersion defined by the conformal ininersion X:S0 + S c such GOX, where G:S -s. S2(1) is the classical Gauss map. that g = IF! IdzI never vanishes on S0. then there exists a conformal
Ininersion X:S0 + S Im
=
for which g factors via a Gauss map if and only if
0.
Let M and N be surfaces with conformal metrics given locally by ds2 = u2!dwI2 respectively. Let g:M N be a smooth map. Then the x21dz12, do2 energy density of g is e(g)
We represent this by the sum of
e(g) =
e'(g)
and
components given by
+ e"(g)
where
e'(g) =
=
i4
lw—I2,
where the map g is represented in the form Is given by
t(g) =
= f(z).
The tension field
{w2j + 2(log
The map g is harmonic if t(g) 0. The following result was first proved by Kenrnotsu [6]: be a simply-connected Riemann surface and let g:S0 S2(1) be a harmonic map which is nowhere anti-conformal. Then there exists a conformal imersion X:S0 + S where S is a surface of constant mean curvature such that Let
g = GoX and G is the Gauss map of S.
This is related to an earlier theorem of Ruh and Vilms [7), [8] which states: A surface s in P3 has constant mean curvature if and only if the Gauss map is harmonic. A similar result due to Kenmotsu [6) shows: 145
If g:S0 + S2(1) is a harmonic map of a simply-connected Riemann surface Into the unit sphere, then there exists a conforinally branched Ininersion where S has constant mean curvature, such that g G:S + S (1) is the standard Gauss map. -, S
X;S
G0X where
Finally we note that the holomorphic energy of the Gauss map is a quantity which has already appeared in a different context in the geometry of ininersed surfaces. In fact, for an arbitrary surface s in R3, it can be proved that the holomorphic component of the energy density of the Gauss map G is equal to the square of the mean curvature of S, that is, e'(G) H2. e'(G) * 1. In these terms a We denote the holomorphic energy of G by well—known conjecture of the author (9] can be expressed in the form:
For all Imersed tori in
the holomorphic energy of the Gauss map has
a universal lower bound equal to 2r2. 4.
SURFACES IN R4
The Grassmannian of oriented 2-planes in R4 may be identified with the quadric in (P3. However, it may be represented as a product of 2-spheres. The map • from C x C given by
(4.1)
+
has the property that = is given by On 4(C x C), (z1,z2,z3,z4)
= o.
=
Hence (tp] takes values in Q2.
(23 + iZ4
-Z3 + U4
z1—izz
zl—i22
Moreover,
(4.2)
gives a bi-holomorphic map C x £ -, Q which extends The equivalence class to a bi-holoinorphic map of (P x (P into Q when we consider (w ,w2)E(x I as homogeneous coordinates on (P x (P . Let (P be equipped with the Fubinisectional curvature 2. Then the induced Study metric of constant can be shown to be metric on Q2, expressed in terms of coordinates 2 I
2 I
—
showing that Q2 is the product of two standard spheres of constant Gaussian 146
curvature equal to 2. This product structure is used to find necessary conditions on the Gauss Let s c be such a surface whose map of an oriented surface ininersed in where each Sk is a onto Sk by standard sphere of Iadius 1/i2. Denote projection of We to parametrize Sk when (Uk is given by (4.2). The conjugate of the use Gauss map a may then be expressed in terms of a local conformal parameter z on S by the pair of functions = f1(z), = f2(z). Then we may write Gauss map is given by G:S + Q2.
We express Q2 = S1
x S2
2
a(z) =
(4.4)
where
•(z) = •(f1(z), f2(z)) (1+f1(z)f2(z), i(1—f1(z)) f2(z), f1(z)—f2(z), — i(f1(z)+f2(z)).
(4.5)
We introduce functions F4 =
(f.)-Z 1+1f11
•
2
(f.) Z 1
F. 1
1+1f11
and on the set where =
,
i
1,2
2
0 we set —
I
= 1,2.
Then we can prove: Theorem F
Let S be an oriented surface ininersed in P4 whose Gauss map G is
given by (4.5) in terms of a pair of functions f1lz), f2(z) where z is a local parameter on S. Then as necessary conditions we have F21, and = whenever 0, 0. As sufficient condition we Im (Ti + T2} = 0 have:
Let S0 be a simply connected Riemann surface and let g:S0 be a map into the complex quadric Q2. Let g be ripresented locally by a pair Theorem 6
of complex functions f1(z), f2(z) by (4.4), (4.5).
Then we have 147
If F1 = F2 0, then g factorizes via the Gauss map of a minimal surface In P' provided that S0 is not compact. 1
If F
are never zero, then g factorizes via the Gauss map of a surface S in P given by a conformal inwnersjon of S0 if and only if and Im + T2} = 0. Moreover, S is uniquely obtained up to a IF1 I translation and homothety of P4. Unlike the situation discovered by Kenmotsu for P3, harmonic maps into Case 2
and F
S2 generally do not represent Gauss maps of surfaces in This fact shows the essential difference between Gauss maps of surfaces immersed in P3 and inP4. s2
Acknowledgement
I am grateful to Dr L. Woodward for helpful discussions. References
(1]
Hoffman, D.A. and Osserman, R.
Thc geometry of the generalized Gauss 236 (1980). Hoffman, D.A. and Osserman, R. The area of the generalized Gaussian map, Mem. Amer. Math. Soc. No.
(2)
image and the stability of minimal surfaces in 5" and P", Math. Ann. 260 (1982) 437-452.
[3] Hoffman, D.A. and Osserman, R.
The Gauss map of surfaces in
.1.
Diff. Geotn. 18 (1983) 733-754. (4] Hoffman, D.A. and Osserman, R. The Oauss map of surfaces in P3 and preprint. [5] Hdrinander, 1. An Introduction to Complex Analyais in Several Van Nostrand, Princeton (1966). (6] Kenmotsu, K. Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979) 89-99. (7] Ruh, E.A. Asymptotic behaviour of non-parametric minimal surfaces, J. Diff. Geo,n. 4 (1970) 509—513. (8] Ruh, E.A. and Vllms, J. The tension field of the Gauss maiS, Trano. Amer. Math. Soc. 149 (1970) 569-473. (9] Wlllmore, T.J. Tight lninerslons and total absolute curvature, Bull.
Lond. Math. Soc. 3
T.J. Wlllmore University of Durham Durham DH1 31E
U.K. 148
(1971) 129-151.
JCWOOD
The Gauss map of a harmonic morphism INTRODUCTION
Harmonic maps between Riemannian manifolds (see, for example, (5,6)) have many special properties when the domain is two-dimensional, for example,
(a) conformal invariance, (b) existence of a holomorphic quadratic differential and (c) (anti-)holomorphicity of the Gauss map (or section) of a minimal branched ininersion. In general, harmonic maps into a two-dimensional Riemannian manifold do not have such however, for a subclass of the harmonic maps, namely the harmonic moi'phiama, properties similar to (a),
(b) and (c) are obtained. The most striking of these is the holomorphicity, in a certain sense, of the Gauss map (or section) of a harmonic morphism from a four-dimensional Riemannian manifold to a Rieinann surface. This is discussed in Section 3. E. and J. Vilms [12] showed that the Gauss map of a minimal Imersion into Euclidean space is always harmonic. In view of its holomorphlcity, it might have been expected that the Gauss map of a harmonic morphism would be always harmonic. However, this Is not the case and In Section 4 we give some conditions for harmonicity of the Gauss map of a harmonic morphism from an open subset N4 of R4 to a Riemann surface. In fact, our results apply to the Gauss Bection of a harmonic morphism from any four-dimensional space form M4 using the concept of harmonic section as studied by C.M. Wood [13). Some further results, examples and full proofs are to be found in [15]. 1.
HARMONIC MAPS FROM A SURFACE
be smooth Riemannian manifolds of dimensions m Let (M",g) and 2 we have the be a smooth map. In the case m and n and let ,:Mm + following well-known properties:
(a)
Conformal invariance [83.
i.e. at each point p
-, (M2,g) be a conformal map,
Let
vX c
= A(p)
M,
positive real number called the conformatity t:(M2,g)
÷
be harmonic.
Then
factor of
is harmonic.
where x(p) is a at p, and let
(In fact,
may be 149
weakly conformal in the sense that A may hate zeros.) As a consequence, • (N",h) depends only on the Conformal harinonicity of a smooth map
equivalence class of the metric g, In particular, the concept of harmoniclty of a smooth map from a
eui'face Is
well—defined.
be a smooth (b) The holomorphic quadratic differential. Let 4:M2 + of = the (2,0)—part of the pull—back map from a Rlemann surface. Set the metric h, thus n Is a tensor of type (2,0) on P12, i.e. a section of
Explicitly, If z is a local complex coordinate, 11 the inner product being the complex bilinear extension to TCN • TN e C of the inner product defined by h. Then (see, for example, (14])n 0 1ff • is weakly conformal and n is holomorphic If 4 is harmonic. This Implies, for example, that any harmonic map from the Rlemann sphere is weakly conformal.
(c) Gauss maps and Gauss sections. Let +:M2
be a non—constant
weakly conformal map from a Riemann surface to a Euclidean space. Let be the Grassnannian of oriented 2-planes in Euclidean n—space, recall
be given the structure of a complex manifold by embedding it in that this of 4 is defined at points (see t4]). The Gauee nrzp y:M2 + Is two-dimensional by y(x) the Image x£N the Gauss .sp y extends smoothly to M2 and Is
Then,
if and only if th. map • Is harmonic. This fact may be generalized as follows: let
denote the bundle be an arbitrary Riemannian manifold and let fibre at q E N is the Grassmannian G0(T N). Given a non-conover N stant weakly conformal map •:N • N define Its Gauae aection y: +4 G2(TN) is two-dimensional by y(x) (Here at points x to N via 4.) Again we see that denotes the pull-back of section P12 and ie an the Gauaa mop 'V extende eircothiy to PP
if
and only if 4 is hcmaonia. This fact is the starting-point for the
twistorial constructions of harmonic maps of, for example, tills and Salamon
(7]. 2.
NORPHISMS
A smooth map
V of monic. 130
-' (N'1,h) between Riemannian manifolds is called a
morphism if, for any harmonic function *:V • from an ppen subset Is non-empty, the composite function o$ is harsuch that From this defililtion we Inunediately deduce: (a) oonferirai invariance:
if 'p:(N2,h)
is
(N'2,h') is a (weakly) conformal map and
a harmonic morphism then
concept of har,nonic
1P°i1
is
morphi8m from
a harmonic morphism.
In particular, the
a Riemannian manifold to a Rlemann surface
is well defined. To proceed further, we recall that, for a smooth map ker its orthogonal is defined by the vertical space at p, in is called the horizontal space at p. Then • is said to complement be horizontal'y (weakly) conformal [10] (or semi-conformal [9]) if, at each either = 0 or maps the horizontal space at p conformally point p B. Fuglede and 1. Ishihara [9, 10] characterized harmonic onto (Mm,g) + (N'\h) is a harmonic morphism morphisms as follows: a smooth map if and only if it is a horizontally weakly conformal harmonic map. In particular, if m < n, any harmonic morphism is constant: if m n, Fuglede showed that the harmonic morphisms are precisely the homothetic maps,'i.e. the conformal maps with constant conformality factor. Let us now consider the case m > n = 2. For any smooth map •:M" N2 let K denote the set of points p N where rank < n. Baird and Eells [3] showed that a non-constant horizontally weakly conformal map •:t41' N2 is a harmonic morphism if and only if the fibres (of are minimal. Let + N be a smooth map with K At points p 1f4( the horizontal space M. • H is two-dimensional and acquires an orientation from that of T N2 by T demanding that d'P :H N be orientation preserving. Define an almostcomplex structure 3 P on by 3 P = rotation through 3 P to linearity and set ® C by the +1 elgenspace of 3 thus H }. Note that the d'} define a bundle H' over H' = X p
p
p
Building on ideas of P. Baird [2] we define a section n of e2 by v seen that n Then it is 0 (on t+-4() if and (Z,Z') only if 'P is horizontally weakly ' (on M) and that n is horiz —iwlomorphic (i.e. V2n = 0 for all p Hi,) if and only if the fundamental form of 'P has zero trace eier horizontal space H for all For example, 11 is horizontally holomorphic if N .. K. p -, N is harmonic and has minimal as an application we note: Proposition 2.1 Let N2 be a harmonic map with minimal fibres and integrable horlzePital distribution whose leaves are hoeieomorphlc to 2-spheres. Then 'P
a harmonic morphism. 151
Proof
The differential
is holomorphic on each horizontal leaf and is thus
zero.
Examples of harmonic morphisms include (1) radial projections (ii) the Hopf maps S3 and (iii) any holomorphic map from a Kähler manifold to a Rietnann surface. All these maps are horizontally weakly conformal and have minimal fibres. For more examples see (II.
defined by x .+ x/Ixt;
3.
HOLOMORPHICITY OF THE GAUSS MAP OF A HARMONIC MORPHISM
Let G0(R4) denote the Grassmannian of oriented 2-planes in the Euclidean
(with its standard o:'ientation). Then it is well known that there isa diffeomorphism which is an isometry if the GrassS2 x space P
mannian and the spheres are given their usual metrics (the spheres having radius 11,2). Further, giving the Grassmannian and the spheres their usual complex structures, i is biholomorphic.. The map i may be described as follows: c given any oriented 2-plane he its orthogonal complement, let P oriented such that IL as oriented vector spaces. Define metric r almost complex structures J p and J p on jj and ri to be rotation through these together gives an almost complex structure J (J P, J P )) on P which is metric and orientation preserving. The set of all such almQst complex structures is a 2—sphere. On the other hand = defines an almost complex structure on P4 which is metric
and orientation reversing; the set of all such almost complex structures is again a 2-sphere. The diffeornorphism i is then given by i(n ) (J1(n ), P2. p 4 2 4 J (no)). Now let H be an open subset of P and let •:tf -, N be a submersion. Then we define its Gauss map y:M4 + G°(P4) by y(p) the vertical space V at p. Under the identification of ) with S x S , y has two p S ; in fact, we see that in the notation above, components y ,y :11
i
124
11(p)
2.
and y2(p) = Let •:M4
Theorem3.1
We study holomorphicity of these:
N2 be a non-constant harmonic morphism from an open
= (p subset of P4 to a Riemann surface. Let K = M : rank d4, < 2), we p 4 may define y and y 2 Then y is holomorphic with respect H 4K as above. 1
to the almost complex structure on M4 and respect to the almost complex structure on Remark 1. 152
1
is antiholomorphic with
The holomorphicity is an easy consequence of the fact that the
foliation defined by the fibres of
has minimal leaves and is conformal.
The result extends to harmonic morphisms from an arbitrary four— dimensional Riemannian manifold if we consider y as a section of the Grass— 2.
Remark
This bundle is the product of the celebrated twistor over M whose fibres at p M consist of all metric almost bundles and which are, respectively, orientation preserving complex structures on are given by and y2:M or reversing, and the sections y1:M mann bundle
y2(p)
=
=
see (15). be a non-constant holomorphic map from an open
Example 3.2
Let 4:M4
subset of
to a Riemann surface.
Note that • is a harmonic morphism. Then (J1(V )} is the almost complex structure on M4 induced from the standard p 2 2 complex structure on and the map y is constant whereas the map y is antiholomorphic in the usual sense. Conversely, let + be a nonconstant trirmonic morphism from an open subset of to a Riemann surface. Then, if y1 constant, the J1(V ) define an orthogonal complex structure p 4 on F with respect to which is holomorphic. 1
4.
HARt4ONICITY OF THE GAUSS MAP OF A HARMONIC MORPHISM
be a smooth map from an open subset of P4 to a Riemann surface. vertical spaces define an integrable foliation F of whose leaves are the fibres of Call this foliation Riemanni.an if the metric Let
-.
the horizontal spaces does not change in the vertical by which we mean that the Lie derivative =0W (see [11)). For example, if is a Riemannian submersion then the corresponding foliation will be Riemannian. Using the fact that the Gauss map is (see Section 3), we can calculate its tension field and prove the following: Theorem 4.1
Let q,:M4
N2 be a non-constant harmonic morphism from an open
whose subset of P4 to a Riemann surface. Let F denote the foliation of -* as in leaves are the fibres of q. Define the Gauss map Section 3 by y(p) = vertical space at p. Ci) If the leaves of F are totally geodesic, then thb Gauss map y is
harmonic.
(ii) If the distribution of horizontal subspaces is integrable, then y is harmonic if and only if the foliation is Riemannian or has totally 153
geodesic leaves.
(iii) If the foliation is Riemannian, then y is harmonic if and only if the distribution of horizontal subspaces is integrable or the foliation has totally geodesic leaves. Thus we cannot expect the Gauss map of a harmonic morphism to be harmonic
except for rather special harmonic morphisms, for example, those with totally geodesic fibres or those which are submersions. The theorem holds for the Gauss section of a non-constant har-
Remark 4.2
monic morphism from a space form M4 defining harmonic
section as
in (13].
References
(t]
Baird, P.
Harmonic Maps with Symetry, Harmonic Mor'phisins and
Deforrmzti_ons of Metrics, Research Notes in Mathematics, 87, Pitman (1983).
(2] Baird, P. The Gauss map of a submersion, Preprint, Centre for Plathematical Analysis, Australian National University, •Canberra (1983). (3] BaIrd, P. and Eells, J. A conservation law for harmonic maps. SWnpoeium, Utreoht 1980, Lecture Notes in Mathematics, Springer—Verlag (1981) 1.25.
(4]
them, S.-S. Minimal surfaces in Euclidean space of N dimensions, Synrpoaiwn in Honor of Marston Morse, Princeton University Press (1965) 181-198.
(5] Ealls, J. and Lemaire, L. A report on harmonic maps, Bull. London Math. Soc. 10 (1978) 1—68. (6] Eells, J. and Lemaire, 1. Selected Topics in Harinonio Naps, Regional
(7]
Conference Series in Mathematics no. 50, Mierican Math. Soc. (1983). Lells, qi. and Salamon, S. Constructions twistorielles des applications harmoniques, C.R.
(8] Eells, J. and
Acad.
Sci. Paris I, 296 (1983) J.H.
685-687.
Harmonic mappings of Rietnannian manifolds,
J. Math. 86 (1964) 109-160.
(9) Fuglede, B.
Harmonic morphisms between Riemannian manifolds, Ann.
Inst.
Fourier 28 (1978) 107-144. (10) Ishlhara, T.. A mapping of Riemannian manifolds which preserves harmonic functions, .1. Math. ICyoto Univ. 19 (1979) 169-174.
154
(11)
Reinhart, B.L. Differentiai
of Foliatione, Springer-Verlag
(1983).
(12]
Ruh, E.A. and Vilms, J.
The tension field of the Gauss map, Trane.
Amer. ltzth. Soc. 149 (1970) 569-573.
Some energy-related functionals and their vertical variational theory, PhD thesis, University of Warwick (198,3).
[13]
Wood, C.M.
[14]
Wood, J.C.
Harmonic maps and complex.analysis, Proc.
in Complex Analysie, Trie8te 1975 (I.A.E.A.,
Szeiner Couree
Vienna, 1976) vol. III,
289-308.
[15]
Harmonic morphisms, foliations and Gauss maps, preprint, University of Leeds (1984).
Wood, J.C.
J.C. Wood School of Mathematics
University of Leeds Leeds LS2 9.JT U.K.
155
R A BLUMENTHAL
Sprays, fibre spaces and product decompositions 1.
INTRODUCTION
In this paper we are concerned with the following two questions. Question 1
When is a mapping between manifolds a fibre bundle?
Given a manifold with some geometric structure, does it decompose as'a topological and geometric product? Of course these questions are closely related since a fibre bundle is locally a product and, when trivial, is so glob3lly. We shall treat these Question 2
questions from the point of view of foliations using as our main technical too,2 the notion of a spray on a manifold. In 1947 C. Ehresmann (3] showed that a submersion defined on a compact
manifold is a fibre bundle and in 1960 R. Hermann (4] proved the following generalization. Theorem
(R. Herniann [4]).
Let 14 and N be connected Riemannian manifolds
&nd let f:M N be a Riemannian submersion. If 14 is complete, then f is a fibre bundle and N is complete.
N. Hicks (5) obtained a similar result for affinely connected manifolds in the case where N and N have the same dimension.
(N. Hicks (5]). Let N and N be connected manifolds of the same dimension each carrying an affine connection. Let 14 be complete and let f be a local diffeoniorphism of 14 into N. Then f is a covering. Let X be a spray on the tangent bundle 1(14) of a manifold 14. We say that Theorem
asubbundle Q of 1(M) is totally geodesic if Q is a union of integral curves of X. In Section 2 we prove: Let N and N be connected manifolds with sprays X and V respectively and let f:M N be a submersion. Let E c T(M) be the kernel of and let Q c T(M) be a complementary totally geodesic sub—bundle such that XIQ
Theorem A
156
is fe-related to V. If XIQ is complete then f is onto, f:M trivial fibre bundle, and V is complete.
N is a locally
It is easy to see that the theorems of Hermann and Hicks are irmuediate corollaries of Theorem A.
Our interest in Question 2 is from the point of view of generalizing the well-known decomposition theorem for Riemannian spaces (see, e.g., [11)) to affine symetric spaces. Let Gill be a simply connected affine symetric space. Let V be the canonical connection on Gui and let R be the curvature tensor of V. In Section 3, using spray-preserving maps and affine foliations, we prove: Theorem B
Let x0
G/H and let E1
of
(G/H).
If R(X,Y)
0 for X
be holonomy invariant subspaces
and E2 xO
xO
,
E1
x0
0
V
then there exist simply
E2
x0
connected affine symetric spaces G1/H1 and G2/H2 such that G/H is isomorphic
to the affine product (G1/H1) x
(G2/H2).
We remark that 1. Ozeki has given an example of a synm*tric linear con-
nection whose linear holonomy group is completely reducible but which is not a product even locally; the curvature condition in B is not satisfied. 2.
PROOF OF THEOREM A
Let M be a smooth manifold and let 71:T(M)-'M be the tangent bundle of Let X be a vector field on T(M). Then X is a spray if 1T* X IdI(M) and 1(M), c ER where p :T(M) 1(M) is multiplication by = CLI (Xv) for v For be the integral 1(M) let of X through v and let = Then X is a spray if and only if civ' = av and in which case =
the curves are the geodesics of X and the exponential map at a point p M given by exp(v) = cxv(1) maps a of 0 in diffeomorph-
ically onto a neighbourhood of p in M.
Since f is a submersion, f(M) is open in N.To show f is onto it suffices to show that f(M) is also closed. Let q f(M). We now prove Theorem A.
Let V be a neighbourhood of q in N and U a that exp:U -* V is a diffeomorphism. let z V fl f(M). exp(u) = z. Then = q, = u, and cL(1) = z.
let v
be the unique vector satisfying
0 in Tq(N) such Let u U- be such that Let p and
=
157
Then p p
is
*(v)
a geodesic in N satisfying p(O)
=fo
since
i, p(O) = -
and V are fe—related.
Now
and so
is defined
fOl) and so f(M) is = q. Then q is complete) and closed. Clearly V is complete. Let q N. Let V be a neighbourhood of q in N and U a neighbourhood of 0 In Tq(N) such that exp:U + V is a diffeómorphisrn. Let L (1{q). Define U be the unique vector x L -, N as foltows. Let (z,p) LV x L. Let u = u, be the unique vector satisfying satisfying exp(u) = z, let v E x 1 + f1(V) and is project'ion and set •(z,p) = exp(v). Clearly onto the first factor. Define V x L as follows. Let x f1(V). V. Then y = exp(u) where u Let t be the unique geodesic U. Let y f(x) In N satisfying t(0) = q, r(0) u. Then i(1) y. Let v be the unique vector =-(1). Let o be the unique goedesic in N satisfying o(O) * v. Then o(l) C L. Indeed, If we let p(t) t(1—t), then
fo p •
and p are geodesics in N satisfying the same initial condition, whence 1 o a and so f(a(1)) = p(1) = q. Set 'y(x) • o(1)). Then is a
Inverse to 0 (this uses that Q is totally geodesic) and so • is a diffeomorphis. which completes the proof of Theorem A.
To obtain Hermann's result, let X and V be the sprays arising from the Then Q is totally geodesic and XIQ is fe—related to V [10) (also c.f. [9]). Hicks' theorem ininediately by taking Q = T(M). Riemannian connections and let Q = E1.
3.
MAPS AND AFFINE FOLIATIONS
An affinely connected manifold is a pair (M,v) where N is a connected smooth manifold and v is a linear connection on M. A map f:(M,v) (M',v') is is X(M'), then affine If, whenever X,Y X(t4) are f-related to X', V' In the case where v and v' are torsion-free, f is affine f—related to
if and only If it preserves the sprays arising from v and v'.
We remark
that, given sprays X and X' on manifolds M and N' and a spray—preserving
map. f:M + M', there exist torsion-free linear connections v and v' on H and N' relative to which f is affine [1]. We say that a codimension q foliation F of an affinely connected manifold (14,V) is affine if F can be defined by an N—cocycle where N is a (not necessarily connected) €A q-dlmenslonal manifold with a linear connection Vt and each N is an affine submersion. 158
In the current literature the tern "affine foliation" is
and V1 is the canonical flat used to designate the situation where N is We are using the term more generally to allow N to be connection on any manifold and v' any linear connection on N
Lemna 3.1
Let (M,v) be a complete affinely connected manifold where v has
parallel torsion and curvature and let F be an affine foliation of N.
Then
F is transversely modelled on a homogeneous space G/H and the developing map
H -. GIN is a Serre fibratlon. Let I and R (respectively, 1' and R') be the torsion anc! curvature A and consider tensors of V (respectively, v'). Fix fa(Ua)cN. Proof
and VIP', An elementary argument shows that vi and yR are f-related to respectively, and hence (v'T')IV = assume that 0. By [8) wc Is an open subset of a simply connected reductive homoge.neoos space (G/H) V . It is and that v'Iv .= V IV where V Is the canonical connection on Cl me transeasy to see that each f8(U n U8) + f(U n U8) is an formation and hence can be uniquely extended to an affine is elementary that (GIN) and (G/H)8 are affinely isomorphic for all A and hence there is a simply connected reductive homogeneous space GIN upon which F is transversely modelled. There is a
• G/H constant along the leaves of the lift of F to the universal cover N of N [2]. Clearly f is affine with respect to the lift of V submersion
The proof of the lemea will be completed once we establish •jhe following claim. to
(*) Let (M,v) and (M',v') be affinely connected manifolds with H complete and let f:M + N' be an affine submersion. Then f is a Serre fibration. We sketch the proof of (*)• Let 1T:F(M) • M be the frame bundle of H, let E • kernel (fe) c 1(M) and let ,r:A(M) + N be the bundle of adapted frames, a There is a natural bundle homoreduction of.F(N) to the group G = morphism F:A(M) • F(M') covering f whose corresponding homomorphism between B. Clearly the connection in F(N) structure groups is given by sends horizontal vectors to horicorresponding to v reduces to A(H) and zontal vectors. For X In the Lie algebra of G let X* be the corresponding (q = dim H') let fundamental vertical vector field on A(M) and for h 8(h) be the basic horizontal vector field on ACM) correspondlng.to dim 14). Since F is a submersion it defines a foliation (O,...,O,h) e (n 159
If
FofA(M).
is the sub—bundle of T(A(M)) and 1s a basis X* where X e gl(q,R), then we may regard Q
spanned by
the normal
of f.
An elementary argument shows that
are parallel along the leaves of I and so F is transversely complete whence of leaves A(M)/F is a smooth Hausdorff manifold and the natural the a locally trivial bundle [7]. Using Theorem A project,on q:A(M) one can show that the oap h:.YM)/F -. 1(1') induced by F is a covering. From -+
:hese facts it is not diff cult to show f has the homotopy lifting property with respect to any finite polyhedron. Before applying Lemma 3.1 to prove Theorem B we make a few remarks con-
cerning affine products. Let (M1,v1) and (M2,v2) be affinely connected manifolds. Let M let 1T:M M2 be the projections, and M1 and p:M let E1 (respectively, E2) be the kernel of (respectively, Let and and E2 respectively. The connections on = = product connection on I'l is the linear connection ' on M defined by for X,Y X(M). Then TT and p are affine submersions, =c +c •
Xl
X2
E1 and E2 are holonomy invariant distributions on M and the curvature R of satisfies R(X,Y) = 0 for X r(E1), V 1(E2). Assume now the of Theorem B. Using the parallel transport determined by V, the subspaces E1 , can be spread out to holonomy x
x
invariant distributions E1, E2 c T(G/H) and since the curvature of the canonical connection on G/H is holonomy invariant it follows that R(X,Y) = 0 and E2 for X £ r(E1), V e 1(E2). Since v is torsion—free, we have that and F2 of G/H are autoare involutive and the corresponding foliations parallel. Let = vIr(E2), a connection on the normal bundle E2 of F1. An elementary argument shows that V2 is basic. Let U be an open set in C/H and let f:U V be a submersion constant along the leaves of F1 where V is an q = codim F1. Since is basic there is a linear connection open set is V2 on V such that and it is easy to see that f:(lJ,v) + =
affine.
is an affine foliation of (G/H,v) and hence, by Lemma 3.1, is a simply connected affine syninetric space and the natural projection is a Serre fibration and an + affine submersion where has its canonical connection Similarly, the space of leaves (G/H)/F2 is a simply connected affine syninetric space G2/H2 and the natural projection ii2:G/H + is a Serre fibration and an affine Thus F1
the space of leaves (G/H)/F1
160
submersion.
Define F:G/H + (G1/H1) x (G2/H2) by F(x) = (,r1(x), 1T2(x)).
Let
a complete Since Li is autoparallel, V induces.on G2/H2 is a local affine isomorphism linear connection such that ,T2111 :
be a leaf of F1.
is a diffeomorphism. Similarly, if 12 is a leaf of F2 , then 12 + G1/H1 is a diffeomorphism. It is now easy to see that F is a and iT2 are affine mappings, V must be the product diffeomorphism. Since and so
connection. References
El]
Ambrose, W., Palais, R.S. and Singer, I.M.
[2]
(1960) Blumenthal, R.A.
Sprays, An.
Acad. Brae.
32
homogeneous foliations, Ann.
Inst.
Fourier (Grenoble) 29 (1979) 143-158. [3)
Ehresmann, C.
Sur les espaces
C.R. Acad. Sci.,
Paris 224 (1947) 1611—1612. (4) Hermann, R. A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc. ii (1960) 236-242. [5] Hicks, N. A theorem on affine connections, Illinois J. of Math. 3
(6]
(1959) 242—254. Kobayashi, S. and Nomizu,. K.
Foundations
of Differential Geometrij,
Interscience Tracts in Pure and Appl. Math., 15,
New
York (1963). (71 Molino, P. Etude des feuilletages transversalement couplets et applications, Ann. Scient. Sc. ,Norm. Sup. 20 (1977) 289-307. [8] Nomlzu, K. Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954) 33-65. [9] O'Neill, B. Submersions and geodesics, Duke J. 34 (1967) 459-469.
(10] Reinhart, B.1.Foliated manifolds with bundle-like netrics, Annals of Math. 69 (1959) 119—132. [11] Wolf, J. Spaces of Constant Curvature, McGraw-Hill, New York (1967). Robert A. Blumenthal Department of Mathematics
Saint Louis University St. Louis, NO 63103 U.S.A.
161
J GIRBAU & M N!COLAU
Deformations of holomorphic foliations and transversely holomorphic foliations main problem we wish to present here will be correctly formulated in Section 4. We shall try to precisely the results we have obtained, giving indications of proofs of some of them. We shall give also some concrete examples to illustrate these results. Detailedproofs will be given The
elsewhere.
First of all let us start by recalling briefly the main facts of the classical theory of deformations of complex structures. 1.
DEFORMATIONS OF COMPLEX STRUCTURES
This theory, initiated by Kodaira and Spencer, was completed by Let N be a compact complex manifold. Denote by J its almost-complex structure. Denote by 8 the sheaf of germs of holomorphic vector on M. Kodaira, Spencer and Niremberg showed that if H2 (14,8) 0, then there is a family of integrable almost-complex structures (that is, complex structures) parametrized by a neighbourhood V of the origin in whose expression Jt(x) in the coordinates of M depends differentiably on x and holomorphically on t, such that J0 = J and, given any other family of complex 'structures in N fulfilling the above conditions and parametrized by a neighbourhood of the origin U in a complex vector space, then there is a map f:O V c H1(M,e), such that in a neighbourhood of 0. More-
over (df)0 is unique. , whoseI existence the theorem asserts, the Let us call the family {J ttEV } — versal family. Kuranishi succeeded in removing the condition H2(M,e) = 0 from the above result, but at the cost of allowing the versal family to
In the Kuranishi theorem the versal family is parametrized by an analytic set, that Is, a set of zeros of a finite number of holomorphic equations. For this reason one must deal, from the beginning of the theory, with deformations be parametrized by more general spaces than complex vector spaces.
parametrized by complex spaces and not by vector spaces as Kodaira and Spencer did at
162
In 1964 Douady gave a short (but very concentrated) v,ØPslon of the Kuranishi theorem (2] using non-reduced analytic spaces as spaces of para-
This version will be our main reference for this theory.
meters. 2.
FOLIATIONS
Denote by rtr the pseudogroup of local differentiable automorphisms of
XRm, f(Za,Xu) = (fa,fu), fulfilling
=
az
and u =
1
0,
wherea,b=1,...q
... m.
Let M be a differentiable manifold of dimension 2q + m. A transversely holomorphic foliation on N can be defined as a - manifold, that is, by an atlas whose changes of coordinates are given by elements of rtr. We are then in the presence of a foliation having a complex structure in the local transverse slices. For example, the foliation In 52q+l whose leaves are the
fibres of the Hopf fibratlon
+
is of this type.
The notion of holomorphic foliation on a complex manifold is well known and need not be defined here. Of course, all holomorphic foliations are, in particular, transversely holoniorphic. The theory of deformations was extended early from complex structures to holomorphic foliations by Kodaira and Spencer (5). A theorem analogous to that of Kuranishi was not given in that paper because the classical theorem
of Kuranishi for complex structures did not yet exist at that time.
How-
ever, such a theorem can be derived now from the paper of Kodaira and Spencer.
For real foliations a theorem of this kind does not work at all, but for transversely holomorphic foliations such a theorem was given by Duchamp and
Kalka [31 and in a little more general form by Girbau, Haefliger and Sundararaman (4] using non-reduced analytic spaces as spaces of parameters, as in Douday's version. 3.
ANALYTIC SPACES
We have to deal, for technical reasons, with non-reduced analytic spaces. From the local point of view, the only one we shall need here, such a space is given by a couple (R,OR) obtained in the following way. We start with an 163
be the sheaf of germs of open neighbourhood U of the origin of Cm. Let holomorphic functions on U. Denote by J a coherent sheaf of ideals of to R and 0R is the R is the support of the quotient sheaf of Let us give you two examples to illustrate this concept.
(I) Take a finite number of holomorphic equations in a neighbourhood U of the origin of
f1 = 0 = 0. Let R be the set of zeros of these ... consisting of those germs of functions functions. Let J be the subsheaf of vanishing on R. J is a coherent sheaf of ideals. Take = (Ou/J)IR. The couples (R,OR) obtained in this way are the classical reduced analytic spaces.
(ii)
Let U be the complex line C.
Take the equation z2 = 0.
Let J be
the ideal of generated by the germs of the function z2. The support R of the quotient is a space consisting of a single point, the origin. The ring 0 = (0 /J)IR is the of numbers of the form a + be with a,b
C and e
= 0.
This is an example where the topological space R is a
single point, but the ring 0R is not trivial. For a concise summary of the basic notions concerning analytic spaces we
refer to (6]. 4.
DEFORMATIONS OF FOLIA1IONS.
STATEMENT OF THE MAIN PROBLEM
The main problem we wish to present here is the following one: given a holomorphic foliation F in a compact complex manifold M, compare the holomorphic deformations of F with the transversely holomorphic deformations of F. To be more concrete, let us define what we mean by holomorphic and transversely holomorphic deformations of F. an analytic space R we shall denote by rR the pseudogroup of local x differentiable autoniorphisms of R x of the form f(r,za,zu) = (r,fa,fu) fU zU, holomorphiwhere ? and depend differentiably on the variables r,
cally on r, and such that afa
U
1
... p.
3fa3fa
We shall denote by r the sub-pseudogroup of r tr consisting of those elements of which depend holomorphically on r, Let rR be the 164
sub-pseudogroup of rR consisting of those elements of such that the fa do not depend on r. Suppose that we have a holomorphic foUation F in a compact complex manifold M. Let R be an analytic space and denote by 0 a distinguished point of
A deformation of F as transversely holomorphic foliation (briefly, a tr-h-deformatjon) parametrized by (R,O) is a topological space X with a proper projection p:X R and a on X such that, In the local coordinates (r,za,zu) of this structure, the projection p is the canonical projection (r,ia,zu) r. This -structure induces on each fibre p1(r) a that is, a transversely holomorphic foliation. We require also that M = p1(D) and that the foliation induced on M by is F. In the same way, but using rR and instead of we define the notions of holomorphic deformation of F (briefly, h-deformation) and deformation of the coir'lex structure of the leaves of F (briefly, HF-deformation). The P.
notions of tr-h-deformation and h-deformation are well known. However the notion of HF-deformation is new. One can show that if p;X + R is one of these
three kinds of deformations of F, all the fibres
in a neighbourhood of In other words, such a deformation changes the foliation as well as the complex structure, but not the manifold from the differentiable point of view. One can also show that an HF-deformation does not deform F from the transversely holomorphic point of view. It only deforms the complex structures of the leaves. The first result we wish to present here is a Kuranishi theorem for HFdeformations of F. This can be stated in the following way. O are diffeomorphic.
There is an analytic space S with a distinguished point 0 S and a S of F parametrized by (S,0) such that, for any other HFdeformation V P of F parametrized by (R,O), there is a holomorphic map of germs of analytic spaces f:(R,O) -' (S,O) such that the germ of the deformation V at 0 is to the germ of the image f*(X) + R. Moreover, the differential d0f of f at 0 is unique. S is called the Kuranishi space Theorem
HF—deformation X
for HF-deformations.
The theorem does not assert that the morphism f is unqiue. In the case that this happens for any HF-deformation of F we shall say that S is universal. Remark
Denote
K the Kuranishi space for the h-deformations of F. Let I be 165
the Kuranishi space for tr—h-deformations of F. These Kuranishi spaces exist by virtue of (3], (4]. Let S be the Kuranishi space for the HFdeformations of F. This space exists by virtue of the above theorem. We examples where K = S x I and give some sufficient conditions shall give
for the splitting K • S x 5.
I.
SKETCH OF PROOF OF THE ABOVE THEOREM
Each proof of a theorem of this type involves a resolution of the fundamental sheaf associated to the problem. Let us show the fundamental sheaf and the resolution used here. Denote by e the sheaf of germs of holomorphic vector fields preserving Za) adapted to F (that is, the Initial foliation F. In a local chart = constant), o is the sheaf of germs of holothe leaves of F are given by morphic vector fields of the form +
U
depend only on the coordinates zb (of courses holomorphically). o is the fundamental sheaf used by Kodaira and Spencer to study the bobmorphic deformations of F. Let us recall the Kodaira-Spencer resolution. Denote by A*(M) the algebra of C-valued differential forms on M. Let F be the sub—bundle of CT(M) which in each adapted local chart (U,Za,Zu) is gena/azU, aiar'i. Denote by 1F the Ideal of A*Q4) consisting erated by of those differential forms whose restriction to F vanishes. is generated by (dZa}. Let 'T be the ideal consisting of those differential forms whose where the
restriction to T°'1 vanishes.
is generated by {dZa, Denote by of A*(M) of degree k the complex vector space of complex derivations According to Kodaira and Spencer, each such that &(IF) 'F and &(IT) c I
derivation of degree k is given in a local chart (U,Xa) by a couple of vector differential forms of degrees k and k+1,
•=
=
9
= is the ordinary k-form •a = It can be and proved that • and is are global differential vector-valued forms not). A derivation belongs to if, in a local chart (tJ,?,ZU)
where
166
We define an operator D:Vk the where d is the exterior derivative. Denoting by • by sheaf of germs of eleme*ts of we have an exact sequence adepted to
belongs
0
•0
0k+1
LF
(1)
The kernel of is the sheaf e of differentiable vector fields whose expression in an adapted local chart is +
a
az
az
a
az
az
where
quotient of
by a fine subsheaf, so the cohomologies of o and ô coincide except in degree 0. The resolution (1) of is essentially due to I(odaina and Spencer and it is used to prove the Kuranishi theorem for holomorphic foliation. a
We need to describe briefly the Duchamp-Kalka resolution for transversely holomorphic foliations.
We can define what we call the space of F-differential forms, as the quotient A*(M)/IF. A*F is locally generated by the classes [dz"), [dr]. The exterior derivative d of M induces a derivative dEin A*F(M) by = Let A(v) be the space of F-forms with values in the normal bundle of type (1,0), V1'0 = T(M)/F. An element of A*r(v) can be expressed locally by
where
is an F-form.
=
@ 1.ajaza].
We define the operator by Denote by the sheaf of germs of elements of Ak(v).
We have a sequence d
The kernel of
v,.
i
F
+
d
F
is the sheaf
•..
°tr
.
(2)
of germs of holomorphic sections of
167
(2) gives us a resolution of o which is the Duchamp-Kalka resolution. in the following way. natural projection ir:Vk We car define locally, where • is a global differential is a couple Let 6 such that
as the element of
We define
form.
oYI_.Yk) = where V
... V are sections of F and p is the canonical projection
CT(M) + v
'
If k = 0 iT is defined by ¶(6) =
.
projection .
such that DoiT = ir°d Denote by ô the kernel of
restricts to an operator
o
.
>
.
-.
Denote by
-, 0
.
the kernel of
The operator
•
In this way we have a resolution of
> •f
0
In this way we have a
(3)
restricts to a bracket in Moreover, the bracket t , j that we have in can be considered as the sheaf of sections of an approEach sheaf priate vector bundle E . Moreover, the complex r(E0)
>
r(E1)
>
r(E2)
is elhptic. With all these ingredients one can prove the Kuranishi theorem for Hr-deformations in a standard way.
For subsequent use let us write here the equations giving the Kuranishi space S corresponding to the HF-deformations of the initial foliation F. of all, take a real analytic Hermltian metric h in M and define Nermitlan product in in the following way. Given 6, 6' vk, suppose Recall that = that *5 Is the cosple locally, and *5' and - £' are globally defined. W. define <6,6'> to be >+ the products induced where the products < ,> In the second the space of differential forms. Denote by by h in the > the restriction to 0k of the product Denote also by < sections of Let D* be the adjoint operator of t) with respect to < , > in < , > In S Is described in the following way. S is given by
168
S
(.5
Nf
H(S,.S]
:
0),
(4)
where (ts
:
5
HtS
(5)
+ (1/2) D*G1.S,SJ},
H is the harmonic projection and G the Green operator. 6.
SOME EXAMPLES
(1) Let us recall first an example of (4). Suppose that we have a compact holomorphic fibre bundle p;X + B over a complex manifold B and suppose that
we have a holomorphic foliation on X transverse to the fibres of the bundle. Denote by KB the Kuramishi space of the deformations of the complex structure of B. If KB is smooth (that is, it is a complex manifold) and if some cohømological conditions are fulfilled (for example H'(M,OM) = 0, where M is the
fibre of p:X -+ B), then it is proved in [4) that K splits Into a product K = T x K8 (see Remark in Section 4 for the definitions of K and 1). Now it is easy to prove that here KB coincides with the Kuranishi space S of HFdeformations we have just introduced. Thus we have here K = T x S.
(II)
Suppose that in the preceding situation we take B LP1 and X = £P1 XIq where Tq is a complex torus of dimension q. Take here the foliation F whose leaves are CP1x (point). In this case the cohomological assumptions of the
previous example are not fulfilled and, in fact, K Is not a product I x KB, as is remarked in (4). However, if we denote by S the Kuranishi space of HF-deformations, even in this case we have K I x S. As this example is a little surprising we shall give in Section 8 the calculations involved here.
(lii)
On a complex torus take the holomorphic foliation whose leaves are
the orbits of a complex vector field In this case the Kuranishi spaces K, S and T are all smooth and K= Sx I. 7.
SUFFICIENT CONDITIONS FOR S AND I TO BE ANALYTIC SUBSPACES OF K
next consider the problem of finding sufficient conditions for the splitting of K into a product S x I. The first difficulty is that, in general, S and I need not even be analytic subspaces of K.
To get this, some supple169
have supposed that the Initial foliation T is such that it has a transverse complementary holomorphic .... on we shall assume this hypothesis. The first F1. folio' be
us
assumption is that, given a tr—h—deformation is a h-deformation V -+ R of F This is a fundamental
,
such
f a section s in the exact
fact. sequence
:.>
..,
C
Take an atlas .. = constant
>0
in the following way. whose local charts are such that the leaves of F and zU = constant are the leaves of he
14
If
h
in a local chart of A is
•
the
aza
defines a global form.
Take the map s:Ak(v)
defined by saying that
s()91s the derivation given by the cojple (4, d4 - d •).
The map s
induces a map s between the corresponding sheafs of gems
-,
• which is
the desired section. 4,k the image of by S. Denote by = We haVe an orthgonal deconlpositlon
the space of sections
by
of
13k
•
.
the adjomt operators of 0 in vk, to the scelir product < , >. It Is easy to see that and
Denote respectwith D
operators, one has decomposition of the
In PaPtt r, we have an The standard 14• ipaces W constructions of the corresponding Kuranlshi spaces show that in this situaT are •nlaytic subspaces of K in a natural way. Unfortunately tlonS and do not behave properly and this is the brackets we have in + At,..
170
the reason why we cannot prove from this situation that K splits automatically Into a product S x 1. Nevertheless from this situation there follows the
result Theorem
8.
If K is smooth, so are S and I and K = S x
1.
THE CALCULATIONS OF EXAMPLE 2 IN SECTION 6
Denote by T the torus Tq' by P the projective space and by X the product the sheaf of germs of holoinorphic vector fields which P x 1. Denote by the coordinates of Tand by are tangent to the leaves. Denote by z1 ... u the inhomogeneous coordinate of P. Take the Fubini metric in P and the In 1. We have e flat 0 where C is the = constant sheaf over P of fibre C. From the KUnneth formula we have H'(X,Of) H1
H'(XiOtr) = H1(P,C)
(1,01).
H0(T,01) e
= H1
We have
a/au; a,b,c C, I
H'(T,OT) H1
=
C, i,k
• a/az1;
(I,er)
C)
We then obtain H1(X,Of)
;
dik
k = 1...q}
t, i,k =
The map which assigns to H'(X,Otr) an isomorphism between H'(XiOtr) and
1
...q}.
derivation
It is easy to see that
is
is 171
and that
= 0.
From the description of the Kuranishi space I for tr—h—deformations = (analogous to that given in (4) and (5) in Section 5), one has I = is the derivation = The map which assigns to In fact, belongs an isomorphism between H1(X,Of) and 4. whose first component is the vector is the unique derivation in —
2-form 2
A.k(u)
j
dik
where Ajk(u) =
3
+
+
(bjck_bkcj)u2.
The description of S given in (4) and (5) tell us that S is the analytic subspace of H (X,Of) = q given by the Since D5
= 0, we have N
H1.
=
equations
aJbk_akbJ =0 aJck — akcJ = 0
(1
j
q)
back - bkcj = 0. If WE H2 (X,Of) and
then
= 0.
Therefore, one has
K=SXT. 9.
SUFFICIENT CONDITIONS FOR THE SPLITTING K = S x T
At the end of Section 7 .we obtained a theorem giving a spl{tting of K into obstructions to such T. Furthermore, we have succeeded in defining S a spPtting. The obstruction of order k is defined in an appropriate way to where is a certain subset of the be a subset of H2(M,Of) e where MSXT is the maximal ideal of 0SXT in the distinquotient guished point 0. The vanishing of H (M,Of) is a sufficient condition for the vanishing of the obstructions of all orders.
Using these obstructions and a result of M. Artin [1] concerning the formal solutions of a holomorphic equation f(x,y) = 0, we have obtained the
following result. 172
Theorem K
Sx
If H2(M,®f) = 0, and if the Kuranishi space K is universal, then T.
Sufficient conditions for K to be universal can be given. 110(11,0)
For example,
= 0.
References
[1) Artln, N.
On the solutions of analytic equations.
Inventiones
Math.
5 (1968) 277—291.
(2]
des modules pour les varlétés analytiques (1964/65) No. 277. (3] Duchamp, 1. and Kalka, N. Deformation theory for holomorphic foliations. J. Diff. Geom. 14 (1979) 317-337. [4] Glrbau, J., Haefllger, A. and Sundararainan, D. On deformations of transversely holomorphic foliations. J. reins anqsw. Math. 345 (1983) Douady, A.
Le
complexes.
Sêminaire Bourbaki,
122—147.
[5] Kodaira, K. and Spencer, D.C. Nultifoliate structures.
Ann. of Math.
?4 (1961) 52—100.
(63
Nalgrange, B.
Analytic spaces.
Math. 117 (1968) 1-28.
J. Glrbau and N. Nicolau liniversitat Autónoma de Barcelona Bellaterra Spain
173
A HAEFLIGER
Pseudogroups of local isometries In this paper we begin a classification of the kind of pseudogroups H arising as holonomy
of Riemannian foliations on a complete manifold
(with a bundle-like metric).
The main inspiration for this work is the theory
of Molino, who has discovered important invariants such as the space W of orbit closures and the locally constant sheaf C of Lle.algebras which can be thought of as the Lie algebra of the closure i¾ of H which turns out to be a Lie pseudogroup.
Another basic invariant is a dense subgroup r of a one—connected Lie group G with Lie algebra which can be detected by looking at the normal bundle
of the closure of any orbit. The precise description of the action of ii on the normal bundle of an orbit closure is given in Section 3.2. There we correct a mistake in the first version of this paper: namely, in general the closure I? of H is 'not locally equivalent to a proper action of a lie group. Nevertheless the action of ii is in many ways similar to proper action of Lie groups. The space W of orbit closures is stratified by the normal type orbit as in Davis [1]) and there is an open dense set which represents the union of principal orbit closures. If the pseudogroup came from a proper action of a Lie group, then the union of orbits of the same normal type would form a locally trivial bundle of which the structural group was the Lie group of self—equivalences of the orbit of the given type. In our case we have a similar situation, but more complicated because the group E of self-equivalences 0f the pseudogroup
restricted to the orbit closure of the given type is no longer in general, a Lie group but has to be replaced by a sheaf of groups. So one has to consider a more general kind of bundle theory, which is the main object of this paper (see Section 2). An interesting feature is that the arithmetic properties of the group F as a subgroup of G plays a role in the computation of E.
We recall briefly in the first section some basic concepts, in particular the notion of equivalence of pseudogroups illustrated by Corollary 1.1.3.2. }n the Section 2, we explain the generalized bundle theory and give 174
examples.
In the Section 3, we first define the notion of complete pseudogroups of isometries which is the natural generalization of holonomy pseudogroups of complete Riemannian foliations. The structure of orbit closures is given and the considerations of Section 2 are applied to describe the union of orbit closures of a given type. The details of the proofs will be given elsewhere. 1.
GENERALITIES ON PSELIDOGROUPS AND FOLIATIONS
EquIvalences of pseudogroups
1.1
DefinitIon A paeudogroup U of transformations of a topological space T is a collection ft of homeonorphisms h:U -# V of open sets of T such that: 1.1.1
(I) if h, h' E H, then the composition, whenever it is defined hoh', belongs to H and h1
H.
The identity
1T of T is in H.
(41) The restriction of any element of H to an open set of I is in H.
(iii) If h:IJ
V is a homeomorphism of open sets in I and if each x
has an open neighbourhood
such that the restriction, of h to
U
is in 0,
then h is In H. For a pseudogroup H, we shall denote by ft the topological groupoid of germs of elements of u with the sheaf topology.
Definition Let ft and H' be pseudogroups of transformations of spaces I and I' respectively. An equivalence •: H + K' is a collection • of honeomorphisms of open sets of T on open sets of T' such that • Is stable under the composition with elements of ft and U' (so • K' e • ° 0) and such that ft (resp. H') is generated by homeomorphisms of the form o •2 (resp. 1.1.2
where
1'2
If I and I' have a differentiable structure (or a Riemannian metric,...) invariant by ft or H', then an equivalence 0:11 W is differentiable (resp. isometric,...) if each element of • Is differentiable (or a local isometry,..). Equivalences of pseudogroups form a category. in particular, the self— equivalences of H form a group E = E(11) depending only on the equivalence
class of K. 1.1.3
Main'example
We shall say that a group r acts quasi—analytically on 175
connected manifold I if any element of r is the identity when its restriction to an open Set of T is the identity. 1.1.3.1
Fundamental lenana
Let H and H' be pseudogroups generated by groups r
and F' acting quasi—analytically on one-connected manifolds I and I'. Then any
B' is generated by a homeomorphism $:T + I' such that •(yx) = a(y)$(x), for each x 1, y F, where is an isoniorphism Of F On
equivalence •:H
r'. Two
such pairs
and
an element y' of r
such
that •'
=
define the same equivalence iff there is y' o • and a'(y) =
The proof of the lema follows easily from the observation that the space
of elements of • is a principal r-covering of I' as well as a principal r'-covering of I. 1.1.3.2 Corollary The group £ of self-equivalences of a pseudogroup H generated by a group r acting quasi-analytically on a one-connected space I is isomorphic to N1'/r, where is the normalizer of r in the group Homeo(T) of homeomorphisms of 1.
Note that the group of self-equivalences of the action of r on I is the centralizer of r in Homeo(T). This is typical of the diffe4nce between the two concepts of group and pseudogroup. The holonomy pseudogroup of a
1.2
let F be a foliation on a manifold X.
ft can be defined by an open covering and submersions f1 of on manifolds T1, the f11s being surJective and with fibres, such that there exist homeo... n f1 on U1 n so that morphisms : f1 (1J1 A 1.2.1
ConstructIon
Let I be the disjoint union of the Ti's.
A representative of the holonomy Is the pseudogroup B generated by the local homeomorphisms
pseudogroup of h.
ofT. •
.
*
S
Mother choice of the submersions f1 would lead to a pseudogroup equivar
lent to H. is equivalent to a pseudogroup generated by a group acting quasi-analytically on the connected manifold I. Let be the universal covering of I and the group of homeomorphisms of 1.2.2 Developable foliations
176
Assume that
T projecting on the elements of r (it is an extension of r' by the fundamental group of 1). Then the foliation on the universal covering of X which is (called the development of the lifting of F is given by a submersion F) surjective with connected fibres and which is equivariant with respect to a representation of the fundamental group of X on r. Note that in general I is not Haussdorff. In case of a Riemannian foliation, on a complete manifold, then f is a locally trivial fibre bundle. Conversely, if F admits a development as above, then the holonomy pseudogroup is equivalent to a pseudogroup generated by a group. So it is important to decide when a pseudogroup is equivalent to a pseudogroup generated by a group.
Existence of a compact total transversal manifold A total transversal manifold for F is an ininersion j of a manifold T of dimension equal to the codimension of F, which is transversal to the leaves and whose image cuts every leaf. To such a j:T X is associated a representative H of the holonomy pseudogroup of F acting on 1; the germs of elements of It of source x 1.2.3
and target y in I are obtained by choosing a path connecting j(x) to j(y) in a leaf of F and constructing the corresponding holonomy element. Conversely, if a pseudogroup It of transformations of I is a representative of the holonomy pseudogroup of F and if the holonomy covering of each leaf is (q—1)-connected, where q is the codimension of F, then H is induced as above
bY a total transversal ininersion j:T + X (cf. t7)). In particular, if a total compact transversal manifold exists, then there is a representative of the holonomy pseudogroup acting on a compact manifold (and conversely in the case where the holonomy coverings of the leaves are (q-t )-connected). 1.3
Compactly generated pseudogroups
Definition Let If be a pseudogroup of transformations of a locally compact space I. Then H is compactly generated if there is a relatively compact open set U in I meeting each orbit of If, and a finite collection of elements h1 of H of source and target In U such that:
(I)
(ii)
the h.'s generate the restriction of H to
each h1 is the restriction of an element of H defined on a neighbourhood of the closure of the source of h1.
One can easily check that this notion is invariant by equivalence, and that the relatively compact open set U meeting every orbit can be chosen foliation on a compact manifold is compactly In fact most theorems concerning foliations on compact manifolds a
generated. can be rephrased as theorems on compactly generated pseudogroups.
Also the growth type of orbits can be defined for compactly generated pseudogroups, and is an invariant of the equivalence class. Minimalizable pseudogroups
1.4
Suppose that H is a pseudogroup of local diffeomorphistns of T such that there is a compact set K meeting all the orbits of it. Then H is said to be minimal— if there is a positive smooth function f with compact support on T, strictly positive on K, elements with of H and 1-forms compact supports contained in the target of such that
df = E
-
This definition is justified by the fact that, for a foliation on a compact manifold X whose holonomy pseudogroup is equivalent to H, then this
condition is satisfied iff there is a Riemannian metric on X such that all the leaves are minimal submanifolds (Cf. 1.5
[6]).
Classifying spaces
For each pseudogroup It, one can construct a classifying space BR; one should
think of BR as a space with a foliation y such that the holonomy covering of each leaf is contractible and such that the holonomy pseudogroup of Fis equivalent to H (see [7]). Of course the leaves of F are in general infinite dimensional.
So, in some generalized sense, the association ii Bi realizes any pseudogroup It as the holonomy pseudogroup of a foliation. In view of 1.2.3 any representative of 11 is induced by a total transversal j:T + B. In general there is no good representative for the classifying space of a pseudogroup in the equivalence class of H. Nevertheless we shall see in some examples that there may exist a very nice representative for BR, in spite of
the fact that one cannot find a nice representative for H. 178
1.6
Covering of pseudogroups and fundamental grout
I be a covering of Let U be a pseudogroup acting on a space I and let p:? each element h:U V of H, there is I on which H acts. This means that, a lifting h which is a homeomorphism and such that if + h1, h2
H,
a pseudogroup
then h1h2 ft
h1h2
acting on
and
h
"-1
=h
The elements
.
called a covering
of
h
generate
it.
It is clear that an equivalence between two pseudogroups induces a correspondence between their coverings. As an example, consider the case where 14 is generated by a group r
1—f
acting
quasi—analytically on a manifold I. Let p:? = T x nc0 + I be the natural projection, where is a subgroup of r. The lifting of the element y of r is defined by ç(.x,gr0) = (yx,ygr0). The corresponding pseudogroup Is equivalent to the pseudogroup generated by r0 acting on I (an equival'nce is generated by the map x
(x,r0) of I In 1).
For a pseudogroup H acting on I with a base point x0 I, one can define the fundamenta1 group ir1(H,x0) of H (Cf. [7]). Its elements are homotopy classes of H—loops based at x0. Such a loop is given by continuous maps T, where 0 = t0
Then there is a one-to—one correspondence between subgroups of ,T1(H,X0) and
isomorphism classes of based covering of H with space of orbits connected. In particular, the universal covering of H can be constructed using based homotopy classes of H-paths. When H is generated by a group r acting quasi-analytically on a connected manifold T, its universal covering is represented by the universal covering of T with the identity pseudogroup.
Conversely, if the universal covering of H is equivalent to the ldentity pseudogroup of a (necessarily 1-connected) manifold T, then it is equivalent to the pseudogroup generated by a group isomorphic to 1r1(R,x1) acting on T.
179
Locally trivial bundle of pseudogroups
2.
Continuous family of
2.1
first consider an important particular case. We denote by T a paraconipact 1-connected manifold on which a group F
opePates quasi-analytically. Let U be a simply connected manifold and let 11(U) be the pseudogroup generated by F acting on T x U by y(x,u) a (yX,U). Let 11(U) be the group of equivalences of 11(U) on Itself projecting on the Identity of U. As in 1.1.3.2, Nr is the normalizer of F In the group Homeo(T) of homeo-
centralizer. Let be the arcwise—cOnnected component of the identity in the group with the compact open topology.
morphisms of T and C1'
DefinItion of a topology on Ni'. On N1' we consider the topology which makes N1' a topological group and such that the normal subgroup with the compact open topology Is open in We denote by N1'(U)'(resp. the group of continuous maps of U In N1' (resp. with this topology. The group N1'(U) is the normalizer of r In the group of homeomorphisms of I x U projecting on the identity of U. As consequence of 1.1.3.1, we get the following description of 11(U), namely the 2.1.1
exact sequence F
1
N(U)
11(U)
(2.1.2)
1.
From this we deduce the exact sequence I
where
U In
+ 11(U)
a
fl
+.I
(2.1.3)
r is identified to the subgroup of C0(U) of constant maps of Is Induced by the projection
The map 11(U)
N1'(U) +N"(U),C1'(U)
=
2.2
2.2.1
Assume that I is a 1-connected Lie group G and that r is a dense sub-
group of G acting by left translations. Let Aut(G,r) be the group of automorphlsms of G mapping r in r (this is a subgroup of the group Aut(O) of the automorphisms of the Lie algebra of G, because G is simply connected). 180
To a pair Then N1' is isomorphic to the semi-direct product G cg(x)g1 of G = 1. The group I' is (g,a) corresponds the homeomorphism x ady). identified to the subgroup of elements of the form So we have the exact sequence
1+r
G
Aut(G,r)
£
(2.2.1.1)
1.
Note that the centralizer of F is the subgroup G x centre of 1'.
1,
and that
is the
So 2.1.3 gives the exact sequence 1
G(U)/r0
E(U)
Aut(G,1')/Int1'
1
(2.2.1.2)
continuous maps of U in G and Intr is the subgroup of Aut(G,r) given by the conjugation with the elements of r. where G(LJ) is the
2.2.1.3. As a specific example, take G = and r a finitely generated dense subgroup of Rn. For a generic 1', Aut(G,r) is the group of order 2 generated But for special r, such as the group o(K) of by the central syrmetryof
integers of a real number field K in R, Aut(R,r) is the group of units of
2.2.2. We now consider the more general case where r is a dense subgroup of a Lie group G (not necessarily connected) acting effectively on the space 1' = G/K, where K is a closed subgroup of 6 and I being 1-connected. As before, we denote by Aut(G,r) the group of automorphisms of 6 mapping r in r and by G '4 Aut(G,r) the corresponding semi-direct product. One can identify K with the subgroup K x and we shall denote by NK(G Aut(G,r)) the normalizer of this subgroup K in G x Aut(G,r). The centralizer C1' of r in'the group of homeomorphisms of I is isomorphic to NK(G)f K and its normalizer N1' is isomorphic to NK(G Aut(G,r))/K. An element (a,a) of NK(G Aut(G,r)) gives the homeomorphism gK + of I = 6/K. The group r is identified to the subgroup of elements of the form (y1, ady) mod K, r. 1
So we have the exact sequence 1
f
NK(G
9 Aut(G,r))/K + E
1
(2.2.2.1)
and the exact 181
E(IJ)
1
(2.2.2.2)
(Aut(G,r)/tntr,
where r0 = r n (NK(G)/K).
2.2.2.3. Again as a concrete example, let G be the subgroup of isometries of Rn generated by all translations and the central mapping x on -x. Then if K Is the subgroup of order 2 generated by a, we to I G/K. Let 1' be a dense subgroup of G generated by a can identify and by a. dense subgroup I" of the group of translations of
In that case, the centralizer of
NK(G)/K
is trivial and the projection
Aut(G,r))/K -I. Aut(G,r) is an isomorphism. The group Aut(G,r) is and F isomorphic to F' F' )412 is identified tothe subgroup 2r' ' Z2, where Z is identified to the subgroup of generated by a. Hence 11 • N /F is a discrete group isomorphic to F' NK(G
Bundle theory
2.3
be a pseudogroup of transformations of a manifold T and let W be a topological manifold considered as a parameter space or a base space. As before, for an open set U in W, we denote by 11(U) the pseudogroup of transand by formations of I x U generated by elements of the form Ii x Idu. h E ECU) the group of self-equivalences of 11(0) projecting on the identity of U. E(IJ) is a sheaf E of groups on W. Then U Let
2.3.1. w:S
We consider a pseudogroup it acting on a manifold S, with a submersion W preserved by ft and such that each point w of 14 has a neighbourhood U
so that there is an equivalence of 11(0) on 111w1(U) projecting on the identity of U. be an open covering of W such that there are equivalences Let {u1)1 + Then for each i,j E I, we get an element by composing the inverse of the restriction of to I x (U1 n UJ) with the to I x fl U3). restriction of This gives a tech cocycle defined on the coverln9 and with values in the sheaf E, namely for £ I, then =
after restriction to 01 n 03 fl UK.
Another cocycle
defined on a covering
is
cohomologous to
if there is a 1-coc cle on the covering which is the union of the covering {U1) and and which induces the given cocycles and The class of 1-cocycles under this equivalence relation form a set denoted by 182
H1(W,E), with a base point.
The following proposition is obvious from the definitions. 2.3.2. Proposition There Is a one-to-one correspondence between elements of H1(W,E) and classes of pseudogroups verifying the above conditIon 2.3.1 under equivalences projecting on the identity of W. The base point in H1(W,E) corresponds to the pseudogroup 11(W).
So we have to study the kind of bundle theory considered by Grothendieck
in [4] or by Frenkel in (2]. 2.4.
Some computations of
Let us come back to the case where
is generated by a group acting quasianalytically on the 1-connected manifold T. According to 2.1.2, the sheaf E is defined by the exact sequence of sheaves 1
r
N1'
E
(2.4.1)
1
where N1' denotes the sheaf of germs of continuous maps of open sets of W in with the topology defined in 2.1.1. Here r is considered as the constant
sheaf on W with stalk r. From 2.1.3 we deduce the exact sequence of sheaves:
r
rr
(2.4.2)
is the sheaf of germs of continuous maps of W in and is the quotient of this sheaf by the constant subsheaf whose stalk is = rfl C0. The sheaf Is the constant sheaf (it is Isomorphic with the quotient where
Nr,CrF)
get some i.nformation on H1(W,E), one can use the cohomology exact sequence associated to 2.4.2 (Cf. Frenkel [2)). We only consider some particular cases. To
2.4.3 W is the circle S1 =R/Z. Then H1(S1,E) is the set df conjugacy classes in the group N1' induces the automorphism of r, then the pseudogroup 14 corresponding to the class of this element in is equivalent to the pseudogroup generated by the group
(semi-direct
duct
to
acting on I x P by (-y,n)(x,t) = 183
then H is equivalent to x or equivalently to the pseudogroup of transformations of '11(w)' I 11 generated by I' x Z acting on T x H by (y,n)(x,t) (yx,t+n). There is we can a continuous path 4, in C1' with 4, the identity element and 4' • An such that extend it as a continuous map t + = 4' ° equivalence is then given by the map (x,t) -'
Let us check for instance that, if 4'
2.4.4 W Is the 2—sphere S2. Then H1(S2,E) is the quotient of the group of homotopy classes of continuous paths in Cr connecting the identity element to an element of 1' by the action of N (operating by conjugation on C ).. If Is such a path in C0, then the corresponding pseudogroup H is equivalent to a pseudogroup generated by a group iff the element r0 acts
freely on I so that the projection of I on the quotient I of by a covering map. Then 11 is equivalent to the psuedogroup generated by a group isomorphic to r/r1 over
with fibre
acting
on a bundle
(Cf. 2.4.7).
2.4.5 Assume W 2—connected. Then H1(W,E) is the set of isomorphisms classes of bundle E on W, modulo an action of Ni'. The corresponding pseudogroup is generated by the group r acting on Lx
by y(e,x) = (e,yx).
2.4.6. Assume r0 discrete in in the group of self—equivalences of the pseudogroup E generated by F acting on I is a topological group and the elements of H1(W,E) correspond bijectively to the isomorphism classes of principal E-bundles on W. One can give a simple description of the classifying space BR of a pseudoBNr be a group H corresponding to an E-principal bundle E Over W. EN1'
universal principal bundle for the group Ni'. As F is discrete in N1', the quotient EN1'/r is a model for the classifying space BF of F. The associated bundle EN1' x1'T is a model for BR (the leaves of the foliation on BR are 0 F the Images of the subspaces EN x {x} ). The topological group N1' acts continuously on EN1'
n(e,x) = [en'1,nx]
184
-
I
by
acts on EN1' x1' and the closed subgroup 1' acts trivially so £ = A representative for BR will be the associated bundle E XE (EPI1' Xr 1).
2.4.7.
Assume that
acts freely and properly on T (this implies that
is discrete in
Then any pseudogroup it corresponding to an element of H1(W,E) is equivalent to a pseudogroup generated by a group.
Indeed, let T be the manifold quotient of I by the action of 1' . Denote the connected component of the identity in the Lie group £ = N /1', we
by H have
=
Consider the manifold I = H
£
;
0
is a disjoint union of connected
it r
components isomorphic to I • Let N IC ; it is the group of connected components of N and one has the exact sequence
The group r acts on 1, the action of the class of n N 1' on [e,x] being This action coninutes with the natural action of H on H x 1 =1. o The inclusion of in I mapping x in (1,x] is an equivalence of the pseudogroup H0 on the pseudogroup generated by the action of I' on 1. Now let E be a principal bundle corresponding to a pseudogroup H. Then H is equivalent to the pseudogroup generated by the group r acting on the associated bundle E X
The hypothesis of 2.4.7 is fulfilled in Examples 2.2.1 when the centre of S is discrete and, of course, in 2.2.2.3 where is trivial. At the end of Section 3 we shall discuss an example where crete. 3.
is not dis-
COMPLETE PSEUDOGROUPS OF LOCAL ISOMETRIES
3.1
DefInitions and general properties
A pseudogroup it of local dlffeomorphisms of a Hausdorft differentiable manifold I is a complete pseudogroup 3.1.1
DefinItion of a complete pseudogrou_p
of local isometries of I if: each point x of T has an H-invariant open neighbourhood V containing a closed invariant neighbourhood of x such that V has an Riemannian metric; (C.1)
185
If x end y are points of T, there are open nelghbourhoods of x and y such that any germ of an element of H with source In is the germ of an element of H defined on the whole of In (C.2)
and
and target
This definition is justified by the following observation which is an easy consequence of the work of Reinhart [10].
3.1.2. Proposition The holonomy pseudogroup of a Riemannian foliation (with a bundle-like metric) on a complete manifold is complete. We are interested in the classification of such complete pseudogroups under equivalence (so the Riemannian metric is not preserved under equivalence). As Ininediate consequence of the definition, we have 3.1.3. PropositIon If H' is differentiably equivalent to a complete pseudogroup H of local isometries, then Il' is also a complete pseudogroup of local isometrics.
3.1.4. Proposition The closures of the orbits of H form a partition of T and the space W of orbit closures is a Hausdorff space (depending of course only on the equivalence class of H). 3.1.5 of the closure of H. Let be the groupoid of 1-jets of diffeomorphisms of T. It is tiaturally a differentiable manifold and a
differentiable groupoid. Let us denote by ft the groupoid of germs of elements of H with the topology of germs.
j
:
The map ft
associating to a germ at x its 1-jet is a continuous homomorphism of topol-
ogical groupolds and it is injective by (C.1). Condition
3.1.5.1
implies easily the following theorem.
There exists a unique pseudogroup it of local isometries is the closure of j(H). We shall call the pseudogroup H the closure of 14. It is also a complete pseudogroup and 11 is a subgroupoid of ii which is open and closed. Moreover ft is equivalent to a covering of Theorem.
of I such that the image of a In
186
Note that on the groupold a we have two topologies; the germ topology and the soft topology Induced by the embedding j In J1. We shall see that ii with
manifold with a foliation so that the the soft topology is a germ topology of ii is the leaf topology. The soft topology gives a topology on 111(a,x0) and (assuming the space of orbits to be connected) the equivalence classes of complete pseudogroups H whose closure is equivalent to ii correspond to th, dense subgroups of
This leads to the following program for classifying complete pseudogroups
of local isometries: 3.1.5.2 Classify first the closed complete pseudogroups, then compute their fundamental group and determine their dense subgroups.
.3.1.5.3 As an example, consider the following complete closed pseudogroup Ii acting on the disjoint union of two copies V I with coordinates denoted respectively by z and w, generated by all the rotations z elez and on both copies, e E P. and by the imiltivalued •w of t-(O} on £-{O}, where Is a strictly positive fixed real number. The fundamental group ir1(i) is Isomorphic to the quotient of P by the subgroup generated by I and o (with the quotient topology). The universal is equivalent to the subpseudogroup generated by and z w Any complete pseudogroup whose closure Is equi-
covering 110 of w
valent to ii correspond to a dense subgroup of P contaIning I and a (if irrational, such a subgroup is always dense). 3.2
is
Local
The analogue
Molino theory (Cf. [9]) in this framework is the following
basic property of complete closed pseudogroups which will be proved in the thesis of Ellane Salem. It is a generalization of the theorem of Myers and Steenrod asserting that the group of isometries of a Riomannian manifold with the compact open topology is a Lie group.
A complete closed pseudogroup a of local isometries is a Lie pseudogroup in the following sense. There is a locally constant sheaf of finite dimensional Lie algebras whith is a subsheaf of the sheaf of germs of vector fields on T such that any elenent of 1 which close enough to the 3.2.1
Theorem
181
identity is of the form
• where
Is a local section of 0 close to zero.
So, assuming the space of orbits to be connected, to is attached a well defined (up to Isomorphism) finite dimensional Lie algebra 0. The locally constant sheaf F is an i-sheaf, namely a sheaf on which fi acts by automorby the exponential of the phisss. The elements of ii of the form adjeint representation. From this, one can deduce the stnucture of the normal neighbourhood of
the orbit Itx under ii of a point x of T. let K be the stability subgroup of x, namely the group of germs at x of elements of ii fixing x. The map associating to an element of tç its 1—jet at x is a bijection of K on a compact subgroup of J1(T). The orbit i%x Is a closed submanifold whose dimension is dimO — dim K. Let V be a K-invariant complement of the tangent space of iix at x In the tangent space of I at x. The group K acts on V via a linear representation p:K Sumning up we get the following characterization of neighbourhoods of orbits of a.
3.2.2 Corollary The equivalence class of the restriction of ii to a small enough neighbourhood of the orbit is characterized by a quintuple where:
(a) (b) (c)
(d)
K is a compact Lie group;
I Is an Injection of the Lie algebra K Of K in 0; : K + Nit 0 is/a/representation such that ad induces on K the adjoint K is the adjoint representation 9'#. K and the derivative of ad at 1 K on e (via the a representation such that its sum with the representation p:K + Aut
ad
on S/K Is faithful (reflecting the fact that the action of Is effe9tive).
Induced by
one can reconstruct an Conversel)1, from such a quintuple action of /closed complete pseudogroup ii on a neighbourhood of an orbit. K0 be the Let K0"be the component of the identity In K and p :
universal covering of KQ and let N be the kernel of p (it is a discrete subin the centre of Let be a simply connected Lie group with Lie algebra S and let j : C0 be the homomorphism induced by the inclusion :K • S and let H be the image of K0 under j (it Is a not necessarily closed Lie subgroup of C0). The image j(N) iS In the centre of G0. '1
As Aut(6) is canonically isomorphic to the group of automorphisms of the representation ad gives an action of K on G0, still denoted by ad. Let be the corresponding semi-direct product group. This group operates x V : the action of (y,k) on (g,v) E x V being (yad(k)(g),p(k)v). image of R0 by the homomorphism Let I be the subgroup of
k0 + (.i(k;'). p(k0)). It is a connected normal subgroup of G0xSK because Its Lie algebra Is an ideal. x V whose leaves are the orbits of L. Denote by F the foliation on As L Is a normal subgroup, G0iK leaves this foliation invariant. Under the projection G0 x V + V, the leaves of F are finite coverings of the cosets gj(k0). The generic leaf of F is isomorphic to L. On we have two structures of Lie group the soft one coming from the Lie structure of G0 and K, and the fine one characterized by the property that L Is the connected component of the Identity.
3.2.3 Theorem For a quintuple (6,K,l,ad,p), the corresponding complete closed pseudogroup ii acting on the normal bundle of an oPbit 'is equivalent to the pseudogroup generated by the action of on the "leaf space" of F. The fundamental group of ii is the quotient (C0 (with the quotient topology). The universal covering of ills equivalent to, the holonomy pseudo— group of F.
precisely, let W be a small neighbourhood of the identity in G0 such to W x V is given by a surjective submersion f:W x V -, 1 with connected fibres. Then Il is generated by all local diffeo— morphisms of T which are projections under f of transformations coming from the action of Another way of expressing this is to say that Il is equivalent in the more 'gefler4l sense of Skandalis (Cf. [7], p. 80) to the topological groupoid given the action of G0xIK with the fine topology on G0 x V. More
that the restriction of F
3.2.4 Corollary The pseudogroup Il assàciated to the quintuple is equivalent to a pseudogroup generated by the action of a group 1ff the
following two conditions are verified: (1)
j(N) is a discrete subgroup of C0 (equivalently j(K0) is closed in
(2)
18!
Under these two conditions, G = (G0xiK)/L is a Lie group containing the action Identified to K and Is equivalent to the subgroup of 6 on the homogeneous vcctor bundle I = 6 xK V determined by p,
ConditIons (1) and (2) are always verified when 6 is solvable (because i
is injective and j(K0) is closed) or verified If dim I 3. Condition (1) is
is 1—connected.
They are always
if the centre of 6 is
trivial. 3.2.5 Examples The smallest dimension of T for which (2) does not occur is As an example, take for 6 the Lie algebra of SU and for K the group of complex numbers of norm 1 with its natural representation £ = V. The
4.
inclusion i of the Lie algebra of K in the Lie algebra of SU2 Is chosen In a way that it induces a homeoniorphism of
in SU2 with kernel generated
where n is a positive integer. occur is S. For The smallest dimension of I for which (1) does SU2 x SU2 x instance, take for 6 the Lie algebra of G0 and for K a 2-
by
torus whose universal convering K0 is embedded by j as a non-closed subgroup o'f.G0 whose projection on SU2 x 5U2 is a maximal torus.
32.6
Remark
From the explicit descriptionof 3.2.4, It follows that for a
countably generated complete pseudogroup of local isometries, Its restriction to the normal neighbourhood of an orbit is the holonomy pseudogroup of a Riemannian foliation on a complete manifold. of H to the normal bundle 'v of an of the restriction of Is described by the quintuple (6,K, orbit closure. if the closure is equivalent to the pseudogroup generated by the action of i,ad,p), then
o dense subgroup A of G0dK containing I on the leaf space of F, in the notations of 3.2.3. The Intersection of A with the normal subgroup = The Isomorphism class of the of G04K Is a dense subgroup r0 of
pair (%,r0) is an invariant of H (assuming the spice of orbit closures connected), namely it Is the same for all normal bundles of orbit closures). are verified, then is equivalent When conditions (1) and (2) of to the pseudogroup generated by the action of the dense A/I = r of V. 6 — (60'oK)/L acting on 6 When K is the trivial group, then H0 is just equivalent to the pseudogroup generated by the action of r0 onG0 x V. This is the case iff leaves 190
invariant a local parallelism. As in Molino [9]),one always has this situation after lifting the action of H on the bundle I I of orthonormal frames
ofT. 3.2.8 Proposition If H is a complete pseudogroup of local isometries of I, one can find on I a global Riemannian metric invariant by H (assuming that the space of orbit closures is paracompact).
So condition (C.1) could be replaced equivalently by the stronger condition that there is a Riemannian metric invariant by H on I. This Riemannian metric can be constructed by using a differentiable partition of unity on the space of orbit closures to average the given local invariant metrics. Note that 3.2.1 implies that the space of orbit closures is locally the quotient of a vector space V by the action of a compact group of linear transformations. Differentiable cohomology and duality Let U be a locally constant sheaf of finite diniensional vector spaces on I on which ii acts by automorphisms. We consider the complex A(T,U) of differential forms on I with of H-invariant forms. The cohomology values in U and the subcomplex 3.2.9
of this complex is an invariant of the equivalence class of U and will be and called the basic cohomology of H.. It is likely to be the differentiable cohomology of ii in the sense of [5]. Let us denote similarly by the subcoinplex of of forms with support contained in a union of orbits meeting a compact set. Let tA'1(efl be the locally constant sheaf of vector spaces of dimension one corresponding to the absolute value of the exterior power of where n is the dimension of Denote by a the orientation sheaf of I. We have a Poincare duality whose proof is the same as that given by Sergiescu in [11]. denoted by
Proposition
There is a canonical isomorphism of on where m is the dimension of T. the dual of Uea 0 is H—trivial. In that case we We call H unimoduiar if the sheaf provided that the orbit space is connected. have = 3.2.9.1
3.3
Stratification by the normal orbit type
In this section we assume that i( is a closed complete pseuøogroup acting on I and such that the space of orbits W = is connected, so that to i( Is 191
attached a well defined Lie algebra G. One can define dn W a stratification
which Is similar to the stratification by the normal orbit type on the orbit space of a proper action of a Lie group (cf. Davis Cl]). Definitions Fixing the Lie algebra G, a normal orbit type is a quintuple (O,K,i,ad,p) verifying the conditions of 3.2.2 with the extra is the vector subspace of elements of V = 0, where assumption that fixed by all elements of K under the representation p. 3.3.1
Two quintuples
and
+ KS and x:V
are isomorphisms
are equivalent If there V' such that:
(ad')•B, o
(adx)•p = p'
orbit
where dB
:
K -.
is the differential of B at 1
K,
° B.
given by an equivalence class-of such quintuples. For such a type v, we denote by ji a representative of the corresponding class of closed complete pseudogroups (cf. 3.2.3) acting on the normal bundle A ,zonnaj
type v is
of an orbit. 3.3.2
W be the natural projection. To the orbit
Stratum of type v. Let
under ii ofa point x of I is attached (Cf. 3.2.2) a quintuple where p:K + Aut(V') is a representation. The normal orbit type of x is the quintuple obtained by replacing V' by the quotient V = V./VSK and p' by the corresponding representation on V.
Given a normal orbit type v, the points in W corresponding to orbits of normal type v form a smooth manifold W in W, called the stratum of orbits of those orbits is an i—invariant smooth of normal type v. The union submanifold of I. iff there is a neighbourhood U of w = lr(x) A point x belongs to in and a neighourhood U' of U in W so that there is an equivalence of with the pseudogroup a
x
(generated by the product of elements
with the identity of U).
of
So above the stratum
of
to the normal bundle of Section 2.3. The particular case for which 192
type v, namely for the restriction of H we have the kind of bundle described in
= C was already studied in [8].
3.3.3 Partial ordering and principal orbit
As for proper actions of Lie groups, one can define a partial ordering (Cf. [1)) on the set of normal orbit type, namely v' v if v appears as a .normal orbit type for an orbit in v. There is a maximal orbit type characterized by a quintuple for which V
0, called the principal orbit type.
3.3.3.1 Principal orbit theorem. Let H be a complete pseudogroup acting on 1; assume the orbit space is connected. Then the union of principal orbits of ii is an open dense set of T and the corresponding stratum in W is an open dense connected manifold W0.
The restriction ft0 of ft to a principal orbit closure is a basic invariant of H which can be seen on the normal bundle of any orbit closure. It is characterized by a quadruple (e,K,i,ad) together with a dense subgroup A of
K containing L (in the notation of 3.2.7). The restriction of H to the union of principal orbit closures is classified by an element of H1(W0,E), where E is the sheaf of of the pseudogroup 110 x projecting on the identity of W0. 0
description of £ is given in Section 2.2.2 in the case where conditions (1) and (2) of Section 3.2.4 are verified. The general case Is analogous, A
but more complicated.
Where we have only one orbit closure type, the coinpu-
tations of Section 2.4 give results on H that we do not repeat here. They give, in particular, conditions under which H is equivalent to a pseudogroup generated by a group action. Recall that for a, foliation having this holo— nomy pseudogroup, this means it is developable (cf. Section 1.2.2). We have, in particular, only one orbit closure type when H preserves a local parallelism; this corresponds to the case studied In Section 2.2.1 (see also the end of 3.2.7). 3.4
An example
3.4.1
We consider here the class of complete pseudogroups with only one
orbit closure type on which the action of C is equivalent to the action of a dense subgroup r of acting by translation on We assume that r Is of finite rank N. We also choose the space W of orbit closures which must be a smooth mani-
fold. The sheaf 11 on W is the semi-direct product
(Cf.
Example
193
Let us fjrst assume W to be simply connected.
Note that when U has only one type of orbit closure and when Bit Is simply connected, then W is 1-connected and the orbit closure type is given by such This is precisely the case studied by in (33. a r acting on because W An element of H1(W,E) is the image of an element of is 1-connected. From the cohomology exact sequence associated to the exact sequence of sheaves
we deduce that
=
H2(W,r), because
is a fine sheaf.
As r is a free abelian group of rank N, the elements x of H2(W,r) classify principal bundles E of which the structural group is the torus of dimension N.
There is a simple description of the classifying space BR of the corresponding pseudogroup H. The torus TN can be identified with the quotient of = RN by the lattice r 1. On TN we have a linear foliation which is r given by the linear map y Q r -t ry of r eR on when lifted to the universal covering. This foliation is Invariant by the translations of Then BR Is the bundle E with the foliation which is the union of those linear
foliations in the fibres. BR will be simply connected 1ff the element x of H2(W,r) which classifies E gives a .surjeétion of ir2(W) on r. The pseudogroup It will be equivalent to apseudogroup generated by a group 1ff the Image of the homomorphism of pfl• i12(W) In r given by X Is a discrete subgroup of More generally, when W is connected, but not simply connected, then ene can construct as model forBit a torus bundle E on W, but not prilKipal in
general, with a foliation
is the union of the linear follatiens descri-
bed above in the fibres. it acts To see this, consider the Lie group I (r eR/Fe on the torus TN = r eR/r e 1 by (t,a) s = t + a(s), preserving the above linear foliation, if we consider the sheaf of germs of continuous maps of W In
we get an exact sequence of sheaves
O-tR N-n 194
A
-t'E+E-tl.
ASRNn Is a fine sheaf, H1(W,E) = H1(W,E).
This last group classifies the I—principal bundles. If E is such a bundle, then a representative for the classifying space BR of the corresponding pseudogroup I Is given by ttie Moreover, ft is equivalent to a pseudogroup generated torus bundle E by a group action 1ff the homomorphism of ir2(W) in r, obtained as before by
lifting to the universal covering of W, has
image in
Considering the cohomology exact sequence associated to the short exact sequence (Cf. 0
[2])
E
Aut(Rm,r) + 1,
—
we see that an element of H1(W,E) is characterized by Its image in which Is described by an homomorphism + and by an element X E H2(W,rT), where is the local system of groups on W
locally isomorphic to F and twisted by r. Assume that r is trivial, so that H Is described by X
H2(W,r).
From X
we can compute the cohomology of BR as well as the basic cohomology of H
(Cf. 3.2.6) as follows. the image of x in
r e R) and by Xb the image of X jn the homomorphism induced by r be a closed 2-form Let on W with value in r eR whose cohomology class if for instance the curvature form of a connection on the principal I -bundle BR + W. Composing with the homomorphism i' we get from a closed 2-form with value Denote by
In
H2(W,
whose cohomology class is Xb.
The real cohomology of the torus
differential algebra A(r
BR is the cohomology of the
eR) e
A(w,R), tensor product of the exterior algebra on the dual of r by the algebra A(W,R) of real valued differential form on W, with the differential twisted as 1 0 1) where u,
(reP)'.
The basic cohomology Iç(H.R) (cf. 3.2.6) is the cohomolegy of tM dlfferentlal subalgebra AiR's') with d(o e 1) = 1 0 e o OifO cah replace the de Rham algebra A(W,R) by any differential to it in the sense of Sullivan. We see that formally the basic cohomology behaves like the cohomology of a Ta—bundle over W, and also that the real cohomology of BR behaves like the cohomology of a TN_n bundle over a whose cohomology is the basic cohomology. This is only formal because the characteristic class of 195
describing this bundle is a real class which cannot be defined over the rationals (like the class of Xb in H (W,R)). The class * is of the homorepresented by a 2-form on W with value in the kernel L = by composing with a linear projection of deduced from morphism r eR-s
reRonL. 3.4.2
The same considerations can be extended to the more general case where
is replaced by a 1-connected nilpotent Lie group N and r by a finitely generated dense subgroup of N.
Then, according to Malcev theory, there is a 1—connected nhlpotent Lie group
and a cocompact subgroup F of
as well as a Lie group homomorphism
-s N whose restriction to F is an isomorphism on r.
we identify r with r.
When N is
By this isomorphism,
N is just r eL
Any atttomorphism of N inducing an automorphism of r lifts uniquely as an automorphism of inducing the same automorphism on F. In this way we get an action of Aut(N,r) on preserving F. Let us consider the tie group N where r Is identified to the discrete subgroup of the semi-direct product of elements of the form r. The Lie group E acts smoothTy on N/r by (n,u)(n'r) y
= na(n')r, where n, n' EN.
This action preserves the foliation on N/r which is given by the submersion when lifted to the universal covering N. 3.4.3
Proposition
Consider complete pseudogroups B having one type:of
orbit closure equivalent to the action of ron N and such that the space of orbit closures Is the maMfold W. Then equivalence classes of such pseudogroups under equivalences projecting on the identity of W are in one-to-one correspondence with isomorphism
of E-principal bundles E over W. The classifying space BR for a pseudogroup corresponding to E can be represented by the associated bundle BR = E N/r with the foliation which
is the union of the foliations on the fibres H is equivalent to a pseudogroup generated by a group action1 (coming from the homotopic exact sequence of the bundle E) is discrete in N (where is the centre of r). Moreover,
1ff the image of the homomorphism 112(W)
Using this explicit construction, as well as a remark of Ghys in [3], one gets easily a generalization of a result of Molino and Sergiescu proved for Riemannian flows.
196
Let T be a Riernannian foliation on a compact manifold, such that the Lie algebra associated (cf. Molino ru and 3.2.1) is nilpotent. The foliation is minimalizable iff the holonomy pseudogroup of ft is uni— 3.4.4 Corollary
modular (cf. 3.2.9). References
[1]
Smooth G-manifolds as collections of fiber bundles, Pacific 77 (1978) 315—363. J. et espaces fibres. Bull. Soc. Frenkel, J. Cohomologie non Davis, M. ftkzth.
[2]
Math.
[3]
de France, 85 (1957)
Ghys, E.
135—220.
Feuilletages riemanniens sur les variatEs slmplement connexes.
Preprint, Lille (1983). [4] Grothendieck, A. A general theory of fiber spaces with structure sheaf, Lawrence, University of Kansas (1955).
[5] Haefliger, A. Differentiable cohomology. CIME course (1976). [6] Haefliger, A. .1. Diff. Geom. 15 269-284. [7] Haefliger, A. Groupotdes d'holonomie et classifiantg, Structuree Traneverees des Feuilletagee, Toulouse (1982) Aeteriaque No. 116 [8]
(1984) 70-97. Haefliger, A. and Quach Ngoc Du. Appendice. une presentation du
groupe fondamental d'une orbifold, ibidem, 98—107. globale des feuifletages rienianniens, Ned. Akad, [9] Molino, P. Serie A, 1 (85) (1982) 45-76. [10] Reinhart, B. Foliated manifolds with bundle-like metrics, Ann. of Math. 69 (1959) 119-132.
(11] Serglescu, V.
Cohomologie basique et
riemanniens,
des feuilletages
IRMA de Lille, VI, Fasc. 3(1) (1984) 1—25.
A. Haefliger Section de Mathematiques Case postale 240 CH-1211 Geneve 24
197
A LICHNEROWICZ
Generalized foliationsand local Lie algebras of Kirillov In 1975 Shlga introduced, for the space of the sections of a vector bundle,
the notion of local Lie algebra, the bracket of two sections having a local character. Kirillov studied in 1976 the case of' the sections of a real line bundle [1]. If 'this bundle is trivial, this is equivalent to studying the local Lie algebras on the space of the functions defined on the base. This "scaler case" is in fact the most interesting case. Methods and results can be extended in a natural way to the Kirillov case. Independently of Kirillov, I introduced in 1975 the notion OX general and in 1977 the notion of genez'aZ Jacobi atñsature [2).
Poisson
Such a structure Is a contravariant combined generalization of or Poisson structures and of contact structures. To each local Llealgebra Is canonically associated such a geometrical structure and conversely. In this paper, I analyse the geometry of these Jacobi structures and the properties of the corresponding Lie algebras of vector fields (3]. 1.
ThE SCHOUTEN BRACKET (4)
Let W be a connected paracompact differentiable manifold of dimension m and class C"; all the considered elements are C". We set N N(W) • C"(WR). We call ansntisyl7mletrical contravariant tensor of order p a p—tenaor. On the space of these tensors, Schouten and Nijenhuis have introduced a
bracket that can be defined in the following way: if A is a p-tensor, B a q—tensor, (A,B] is a (p+q—1)-tensor such that, if 8 is an arbitrary cioeed (p+q-1)-form, we have
= (_1)pq÷q i(A) d i(B)8 + (..1)P i(B) d i(A)8 is the inner product. If p = 1, [A,B] £(A)B, where c( the Lie derivative. The Schouten bracket given by (1.1) has the following properties: where i(
•
[A,BJ
198
)
(_1)pq [B,AJ.
(1.1) •
)
is
(1.2)
If C is a r—tensor, we have: (1.3)
=0
where S is the sunmiation after cyclic permutation. A C]
(A,B] A C +
Moreover,
(1.4)
A [A,C).
(1.2) and (1.3) show that the Schouten bracket defines, on the space of the anti—syimnetrical tensors, astructure of graded Lie algebra. 2.
JACOBI STRUCTURE AND CONFORMAL JACOBI STRUCTURE ASSOCIATED WITH A LOCAL ALGEBRA
(a) A Lie algebra (N,( u,V E N, Sup
,
(u,v) c Sup (u)
]) defined on N is said to be loaal if, for any
n
Sup v.
Let U be the domain of a chart {x1} of W.
It follows from the Peetre theorem
that (u,v] Is given on U by a differential bilinear operator. If this operator Is of bidifferential type (r,s) (with r s), its principal symbol in XE U can be written in the chart A
-
r
...
—
1
s
s
1
The principal of the left member of the Jacobi Identity Is, for r of type (2r—1), s+1, s). The annulment of this symbol for r 2 implIes A
0.
2,
There •follows:
Proposition Each local Lie algebra (N,( , )) is given by a differential operator of maxlmi order 1 in each argument. (b)
There exist thus a 2-tensor It and a vector E such that
(u,vl B 1(A) du A dv + u i(E) dv — v 1(E) du.
(2.1)
The Jacobi identity can be translated in terms of Schouten brackets by [A,A] B 2 E A ft
[E,A] = £(E)/t = 0.
A pair (A,E) satisfying (2.2) defines on W a
(2.2) structure. 199
uEN÷huCN. Two local Lie algebras (N,(
,
]) and (N,(
,
3') are said to be equivalent
if
we
have
(u,v
=
ph[u,vJ
(2.3)
N).
The corresponding Jacobi structures are connected by As
=
E'
A
hE+
(2.4)
[A,h].
The equivalence class of structures defined by (2.4) is a oonformvil etructure.
We
Jacobi
denote by I the lie algebra of the infinitesimal , 3). A vector field Z of
of the conformal Jacobi structure defined by (N,( W belongs to I 1ff there Is aZ = a N such that £(Z)E = a E +
c(Z)A = a A
(2.5)
[A,a).
If E = 0, that is, if A satisfies [A,AJ = 0, A defines on W a PoisBon
Btru—
ctlLre.
(c) If u N, consider the vector field (called a generalized Hamiltonian vector field, or shortly, Ho.miltonian vector field) given by uE + [A,u].
It is easy to see that, if u,v
N,
= X[uv) are thus thus elesatisfies (2.5) with a = - 1(E) du. The and that we have ments of a Lie subalgebra L* at' L. Moreover if Z I and (2.6)
= (c(2)u + au)E + IA,c(Z)u + au] and L* is an
of L. L* is a homomorphism of (N,[ The bracket (2.1) can be written
ideal
The map a:u
kernel C of a.
N
[u,v]
— v i(E) du.
,
3) onto L*.
Study the
(2.7)
= 0 and [u,v] = -v i(E) du for each v. In particular, u i(E)du = 0. If x E W is such that u(x) ,& 0, we have (i(E)du)(x) = 0 and' so [u,v](x) = 0, for each x Sup(u) and for any v.
If u
200
C, we have
It follows from the locality of [ centre of (N,[
•
3).
,
3 that If u
The converse Is evident.
C, u belongs to the
We have:
Proposition The Lie algebra L* of the Hamiltonian vector fields is an.ideal ]) onto 1* and L* is a homomorphism of (N.E of I. The map o:u EN + C, we Moreover, for each u the ternel of is the centre C of (N;( , ]). have 1(E) du = 0. 3.
GENERALIZED FOLIATIONS
(a) Let M be a smooth manifold of dimension in. A field of planes P is a const. P is regular if dim subspace of map: x £ M +
A C'°-vector X of M belongs to P if, for each x
M, the value X(x) of x
P is said to be of class C if there is a set of belongs to for each x of ii. We say that C -vectors belonging to P and generating this set generates P. An integral manifold of P is an ininersed connected c An integral submanifold S of M such that, for each x S, we have manifold S is maxinvil if each integral manifold that includes S coincides with S. We say that P is invariant under a vector X if it is invariant under the flow of X. Sussmann has proved [5): Theorem
Let P be a field of planes of class
generated by a set E of Ct0—
vectors. Suppose that P is invariant under each vector of E. Then,through every point x E II, there passes a unique maximal integral manifold S(x) of P. For each y S(x), we have
and the dimension of the elements of P is constant along 5(x). Conversely if such is the situation, P is invariant under each C -vector belonging to P. (b)
In this case P. Stefan and P. Dazord have given more precise results.
We are led to the following definition [5): Definition A generalized foliation (or, shortly a foliation) on N is defined by a field P of planes of class C invariant under each C -vector field belonging to P. For x M, a leaf S(x) is the unique maximal integral manifold of P passing through x; S(x) is a weakly imbedded submanifold of M. 201
A usual Reeb foliation is called here a
foliation.
We
have:
of H for Let P be a (generalized) foliation on M. Each point which S(x0) has the dimension p admits a chart f of domain U with values in V2 is a neighbourhood of 0 where V1 is a neighbourhood of 0 in V1 x = in) and we have the following f(x0) 0 and the image induced on U is the foliation V1 x Q, where Q is a by f of the foliation Theorem
foliation of V2 null at the origin. We say that f is an adapted where
chart,
U an adapted domain; for x
= 1,...,p and a,b = 1,...,q.
U, f(x) =
The connected components of
in S(x0) n U is isomorphic with V1 and satisfies {Xa} We see that the leaves S(x) corresponding to the points x in the neighbourhood of x0 have a dimension superior or equal to dim. S(x0). The equivalence relation on N defined by P is open and so each closed leaf is proper. A point x is called regular if there is a neighbourhood U of x for which PU
is regular; if x is regular, all the points of S(x) are regular.
We obtain
thus the notions of regular leaf and singular leaf. The set of the regular points of N is an open set, dense on K; on each connected component of this
set, P defines a regular foliation. 4.
THE KIRILLOV THEOREM
be a local Lie algebra on N = N(W), (A,E) the corresponding Let (N,( , Jacobi structure. If x W, let be the plane generated by the values at x of the Hamiltonian vectpr fields, elements of L*; is the plane determined by E(x) and A(x). We call characteristic field of (N,( , 3) the field P of the planes structure
P depends only upon the
determined by (N,[
= Const., transitive
dim.
,
if
conformal Jacobi
]). This algebra is said to be regular
if
= TRW.
Me have the following theorem which is substantially equivalent to a
(a)
statement of Kirillov [1): Let (N,[ , 3) be a local Lie algebra of characteristic field P. The field P defines on W a (generalized) foliation. For each leaf S, the restrictions of A and E to S determine on S a transitive local Lie algebra which is said to be induced by (N,( ,
Theorem
(1)
(ii)
]).
According to the Sussmann condition, it is sufficient to prove that P is 202
Invariant under each HaMitonian vector field. quence of the existence of adapted charts. The geometrical structure of the
(b)
(ii) is an iziediate conseP is simple. A leaf S of
satisfies the contravariant definition of a Pfaffian with a contact 1-form). A leaf S of even diawneion for which
odd
E = 0 is sjjmpiecti_o ([A,A] = 0); It Is globally conformally symplectic if we can reduce the situation to E = 0 by equivalence. If sudi is not the case,
the leaf is a true
locally conforinally 8ympieotic w4nifol4 (t.l.c.s.)
[3].
The origin of this denomination is the consider a manifold W of dimension 2p with a 2-tensor A of rank 2p and a vector E satisfying (2.2). Let F be the 2—form obtained by inversion of' A and set w
— i(E)F.
For
(W,A,E), the relations (2.2) are equivalent to the following relations: WAF=Q
dw=0.
(4.1)
Let U be a contractible domain of W. There is on U a function $ > 0 such that = (4.1) shows that the 2-form is syniplectic on = U. We see why (W,A,E) is Lc.s.
If w is exact
where • is > 0. The 2-tensor defines on W a symplectic structure and (W,A,E) is globally conforinally sym— on
W, we have w =
p1 ecti C.
(c) Suppose that (N,[ , ]) is an arbitrary local Lie algebra on W. There is on W an open submanifold, on W, such that, for each of its connected
the restriction of N to
is regular. If the leaves of have an odd dimension, is a regular Jacobi manifold. If the leaves have an even dimension, they are locally conforinally 4nnpiectic manifolds, true or components
not. 5.
TRANSITIVE LOCAL LIE ALGEBRAS
(a) There follows from the results concerning Pfaffian or cohtact manifolds the following:
Let (N,t , ]) be a transitive local Lie algebra on the manifold W of odd dimension and (A,E) the corresponding Pfaffian structure. The algebra (N,[ , ] is the Jacobi Lie algebra of (W,A,E) isomorphic by a to the Lie algebra L* of the contact infinitesimal transformations and its centre is reduced Theorem
1
203
•
to (O}. The Lie algebra L* (resp. N) coincides with its conmiutator ideal and the derivations of L* (resp. N) are all Inner derivations. We have (0) and the Lie algebra (N.E , ]) is rigid. is here the second differential cohomology space of (N,[ , 3) for the Cheval Icy cohomology with values in N, associated to the adjoint representation.. Saying that (Pl,[ , )) is rigid is to say that each formal differential defoflnation of (N,[ , ]), in the sense of Gerstenhaber [63, is
differentlably trivial. (b)
In a similar way, the results concerning the symplectic manifolds give
[7): Theorem 2
Let (N,(
,
]) be a transitive local Lie algebra on the manifold W
corresponding symplectic of even dimension such that E 0 and let A be structure tensor. The algebra (N.E , ]) is the Poisson Lie algebra of (W,A) of centre R and a is the homomorphism of (N,[ , 3) on the Lie algebra L* of
the Hamiltonian vector fields; L* coincides with its coninutator ideal. The local derivations of (N,[ , ]) are given by (c(Z) + a ), where Z L and aZ = const. We have Hd.f(N;N) (0) and (N.E , 3) admits always non-trivial deformations (given by antisyninettization of star-products). Strictly similar results are valid for the Lie algebra (N.E , ]) corresponding to a globally conformally symplectic manifold. It is necessary to study the new case where (W,A,E) is a true locaily conformaily aympleotic mani(oZ4. To obtain results of the type of Theorem 1 (c)
or 2, a long autonomous study is necessary where we use cohomology operators on the forms given by d+e(w), where e(u) is the exterior product by a non-exact closed 1-form w. We have:
3) be a transitive local Lie algebra on the manifold W of even dimension such that the corresponding structure (A,E) Is true locally conformally symplectic. The algebra (N,( , ]) is isomorphic by a to L* dhich is an algebra of locally conformally symplectic infinitesimal transformations of (W,A,E) and its center is reduced to (0); L* (resp. N) coincides with its 'coninutator ideal. The derivations of (N,[ , ]) are given by where Theorem 3
Z
L.
Let (N.E
The space
,
is generated by 1-differentiable 2—cocycles and
there may or may not exist non-trivial infin{tesimal deformations of (N.E necessarily equivalent to 1-differentiable deformations. 204
,
3),
We see that this situation is intermediary between the situation corresponding tothe Pfaffian manifolds and the situation corresponding to the symplectic manifolds. 6.
THE ASSOCIATE POISSON MANIFOLD AND THE DISTINGUISHED CHARTS
Let (W,A,E) be an arbitrary Jacobi manifold. Consider the manifold W = W x P. we denote by z the canonical variable of P, by Z the vector field of W corresponding to a/az; by p the projection p : W W; and E define on W elements, denoted by the same notation, such that (a)
c(Z)E=O. Introduce on W the 2-tensors
r=A+ZAE
A:
(6.1)
It is easy to see that A satisfies =0 We
=
(6.2)
àbtain:
Proposition To each Jacobi structure (A,E) on W is canonically associated on W = W x P a Poisson structure A homogeneous of degree -1 with respect to Z, and conversely. The analysis of the leaves of (W,A) gives: Proposition Let S(x) be the leaf of (W,A,E) passing through x and be the leaf of (W,A) passing through such that p = x; we have p S(x) = and
(1) (ii)
if 5(x) is of odd dimension, coincides with S(x) x if S(x) is of even dimension, p defines as a covering of S(x).
We see that a l.c.s. leaf admits an interesting symplectic covering. (b)
Let (W,A) be an arbitrary Poisson manifold. A Weinstein (8) has proved:
Proposition Let x be an arbitrary point of W_such that S(x) has the dimension 2p. There is a chart {x13 = {xa;xa) = {XA;X A;xa) p A = A+ p;
a,b,c = 2p+1,...,m) of W, of domain U containing x such that the non-vanishing 205
components of A are
-
Aab(XC)
1
locally Iso-
are null at x. For U, the-Poisson
where the
manifold and of a Poisson manifold of morphlc with the product of a rank 0 at x. The symplectic factor is described by xa . 0 and corresponds to the connected component of x of S(x) n U. Such a chart {Xa;Xa} is said to be a distinguished chart in the neighbourhood of x. Using these propositions, we can obtain a notion of distinguished chart for the neighbourhood of a point of an arbitrary Jacobi manifold (W,A,E).
Let x be a point of W for which the leaf S(x) has an even dimenejon. It for a domain U containing x, the Is possible by equivalence to suppose connected component of x on S(x) n U is andb:
We
can deduce from a
1
For each point x such that the leaf S(x) has the even dimension 2p and for a structure (A,E) such that S(x) is symplectic in the neighbour{xa;xa) {xAx A.,Xa} (a = x, x=1,...,p; hood of x, there is a chart x + p; a,b,c = 2p+1,...,m) of W, of domain U containing x such that the components of A and t are respectively Proposition
-
AAX
AXA =
I
Aab(Xc)
:-
and
Ea(Xc)
and Ea are null at x. For U, the manifold is locally Isomorphic with the product of a symplectic manifold and of a Jacobi manifold of rank 0 = 0 and corresponds to the at x. The symplectic factor is described by x of S(x) n U. connected component where the
(d)
Aab
Let x be a point of W for which the leaf S(x) has an odd dimension.
We
have:
Proposition
For each point x such that the leaf Six) has the odd dimension
(2p+1) there is a chart {x1) = {xa;xaj = A = A + p; a,b,c = 2p+2,...,m) of W, of domain U containing x, such that the non—vanishing components of E and A are 206
=
=
_Aao(Xc)
Aab(Xc)
are_null at x. For 1), the manifold is locally iso= x0 of the product of the Pfaff Ian leaf S(x) morphic to the diagonal described by = 0 by the Jacobi manifold (AabAo'a E° = 1) of rank I at where the
and Aab
x described by x
A
= xA
0.
We have obtained a notion of distinguished chart in the neighbourhood of each point of an arbitrary Jacobi manifold. We suppose in the following everywhere.
ft
0
7.
DERIVATIONS OF THE LIE ALGEBRAS ASSOCIATED WITH A LbCAL LIE ALGEBRA
We denote by Der(L*) (resp. Der(L)) the algebra of the derivations of L* (resp. L) for an arbitrary Jacobi manifold (W,A,E). It is easy to see that such derivations are always local. I have proved some years ago if (W,A,E) is a regular Jacobi manifold or Poisson manifold the elements of Der(L*) or Der(L) are given by a vector Z of 1. Using on the one hand the regular components W of W, on the other hand distinguished charts in the neighbourhood of limit points of regular we can extend the previous results to a generaiJacobi manifold. (a)
Let I be the Lie algebra of the infinitesimal automorphisms of the conformal Jacobi structure determined by a local Lie (N,[ , ]) and L* the Ideal of I given by the corresponding Hamiltonian vector fields. Each derivation 0* of L* is given by D* = X E 1* where Z EZ,X) L. Each Theorem
derivation D of I is an inner derivation. Let Oer(N) be the algebra of all the derivations of (N,( , ]); the algebra of the local derivations. It is possible to prove that if V V is given necessarily by a differential operator of order 1. and there is a vector field Z and a scalar a such that (b)
Vu = c(Z) u + au.
(7.1)
Study the conditions under which (7.1) is a derivation of (N,[
•
]).
We
obtain: Theorem
Each local derivation V of (Nd
and where a is the corresponding scalar.
,
)) is given by The first space
I
where Z
of the 207
(P4,{
local
,
is isomorphic to L/L*.
3)
N, associate to the Hamfitonian
(c) let P 5€ an element of Der(N); if u = c(u) the Haniiltonian vector vector
(7.2)
= cj(Pu) = Pu £ + [A,puj.
(7.2) defines a algebra
Der(N)
p:D
defines D*, it defines a local derivation :ar
D*
Der(L*).
If Z E L
of N such that
= D*.
in a uniquo (7.3)
that p(D1) = 0, that is, with values
where
in the
,
P1;N
3).
C is a derivation iff
vanishes
or EN,H).
Let v be an element of C.
We know that if S is a Pfaffian leaf or a
t.l.c.s leaf we nave
0 only on the largest open v submanifolc4 W, for which all the kv.es are globally conformally symplectic. Wehave
Propositton
Each derivation D of (N,[
Du = c(Z) u + au + D1u where
,
]) is given by
(Z E L)
is a non local operator with values in C vanishing on {N,N).
We
have Sup CD1)
that (W,A) jS a The typical case for D1 0 is the following : regular Poisson manifold with conrpact symplectic leaves that define a fibration of W, We have D1u : A(fU) where A is an arbitrary linear operator on is given by (n symplectic the centre C of (N,{ , }) and where = JS(x) volume form'of the leaves). 8.
THE CASE OF LINE BUNDLES
a local Lie W be a Une bundle on W (r(K), (a) Let ¶ = , be a section of K without on the space of the sections of K. zero over a dcmain U of W. s r(K), we have (where = N(U)); Lie algebra ( , I), [ )),induces on UU for which [UU,VO] is 208
by a bidifferential operator of maximum order 1 for
We have with clear notation:
each argument.
= i(Au)(duu A
4
dvu)
i(Eu) dvu
without zero. Let V be anOther domain of W with a section 0) and there follows we have huv °u (with huv Ev = huv Eu +
Itv
(r(K),E
,
1) define thus on W a
(An,
For U n V #
huv).
general conformal Jacobi structure associated
of this structure.
with K; L is the algebra of the infinitesimal To each section s
(8.1)
duu.
r(K) corresponds a Jiamiltonian
element
Vector Field
of an ideal L* of L, such that Eu +
We have a homomorphism a of (r(K),[
,
]) onto L*; its kernel is the centre
C of the considered local Lie algebra. (b)
The conformal Jacobi structure defines by means of L* a (generalized) Let S be a leaf of W. If K is trivial, the geometri-
foliation of W.
cal structure of S is exactly the same as in the scalar case. If K is non—trivial, we have a different situation. ,r
If S has an
CS)
dimen8ian, S has a non-orientable contact structure (not generated by a global contact 1-form). If S has an even dimension, a covering of S is symplectic and, according to its dimension, S may or may not be orientable; we obtain thus an extension of the notion of l.c.s. manifold. In the two cases, the corresponding transitive local Lie algebras have centres reduced to {O}. The corresponding deformations can be studied; if odd
1
trivial,
(S)
(r(K),[ ,
]) admits non-trivial deformations.
be the main field of the vector bundle K + W. It is possible to prove that each local Lie algebra (r(K),( , ]) is canonically associated with a Poisson structure on the manifold K, homogeneous of degree -l with respect to Z, and conversely, The results concerning the d-stinguished charts can be extended in a natural way. It follows that it is the same for the results concerning the (c)
Let
derivations of the Lie algebras L*, L, (r(K),[ derivation P of (r(K), C , ]) is given by
,
]). In particular, a local 2'09
a.s.
Os = c(l) s
where Z E L and a is the corresponding scalar. References
El] Kirillov, A.A. Th.i.ssian Math. Surveys [2] Lichnerowicz, A. Comptes Rendus Acad Math. Puree et Appi. 57 (1978) 453. (3)
(4)
So.
(1976) 55.
Paris 2851 (1977) 455; J.
2971 Lichnerowlcz, A. Conrptes Rendus Acad Sc. Paris 2961 (1983). (1983) 261. F. Guedira and A. Lichnerowicz, J. I4att Puree et Appi. (to appear). (1954) A. Nijenhuie, Schouten, J.L. Cony. mt. Geom. 01ff. Rome Indag.
ES]
31
Math. 1? (1955)
Sussmann, R.S. Amer. Math. Paris
390.
Trans. Amer. Math. Soc. 180 (1973) 171k P. Stefan, BuZZ.
Soc. 80 (1974) 1142k P. Dazord, Comptee
[6] Gerstenhaber, M. Ann of Math. 79 (1964) 59. [7] Lichnerowicz, A. J. Math Puree et Appi. 53 (1974)
(8]
Acad.
So.
for the star—
products, see Ann. Inst. Fourier 32 (1982) 157 and N. De Wilde, P. Lecointe Lett.in Math. Thye. 7 (1983) 487. Weinstein, A. J. of Diff. Geom. 18 (1983) 523.
Andre Lichnerowicz de France Paris Frapce
210
R,nthsa
2971 (1983) 305.
XMASA
Cohomology of Liç foliations Let M be a smooth closed manifold which carries a smooth foliation F. A smooth form ü of degree i is said to be of filtration k a if it vanishes
whenever i—a+1 of the vectors are tangent to the foliation.
In this way the
de Rham complex of smooth forms becomes a filtered complex and we have the
spectral sequence Ek(F) which converges to the real cohomology of N. where is the cohomology of 14 with coefficients in the
sheaf of locally conStant differential functions along the leaves of the
foliation. In this work, we compute this spectral sequence for the lie foliations t2] and we obtain some consequences.
We begin by defining an action of the Lie algebra g on 4. Let X(M,F) be the algebra of infinitesimal transformations of F; we consider the X(N,F) on the algebra of differential forms on N, (1)
=
denotes the usual Lie derivative. Let x(M,F), • Q*(M), where r(F) C X(M,F) be the algebra of the vector fields tangent to the foliation. The factor X(M,F)/r(F) is, in this case, isomorphic to the Lie algebra g, and the action (1) defines an action 0 of g on 4. x
Proposition 1
The cohomology spectral sequence of a Lie foliation is
= Ha(9,4)
:
To prove this proposition, one defines a bigradation of the smooth forms of M -
by considering a complement to TI in TM. The first degreeis transversal and the second is tangent. The differential operator decomposes into three is the homogeneous parts, d = d01 4 d10 + d2,_1 and 211
Both the complement and the foliation deténnine a map
cohomology. 9
>x(M)
F;
)XF;.
Now, g-action on 4 may be stated as = (Lx
d1
F;'
There exists an Isounorphism between Aa9* a
and
and the
proof is straightforward. Coroflary 1 (1] If the leaves of the foliation are dense then the basic .cohoinology of F Is the cohomology of the Lie algebra. The basic cohomologyis In the case of dense leaves, H isR. We now consider the particular case of unünodukzr Lie foliations, looking
for a condition for minimality. be dual bases of g and g*, respectively. If g is a unimodular p-dimensional Lie ajgebra, the differential operator in term of the spectral sequence the Let
and
•
•4
)
4
is defined by d1(Wa
wherew
&
e {s} ) = ± I
...
A...Wa—i Awa+i
a p A...W.
Let 4 be the vector spaée generated by the vectors of the form • F; E c 4. We consider in 4 the topology induced by the In this way it becomes a locally convex topotopology of usual logical vector space.
4/4. Moreover, 4 has finite codimension. The second part of this lema comes from the fact that all Lie foliations are transitive (i.e., the infinitesimal transformationg of F generate TXM for Lesana I
each point) and we can construct a 2-parametrix [7] and conclude that are finite—dimensional vector spaces. 212
For the maximum dimension case, u = q, the leaf dimension
is a closed
It is the kernel of the continuous linear form
hyperplane.
JM A)A...AA$. P e 4.
the transversal holonomy invariant form defined [5). If the leaves of F are dense, by Haefliger [3], Is isomorphic to then the topological dual of is P [3], and 4 is just the closure of 0 In We have
One can now give a minimality condition. cohomology defined by Rummier [6),
In fact, the de Rham F—relative term of the is just the
spectral sequence [5].
If the leaves of the foliation are dense and the Lie algebra Is unimodular, then the foliatIon is minimal If and,only if the topological is P. In another case, this dual of dual Is 0. Corolla!y 2
We have a short exact sequence of complexes
0+ This splits as a sequence of complexes if and only if the Ruuutler condition
Is satIsfied. We do not know whether Acknowledgement.
= 0 is possible for uftimedular Lie foliations.
During the realization of this work the auther took part In
an Integrated Action with the UER de (Unlversitd des Sciences et Techniques de Lille), which allowed him to exchange views on the subject with some of its members, especially with G. Hector. References
(1)
[2)
El Kaciml-Alaoui, A. and Hector, G,
Analyse globale sur l'espace des feuilles d'un feullietage rlemannien. Preprint (Univ. Lille I). Fedida, E. Sur les feullietages de Lie. C.R.A.S. Paris 2?2 (1971) 9991002.
[3]
Haefliger, A.
Some remarks on foliations with minimal leaves. J.
Geom. 15 (1980) 269—284.
[4]
Kaiit,er, F.W. and Tondeur, P.
Duality theorems for foliations.
Aetérieque 116 (1984) 108-116. 213
(5)
I4acias, E.
Diferenciable,
Continua ti Diecreta
FoZiada. Pubi. Dpto. Geometrl'a y Topologfa. Santiago de Coinpostela 60 (1983). rietnannienne et Runinler, H. Quelques notions simples en leurs applications aux feufiletages compactes. Ccvin. Math. Rely. 54 Variedad
(6]
(1979) 224—239. (73
Sarkarla, K.S. A finiteness theorem for foliated manifolds. Soc. Japan 30 (1978) 687-696.
Xosé Masa
Dpto.Xeometrfa e Topoloxia Universidade de Santiago
Spain.
214
de una
.r.
J PRADINES
Graph and holonomy of singular foliations any regular foliation is associated its hoionomy peeudogroup, well defined up to a suitable equivalence (considered independently by W.T. van Est and by A. Haefliger (12]),whlch bears all the topological and differentiable information on the transverse structure of this foliation, In other words To
on its leaf space, which in general falls to exist as a manifold. As a matter of fact, this holonomy pseudogroup may be viewed as a pseudo-
group representative, under a suitable equivalence relation between differentiable groupoids of the holoiwmy groupoi4 introduced by C. Ehresmann in 1961 [9) as a topological groupold (and considered by the author, in a wider context, with its manifold structure in 1966 (15)), later rediscovered, using a different construction, by Wlnkeinkemper (23) and popularized by A. Connes [5], under the 'name of graph of the foUati,on. On the other hand, as explained in the address by A. Lichflerowlcz (13], the study of eympleetio geometry has focused attention on those foliations with possible eingul.aritiee, which are generated by families of vector
fields. Such foliations were encountered previously by H. -Sussmann In the context of càntrol theory (21], and a nice geometric characterization was discovered Independently by P. [20]. Roughly speaking, foliations may be viewed as "lower semi-regular" foliations and genfoliations. There is a basic theorem connecting eral differentiable groupoids (in a way that is not Implied by nor implies theorem), which completes a result. stated by the author in 1966 (15]. It asserts that the transitive components of the groupold define a canonical
foliation. If we drop the differentiable structure, foliations may be viewed as special examples of topoiogicai foliatione In the sense of Ehresmann (9) and as a consequence their holonomy groupold has been defined as a topol.o—
gical by this author, at least under the assumption of (topological) "local einipUoity", which ensures the existence of a "germ of leaf space". It should be noticed that the much more restrictive assumptions of local 215
stability and ualnost regularity", recently used by P. Dazord (6] and M. Bauer f I] respectively, imply Ehresmann's local simplicity. Under these assumptions these authors show that the Ehresmann holonomy characterizes the
structure of the foliation around a singular leaf. However we point out that in general the above process involves a considerable bee as follows from the two subsequent remarks concerning the singular case:
(I) the loss of differentiability is irreversible, the
lay groupoid
'being no longer a manifold, which is related to the fact that even
the
local
quotient epacee fail to ecist ae manifolde;
(Ii) the holonomy group of any singular leaf which is reduced to a single point will. always vanish and therefore tells us nothingabout the vicinity of such a 1sf, which contradicts the .intuition that a kind of whirl should be associated with such leaves, involving a continuous local group action in soma loose sense. Our
is to extend, by means of completely new methods, the con-
structionof the differeitlable graph to a rather wide class of
folia-
tions to be generic in some suitable sense) and to derive from this constructlqn the holononiy groups, which, for singular leaves, will be con— tinuous, and will not arise from the fundamental groups of the leaves. This implies that the "graph" is no longer equivalent in any sense (even the algebraic one) to a pseudogroup, the full definition of differentlable groupoids cannot be avoided. More precisely, we a principal "hoioncvity bundle" which the holonomy covering of a regular leaf. By Nsqueazingll the connected components the fibres, we recover Ehresaann's groups and the associated covering. This enables us to define and study the transverse structure of a singular foliation as the equivalence class (In a suitable sense) of its TMdlfferentlable graph; this Is to future papers. We just sketch here the construction detailed in (25]. Our starting'p.lnt is a new construction of the TMgraphM of a regular foliation published in (19], where we charactertze such graphs as differentiable groupolds shich are locally embedded in the "coarse" groupoid x B, with an extra condition of connectedness. The composition law is then uniquely determined by the differentiable graph structure, i.e. by the pair 216
of submersions 'source" and "target". All these graphs are embedded in a big groupoid called the univereal graphoid. Now the local embedding cpnditlon may be weakened in order to allow singularities, by considering the "sous-trucs" of BxB, in the sense of [10), inthe category-of differentiable graphs with fixed base B(i.e. manifold of vertices): such graphs are called monographs.
It is much more difficult to define the suitable composition law: the obvious one, defined by fibre product of source and target, is "redundant" and is clearly not satisfactory. In the regular case, the composition law is easily derived from the existence of local quotients, which fail to exist to use an intrinsic property of when singularities are allowed. So the underlying graph of a groupoid, which consists In the existence of a
local calculus of fractions t = zy1x, which we call rule
of three.
The
monographs admitting such a local rule of three, which Is necessarily unique, are called conveatora and the units are called convections. It is a highly remarkable but not an obvious fact that these germs of convectors may be conrpoaed in a way which Is the exact generalization of the set-theoretic composition of graphs, but cannot be expressed but in terms In the category of of differentiable maps. It is necessary of to develop a preliminary machinery for differentiable graphs and double graphs, which has Its own interest, and will be used in other papers as well. This leads to a differentiable groupold, called the universal convector, whose units are the germs of convections, which contains the universal graphold as an open subgroupoid. The essential property of this groupold is that
its structure is uniquely determined by its underlying graph structure, —
though it has non—trivial Isotropy groups. However, as the terminology suggests, a convection Is something more than
foliation, and may be viewed intuitively as a certain the underlying class of (multidimensional) flows along the leaves, which are the stream
lines. Unfortunately It may be that there is no way, or at least no canonical way, of associating a convection toa singular foliation. So to match the exact fit of the regular case, we have to limit ourselves to the class of foliations (called 'halonomous") which admit an Intrinsic or those convection, in a sense to be made precise. It seems that this class Is broad enough for a wide nuther of applications, 217
though we must confess that at present we cannot be precise as to the extent
to which it Is 'generic1' (if it Is), and It may be that the definition of "holonomous" that we use here will require some further adjustment In the light of new examples.
this reservation, we have an exact generoiieatian of the different— jabie graph of the regular case. With
Note
This text will be published in its full version (50 pages) In (25].
A sumary Is being published in [24]. We just sketch here the leading ideas, and we refer to (25] for a full account. The present work was achieved in partial collaboration with my student B. Bigonnet, In whose thesis examples of the holonoinu bundle may be found. References
[1] Bauer, M.
Feuilletages presque
C.R.A.S.,
299 (9) (1984)
387-390.
[2] Bourbakl, N. Theorie dee Ensembles. (3] Bourbakl, N. Var (4] Chevalley, C. Theory of Lie Groups. (5] Connes, A. A survey of foliations and operator algebras, I.l?.E.s. (1981). (6)
Dazord, P.
Holonoinfe des feuilletages singullers, C.R.A.S.,
298
(2)
(1984) 27—30.
(73
Ehresmann, C.
Coil.
(8]
Categories topologiques et categories
Diff. Glob. Bruxeilee, C.B.R.M.
Sur les categories Geom. Diff. Bologna, 1967, 31-40. (9] Ehresmann, C. Structures Ehresmann, C.
(1959) 137—150.
Atti Cony. mt.
Proc. 5th Can. Math. Cong., (1961) 109-172. (10] Grothendieck, A. Sur quelques points d'algèbre homologique, Tohoku
Math. J.,
2, 9 (1957) 119—221.
GroupoYdes d'holonomie et classifiants, 116, 70—97 (Journdes S.M.F., Toulouse, 1982). (12] HaêlIiger,A. Pseudogroups of local Isometries (these proceedings). (11]
A.
[13] Llchnerowlcz,A. Generalized foliations and local Lie algebras of Klrillov (these proceedings). 218
(14) [15)
Categorice for the Working Mathematician, Sprinäer, 1971. Pradines, J. Th6orie de Lie pour les groupoYdes différentlables,
MacLane, S.
C.R.A.S.,
263
(1966) 907—910. dans Ia Calcul C.R.A.S., 264 (1967) 245—248.
[16]
Pradines, J.
(17]
Pradines, ). 266 (1968) 1194—1196.
au-dessus
des groupoTdes
groupolde,C.R.A.s.
[18) Pradlnes, J. Building categories in which a Godement theorem Is Diff., Milens (1976). available, Cahiera Topol. Holonomie et graphes locaux, C.R.A.S. 298 (13) (1984) Pradines, J. [19] 297-300.
(20]
P.
Accessible sets, orbits, and foliations with singularities,
Proc. London M2th. Soc. 29 (1974) 699-713.
Orbits of faml.lies of vector fields and integrability of distributions, Trans A.M.S. 180 (1973) 171-188.
[21]
Sussmann, H.
[22)
Van Est, W.T.
Rapport sur les S—atlas, Aetérisque (Journies S.M.F., Toulouse, 1982).
(23]
H.E.
116
(1984)
me graph of a foliation, Aim. Glob. Ann.
236—292.
Geom.
(1982), (24] Pradines, J. and Bigonnet, B. Graphe d'un feuilletage singulier, C.R.A.S., Sknce du 14 Janvier 1985 (to appear). (25] Pradlnes, How to define the differentiable graph of a singular
foliation, Cahisre de Topologie et Vol. 26,
de Geometrie
(to appear).
Jean Pradines
Laboratoire dAnalyse sur les Varietes UniverSite Paul Sabatier 118 Route de Narbonne Toulouse Cedex France
219
M SARALEGUL
The Euler class for flows of isometries 1.
FLOWS OF ISOMETRIES
Let (M,F) be a compact (n÷1)-manifold provided with an orientable Riemannian foliation F of dimension one. The leaves of F are the orbits of a vector field without singularities. We say that F if a flow of isometriee if there exists a Riemannian metric g on M and a vector field Z tangent to F We always assume that Z is which generates a group of isonietries 1.1
a unit vector field (we only have to replace g by (g(Z,Z))1g). 1.2
Recall that a differential form w
is baae—iike for F
if
The cohomology of the complex of base-like forms is the baBe—like oohcwiology of (M,F) denoted by H*(M/F); it depends only on (M,F) and not on the choice
of (g,Z). For a flow F of isometries the oharacterietic respect to (g,Z) satisfies the equations 1.3
x(Z) =
1—form x
of F with,
= 0;
I and
in particular, the form dx is base-like for F. By traneverse (with respect to g) we mean the unique form v the volume form of (M,g). F
volume form of
such that v
xis
According to (7], an orientable Riemannian foliation (M,F) of dimension one is a flow of isometries if and only if one of the two following equivalent conditions holds: 1.4
0;
(a) (b)
0
[vJ
H'1(M/F).
Supported by the Government of the Basque Country - Eusko Jaurlaritza. 220
2.
INTEGRATION ALONG THE LEAVES OF A FLOW OF ISOMETRIES
be the one—parameter group of Let (M,F,g,Z) be as above and let isometrles generated by Z. Since the group of isometries Isom(M,g) of a compact Riemannian manifold is a compact Lie group, the closure I of In Isom(M,g) is a compact coninutative Lie subgroup; thus, a torus. 2.1
2.2
be the subcomplex of forms invariant by the action
c
Let
According to the definition of I, it appears that
of 1.
0).
=
in cZ*(M) is a homotopy equivalence (see (4)), thus
The inclusion of
(M) the form IZU) is invariant by T and base-like for of F. This enables us to construct an integration operator aiong the F (see (5]): For any form
E
which coninutes with d and satisfies the two following properties: (a)
it is onto; indeed, for any a
thus a A X
and
AX + (..l)r_la A izX)
a AX = (b)
Ker
indeed, if
3.1
Is obvious.
To sum up, we have constructed the following snort exact sequence: >
3.
= 0 for The other
L2a - di2a = 0, and a 2.3
we get
0.
GYSIN SEQUENCE AND EULER CLASS
From the above sHort sequence we get the following cohomology sequence: 221
* 4
I
Hr(M,F)
1
>H"(M)
which we will call the Gysin As for any a E
)Hru(M,F) ._.9,+Hr+1(M,F)
sequence of F.
we have
a A X = a, the definition of the
gives
connecting homomorphism
6g[a) = [d(a A x)) =
* [dx].
A dx]
Then, as for Seifert fibrations, we define the Euler class of to g
F
respect
by: e9(F) = (dx] £ H2(M/F).
Up to a non-zero factor, this class does not depend on,the metric g. Indeed, let (M,F,g1,Z1) and (M,F,g2,F2) be two flows of Isometries with the same underlying Rieniannian foliation. The two Gysin sequences give 3.2
f59
)
+ H0(M/F)
>
...
j
1,2.
The space H0(N/F) is of dimension one, thus, by exactness, dim ker
1
and
Now e
(F) and e
such that e
(F)
(F) are simultaneously zero or there exists x E R - (0) xe
(F).
In particular, the fact that the Euler class
of F with respect to the metric g vanishes does the metric g.
depend on the choice of
In the particular case of Seifert fibratlons our Euler class coincides with the usual one, up to a non-zero factor. This factor is exactly the length of the generic leaf of F. Then for a suitable metric we obtain the usual Euler class of F. VAIliSHING OF THE EULER CLASS
Next we obtain a geometrical interpretation of the vanishing of the Euler class of F which generalizes that of (8). Some of our results are also proved in [7) by means of invariant currents and foliated cycles. 4.1
Theorem 222
Let M be a compact manifold with an
Riemannian foliation
F: Then the following statements are equivalent: (1)
(M,F) is a foliated bundle;
(Ii)
F
is a flow of isometries and eg(F)
(iii)
0 and 1*
0 for any suitable metric g;
+ H"(14) is injective.
:
Suppose that (14,F) is a foliated bundle and let F + 14 B be diagram: the fibration transverse to F. We have the following Proof
(a)
pr
FxR
>14
F
>R
p
and it is the covering given by the
where q is the canonical covering of
suspension of the holonomy diffeomorphism h of F. Because F is Riemannian there exists a metric Cdt)2 is on F Invariant by h. The metric = equivarlant with respect to thus induces a metric g on N which Is a bundle-
like metric for it and makes F orthogonal to n. Therefore F Is a flow of isometries and the characteristic form x Is equal to p*(dt). Then dx 0 and the Euler class of F with respect to g vanishes. This proves (Ii). Conversely, let us suppose that F is a flow of Isometries with respect to a metric g. If e9(F) 0 there exists v E suLh that dx di. The form w = X-y satisfies w(Z) = 1 and dw = dr-dy = 0.
By Tischler's theorem (see [9]) the foliation defined by the closed form w can be approximated by a fibration which will again be transverse to F. (b) Now we show that (ii) is equivalent to (iii). We first note that 0 (see a Rieniannian flow F is a flow of isometries if and only if
1.4).
ConsIder the sequence +
>
for any suitable metric g. homomorphism is zero.
H"(M/F)
+ ...
If (ii) holds, e9(F) is zero and the connecting
Therefore
is injective and (iii) is 223
On the other hand, recall that for any Rietnannian foliation there exists, complex of base-like forms of F, a Hodge theory similar to that of theDe Rham complex of a compact manifold (see (3]). It Includes a base-like Hodge operator * which enjoys the usual properties. Thus if we fix a suitable metric g on N, dx is base-like cohomologous to a base-like harmonic that is, there exists y E n1(M/F) such that dx = a + dY. form a
To prove that (iii) implies (ii), it is enough to show that a
0.
So
assume that (ill) holds and a is different from zero. The base-like harmonic form *a positive, Xv, A dual to cx satisfies a A A dX = *a A cx +
A dy =
Then t*ct A dx] =(Xv]is a non-zero class in
A y).
But, by exactness of the Gysin sequence of F, we also get = 0, which (1)n2 Implies A dx] 61*a] = 0. This contradiction ends the proof. 4.2
Remark
(see 1.4).
The condition (ill) is equivalent to the fact that the class
Cv) is different from zero in H"(M), where v is the transverse volume form of (14,F,g) for a suitable metric g on Corollary Let 14 be a compact (n+1)-manlfold provided with a flow F of isometries. If H1(M) = 0 then the Euler class of F is non—zero. 4.3
4.4 Corollary Let N be as above. If the Euler class of F Is zero then there exists a finite covering M of 14 which is diffeomorphic to the product Fx
51
It follows from Theorem 4.1 that if the Euler class is zero then there exists F • N _.2_4 a transverse fibration to F, defined by suspension of a diffeomorphism h of F. We can assume that F is a Seifert fibration (see (2]). Then the holonomy of any leaf is finite, i.e., a periodic Proof
map at any point. Now it is not difficult to see that there exists p 'such that is the identity. Consider -, Z[h] c Diff(F) the holonomy homomorphism of F. Because is the identity we have an induced homomorphism Z/pZ. The associated covering 14 is a foliated bundle F x holonomy is generated by = IdF; thus
The integration operator constructed in Section 2 is a particular case of an integration operator defined for any taut foliation. 4.5
224
Remarks (1)
(ii)
flow may admit The example of Carriere [1] shows that a a transverse foliation G without being a flow os Isometries. In this case G is not Riemannian.
(iii) There exist flows of isometries which admit non—Riemannian transverse foliations and which have non—trivial Euler class [10). 5.
CONTACT FLOWS AND FLOWS OF ISOMETRIES
A flow F (i.e., an orientable foliation of dimension one) defined on a compact (2k+1)-rnanifold M is a contact flow if there exists a form w such that: a contact
is a volume form on
(a)
w is
(b)
the unique vector field defined by w(Y) =
form, I.e.,
A
I
and iydw = 0 is tangent
toF. means of the Euler class, we get a partial characterization of the flows of isometries which are contact flows (see [8] for the compact case 5.1
and
[6]).
Theorem
Let M be a compact (2k+1)-manifold with a Riemannian contact flow
Then F is a flow of isometries and the Euler class of F is different from
F.
zero. Proof
Let g be a bundle-like metric on (M,F).
We can write g
+
(resp. is the restriction of g to the tangent bundle (resp. the normal bundle) of F. We define a new bur.dle-'Iike metric on (M,F) by
where
where w is the contact form given by (a). It Is not difficult to see 0 and therefore F is a flow of isometries with respect Furthermore, dw is a base-like form and [dw) belongs to Ker If Euler class ofF is zero, then [dw] = 0 in H2(M/F) (see Theorem 4.1.), •and there exists y &(M/F) such that = dy. Because dw A y A =,{O},we get =w
w
ft *y •
A
(dw)k_l = d(w A y A (dw)k_l)
'and
wA
dçth A y A 225
Then
is not a volume form which Is In contradiction with a).
A
5.2 We find in [6] a converse statement for the case of geodesible flows In three—dimensional manifolds (Including flows of isometries). Now this
gives a complete characterisation. Theorem
Let (M,F,g) be a flow of isometries on a compact Riemannian 3-mani-
fold, then the following statements are equivalent:
(I) the Euler class of F Is zero; (ii) F Is not a contact flow. 0, then any Corollary If M is a compact Riemannian manifold with H1(M) Riemannian flow F on 14 Is a flow of Isometrles and a contact flow. 6.
FLOWS OF ISOIIETRIES ON S3
.The Euler class enables us to classify partially the flows of isometries on a given dompact manifold. For example, consider the family of all flows of isometries on The Seifert fibrations have been analyzed in (8], therefore in order to get a complete description of these flows, it remains only to study a one-parameter family {F, a [0,1]) which can be described as follows. For ci E ]O,i], F Is the foliation defined in complex coordinates by the flow with
=
z1,elt z2).
is a group of isometries of
For any a,
53
with
to the usual
metric g. As we pointed out in Section F is also a flow of Isometries with respect to the metric g = (g(Z ,Z)Ytg, where is the vector field defined by If is tIle transverse volume form of the Euler class eg (F) is determined by the number r # 0 such that eg (F) = r by a the formula
r=
1
dXAX, a cx
which
gives
r = (isa). 226
It is clear that this nunter r classifies completely the elements of the family {Faict £
If e
p/q Is rational4 F is a Seifert (F) is related to the Euler class c (F)
and our Euler class by Nicolau—Revent6s by the
9a
•
formula
*
qc.(F) = e9(F),
$
onM whose for any Indeed, £a(F) = [dc) the cohstant function 1. On the other hand, the along the fibres of F inte9ral of the characteristic form xa of F is the length of a regular leaf follows by taking of F, that is q. The =
Referends
ti:j Carriere, Y. Flots riemanniens, [2] (3k)
sur les structures transverses des feuilIet4es, Toulouse 1982, Aster'iaque, 116 (1984) 31-51. An. Acad. V. and Ghys, E. Feuilletages totalement Bi'asil. 53 (3) (1981) 427—432. A. and Hector, G. Analyse globale sur l'espace des feuiIles
Preprint (Universite de Lille 1).
feui1Ietage,
[4)
W., Halperin, S. C?ihomology. Academic
[5)
Vanstone, R.
Connectiona' Curvature and
(1973-1975).
Foliations and in P-regress in Mathematics, 32 (1983) 103-152. Manna, G. Feuilletage de contact et cohomologie basique. Preprint. [7] Molino, P. and Sergiescu, V. Deux remarques sur les 'flots riemanniens. Namber, F. and Tondeur, K.
I
[8]
anuscripta !.zath. (to apliear). Plicolau, 11. and Reventos, A. On some geometrical properties of Seifert
bundles. IBr.
[9)
Math., 4? (1984) 323-334.
J.
Tischler, 0. On fibering certain foliated manifolds over S'.
TopoZogy,
9 (1970) 153—154.
[10) Wood, J.W, Math.
•
Bundles with totally disconnected.structure group.
Rely., 46
Connent.
(1971) 257—273.
Martin Saralegui University del Pals Vasco Bilbao,
-
Spain 227
H SUZUKf
An interpretation of the Weil operator X(y1) INTRODUCTION
In this article, we give a cohoinology theoretic meaning to the differential form h1 corresponding to y1 of the secondary characteristic class y1c1 (see, e.g., [5, p. 154] of foliation. Let F be a C°—foliatton on a manifold N. Let vB (VR) be a Bott (Riemannian) connection on the normal bundle v(F) of F Let dE denote the exterior differential along leaves and HD'R5(N) the foliation de Rham cohomology vector space (Cf. [8], [10) and [11)). (Theorem 3.3) For
a
and
VD and V
2j The Well operator
.
(N) does not depend on the choice of
HF6R
of (4] Is regarded as a multiplication
by [(hj)02J%]. l(N). In this sense, the operator x(y1) is essentially an element of In other words, the notion of the Well operators is expanded to that of cohomology classes of O,*(M)
(U,
For a
on
n U8
= log
and satisfy the cocycle condition
are constant along leaves of a c
By
-cay
B
+c aB =0
denote the sheaf of germs of C°'-functions constant deterThe tech cohomology class m(N,F) e mined by ca'led the modular cohomology class of F (see, e.g., (12]) which Is closely related to the modular function of transverse measure on type holonmay groupoid of F by (2, p.41]. One can eastablish a de on
fl
along leaves
U
FJ
.
Let
.
Then we have:
Let (NJ) be a C'°-foliation on a compact Hausdorffpanlfold. Then, for ((h1)01] corresponding to x(y1), we have
(flteorme 5.4)
228
•(2ii{(h1)0,
=
The above formula is regarded as a new interpretation of x(y1) and also similar meaning of X(Yj). 3 (odd) 3 Is expected. In Section 1, we review the Well operator introduced by Heitsch and 5(M) Hurder. In Section 2, we explain the foliation de Rhani cohomology and in Section 3, we prove that the homomorphism induced by leaf preserving transverse map is invariant under the leaf preserving homotopy through these maps.
In particular, we obtain the Poincare Iemaa for
Then we
prove Theorem 3.3.
In Section 4, the notion of F-simple cover Is introduced and then de Rham is proved. In the last section, type isomorphism for tech cohomology a natural Isomorphism from to the differentiable singular cohomology restricted to leaves is obtained. Finally Theorem 5.4 is proved. A1l1 manifolds, maps and foliations are assumed to be class 1.
i
THE WElL OPERATORS
Let
on a paracompact Hausdorff
be a
manifold. For each point m E H, there is an open neighbourhood U of m and we have linearly Independent on U defining F. Let ADI) be the vector space of C°'-forms on H that is the de Rham complex of M and let be the restriction of £ A04) to U. We set A(M,F)
£
A
0,
1
One can see easily that A(U,F) = A(U) A
A ... A
1(M) be the tangent bundle of H. By the integrability condition for tangent sub-bundle 1(F) F of 1(M) corresponding to F, we have 1-forms on U such that Let
q do,.
'
=
E
U
U
Since we have
- 229
o
A
q
* d(n111) A U. +
A
(
E
u
U
U.4 A w4) Li
1
U d(nlu) A Up
the exterior differentiation is closed in A(M,F) and thus A(M,F) is a differential subcomplex of A(M). We denote the cohomology vector space of A(M,F) by H*(M,F). Let K be a local curvature matrix of a connection v on Cm—vector bundle JI Jq V on M. For any Chern monomial Cj = C1 •• of degree k on the Lie algebra gl(q,R) of GL(q;R), we set A2k(M).
=
It is well known that, for any connection V,
= 0, (see, e.g., [7, pp.
296-298]). Let r(V) be the set of Cm—sections of a Cm-vector bundle V. Let 'v(F) be the normal bundle of F, that is, the dual bundle of = 1(M)/F and v(i). Let be a Bott (Riemannian) connection 'on v(F). Then we have
r(Ak(v(F)*)
A
Ak(M),
which Is the essential part of Bott vanishing (1, pp. 34-35]. Let ir:M x + M be the first factor projection and
VBR=Cl_t)vB+tvft which is a connection on the vector bundle ii*v(F) 01) by
£
=
where 1(a/at)
X
is
the substitution operator of a/at and it,,
over the fibre for it( c3 Cv ).
For
.
j odd, we have
For each k, let part of degree k of
230
Define
v(F) x P.
is
the integrat4on
A standard computation shows that ) = 0
=
Cj(78) -
and hence dhj = Cj (V ).
(r =
2((q+1)/2]-1)denote the homogeneous
(see, e.g., [5, p. 140]).
We define a homomorphism Hom(H*(14;F),
X:(A(yl,y3,...,yr)]k by the formula
x(Y)(n]
(h A
* ... *y.is,
h
* ... * h.is
h.
S E
(2j
— 1) = k 9-
for ii
A*(N,F) with dii = 0. The right side is well defined because we ha'e =
c3(v8)
r(v(F)*) A
therefore
d(h An) = dh A
+
(_l)kh
A
= 0,
and for n
0, j =
dx with A A
d(h A x) — dh =
x + (_I)kh A dx A dx.
The latter formula means that [h A ri] does not depend on the represe of (n]. By making use of affine combinations of different Bott C. and Riemannian connections, one can also show that (h A n] does noi thechoices of arid is the Weil operator associ
c" y
(Cf. (4]). 2.
THE FOLIATION DR COHOMOLOGY
Let (M,F) be a codimension q Cm-foliation on a paracompact Hausdorff manifold. By taking a Rieinannian metric on M, one can split the tangent bundle T(M) into the Whitney sum 231
F 0 V,
104)
where V Is the orthogonal complement of F and we have the splitting of the dual tangent bundle, T*(M)
is clearly +somorphic to the normal bundle v(F). be-a local foliation chart of F. Let (x,u):U r(F*)Iu so that C"-l-fonns on U, V
is
basis of for each m V1i•••iVq r(V)Iu so that
a/au
j
E
1
P,
q,
a
+
aj
where
+
Weset A
AS(r(F*))
end we have n
A(M)
k(r(T*(M))
E
k=0 n =
Ak(r(v*)
E
•
k=O
n =
Z
=
232
Ar(T(V*))
r+s=k
k=O £
Ar, S(M)
fields
Then we obtain i
=
v
And one can also choose
U.
for
is the dual basis of
One can choose
S
A A (T(F*))
0.
____ An element of A''
We denote Ar, SCM) simply by A''
is the sum of diff-
erential forms of pure Cr, s)—type.
w*fdu. A...AdU. AOk A...AOk s
1
splits uniquely into the
and the exterior derivative
r, s+1'
s
Ar42, s-I
s-i
iwr+l,sE
Arli, S
' r, s+i
£
Ar. s+I
This spllttihg defines operators d1
:
Ar,
S
Ar,
S
S
s-i Ar+i. 5, + Ar, s+i
From the relation (d1 + d2 and others. For fixed r(q
O,At, 0 We set
Zr. s
then Br,
s
c
vaotor apace
Ar.
= Ker(dF: Zr,
of
c
+
dF)2=d2=0, it follows r 0), one obtains a cochain complex, d
I
d
•••
1_> Ar. p
Ar, s+i), 6r, S
Ar, s
Ar, 5
F
+
= Im(dF: Ar, s—I + Ar, 5),
We define the foliation
DR
(r, s)-cohomoiogy
(M,F) by
Z'' s/Br, s For one leaf foliation (q = 0) on 14, we obtain clearly the ordinary sdimensional de Rham cohomology of II, (14)
We define a homomorphism I4om(H*(P4,F),
[z A Ti] where z Zr, S and '1 C A(M.F) with The right side is well defined as follows: we have
by the formula
dri • 0.
233
d(z
If ri
=
(d1 +
=
0.
d2)z A
+ (
1)r+s1
A
fl
= 0, j = 1,...,q then one obtains
xA
=
d2 + dF)z
= Cd1
r+s d(zAA) =dzAA+ (-1) ZAdX
=(—1) and
r+s
ZAn
if z = dFa, then one gets d(a
(d1 + d2
=
dF)a Afl
+
(
l)r+s-la
A
dfl
ZAfl. does not depend on the choice of the representative an,d the representative z of (z). Hence
3.
(z A
of En)
THE FOLIATION DR COHOMOLOGY CLASS OF WElL OPERATOR
Let (M,F) and(M',F') be codimension q foliati'ns, and let f:M -* N' be a transverse to F' so that F f*F'. For any point in EM, we set in'
f(m).
Let (x',u') be a local foliation chart around in' EM'.
choose a local foliation chart (x,u) around in such that
j
i,...,q.
One can
f*du,
We have
q*(Ar. S(MI)) EAr, S(M) Since f*d
df*, by comparing components of pure type (r, $
+ 1),
it follows
d,f*.
that f*dF Let f0, f1: H +
H'
be C°°—maps transverse to F' so that
If there is a C°'-map H:M x R -, f1(m)
H(m,i)
i
N'
=
= F.
transverse to F' such that
0,1,
H*FI =
where pr:H x R. N Is the first factor projection, then f0, f1 are called and denoted by f0 F',F f1. H is called by leaf preeerving 234
a leaf preeei'ving Lenina 3.1
-
If f0, f1: M -, M' are
by leaf preserving map, then
are cochain homotopic.
Let (x,u): U
Proof
x
chart. Local charts in define a codimenslon q One can take a basis of
pq be a
x P) x of the type ((x,t),u): U x P It is clear that foliation 'In M x P. = foliation chart of tangent vector fields in the
Mx
P
a/ax1,...,
a/at,
and its dual basis of 1-forms
Let 10, ii: M
Mx P be maps defined by
j
= (m,j)
= 0,1.
is transverse to is defined and equal to F. Ar, 5(M x is written uniquely as
Since Any
=p+
a A dt,
Ar, S(M
where p
Ar"
homomorphism p:A"' 5(M 0,
A dt)
=(
do not contain dt.
x
R) -' Ar, S_l(M) 1)r+s-1
Define a
by
dt.
Then we have a A dt), J +
(_1)r+s(ao,at) A dt +
A dt)
F
+
(1)r+s
A dt,
J1
and therefore +
235
j 0,1 and Since H: Mx I + K' defines f0 FT,F f1, we have Hij Ar, S(M) + Ar. S_l(MI) = H*F' = P. Then we obtain a homomorphism Ar, S(M.)
satisfying for -
=
dFIVH*w
=
+
ithich Is As a corollary of Leema 3.1, we get the following Poincare as a detailed version of (10, Theorem 3.1]. A codimenslon q foliation (M,F) is called F—contractible if there exists a q-dimensional submaitifold N of 14 transverse to F and a map f:M+NcM transverse to F such that The leaf preserving homotopy of this is called F-contraction
tof. Corollary 3.2 Suppose that (M,F) is F—contraotibl9. If such that 0, then there exists and Proof
w
Ar, S(14) (s
1)
a
By Lenmia 3.1, it follows that
f*w - w a But f Is factored by 1:14+ N and the inclusion map 1: N + 14.
is the point foliation and w E A''
5(14)
s
1, we have
a 0. Therefore one obtains w a = In SectIon 2, we have constructed an operator 5(14)
Since i*F = F0 0 and hence
-nw.
HoIa(H*(M, F),
The Well operator X(y) and the homomorphism
are related by the following
theorem.
For any vB and yR on v(F), the (0, 2j - 1)-component
Theorem 3.3 2J—1
of
2j—11
Clearly we have
236
jodd > 0 is a
and the cohoinologyctass R CM) does not depend on the choices of V and V
2j_1])[n] for each (n] £ H*(M,F). In Section 1, we have shown that
Proof
EA"
=
2i—r
r
By the definition of dF, it follows that dF(hj)Q
2j—1
and hence dF(hj)O,
=
2j'
2j—1
= 0.
denote h. for the Bott connection v (0) for the Bott connection (1 - t) v B +
Let denote
k = 0,1 (1) B v
on
on
v(F) and h x R.
Let .ik:N + M x R be maps defined by lk(m)
k = 0,1.
(m,k)
Then, by the proof of i%hj
—
'0, 2j—1
3.1, we have
"j
14* —
'0, 2j—1
''1
10'''j'O, 2j—1
= (dF'I' + Since
"J
0, it follows that
2j-1 '0, 2j—1
"J
'0, 2j_1
and hence
By a similar method,
2j—P
J
d wfFj
F" j'0,2j—l'
does not dePend on the choice of v6. elso does not depend on the choice of
on v(F). The last statement of the theorem is obvious. connection
4.
FOLIATION DE RHAI4 ISOMORPHISM
a cover of H by Let (M,F) be a codimension q open sets. If an Intersection of finite open sets of El is F-contractible, we call U an OOVQP. 237
be a foliation on a paracompact Hausdorff manifold. Lema 4.1 which is F-simple. Every open cover U of M admits a refinement U' = The tangent bundle T(M) splits into the Whitney sum 1(M) = F (resp. v") on the vector bundles We take connection F 1(F), V F (resp. V) and we define a connection V on 1(M) by V = vF • Vs". We call a Proof
curve 1(t) in N v—geodesic if it satisfies vdY,dt(dY/dt) = o. VF_geodesic on a leaf is necessarily v-geodesic on II and hence v-geodesic tangent to a leaf Is contained In the leaf. One can assume that every U is a neighbourhood of local foliation chart x •:U and that, for each m N, •(m) (0,0) with some a. We take a small q—disk c (0) x contained in U, and then take a sufficiently small normal open p-disk bundle E on 0q consisting of vectors tangent to leaves such that the image Exp(E) of E by the exponential map is contained in U
U
Let By
Ii' = (U.) be an open cover by
Exp(E)
of H and Q =
n ... n
the property of v-geodesic stated in the above, a connected component of
the intersection of a leaf and Q is v-geodesically convex (cf. (3, p. 34)). One can assume that Uj
Exp(E) c U.
+
be the natural projection. Obviously, it(Q) = B is an of Since each fibre of ir:Q B is contractible, one can conopen struct a cross—section s:B Q of and by Cm-approximation argument, one Let
x
0q•
can assume that $
is
a Cm-map.
is a q-dimensional submanifold transverse to F. The conto the point of N along v-geodesic with respect to its parameter gives a C'° F-contraction to N. Q Since (U' ,. .. ,U'} is an arbitrary finite set of U' with non-empty is F-simple, intersection'and is F-contractible, U' = n ... n N = s,r(Q) c
Q
traction of each fibre of
is obviously a refinement of U from its construction. Let C denote the sheaf of germs of real valued C"-function on N, constant cohomology denote the s-dimensional along leaves of F, and let vector space of M with coefficient C. We have the following de Rham type and U'
isomorphism which is a special case of [10, Theorem 3.2) and is proved here
briefly by Theorem 4.2 238
of Cl). There is an isomorphism
C).
be the sheaf of germs of differential (0, s)—form. For open
Let
Proof
cover U of M, we have a k-dimensional tech cochain vector space
of U with coefficient 4°' Kk, S(u) Kr(U)
40* S)
and we set
=
=
5(U). k+s=r
Let LKk, 5(U)
K1' 5(u) denote the coboundary operator of cochain. On the other hand, dF: A°' A°' s41 defines another operator Kk, 5(u) + Kk, S+l(U) such that 0 and We set = DI = S(u)
0' = (_1)kdF
D' + D'. One can easily see that U: Kr(U) 9. Kr+l(U) is a coboundary operator, that is, u
=0. maps
0* CM) =
p
a:A '
-
E
B:
K
5=0
s=O •
P0 '
AOs '
cE K (U) r
K(U).
z k
k
are defined by the natural Inclusion maps. By making use of Lenina 4.1 and by the parallel argument of (1, pp. 16-21], we obtain isomorphisms CM) 2 R*(K(u),
8*:H*(C(U; n;)) 2H*(K(U)'
and then by taking limit of H*(C(U;C)) for Ii, (8*Yia* defines the isomorphism
2
C).
239
MODULAR COHOMOLOGY CLASS AND X(y1)
5..
Let (NJ) be a codimension q foliation, 0 a positive C°'-density along leaves a positive Cm-density on N. For a local foliation chart I p, j q, we set
of F and
1
1
=
=
Since we have =
Is constant along leaves, it follows that
and
= dF(log(1JB/DB))
on U3fl 1J8.
Therefore
defines a global 1-form on H which Is obviously dF-closed. Therefore, we obtain E On the other hand, we set
C on U ciy By a U B n Uy , the tech cochain ctB CF) is a cocycle and, by taking limit for Li, Its cohomology class defines an element c which Is called the modular claee of F and is denoted by m(M, F) (cf. [9, p. 9)).
{C8) E
Letmia 5.1
Let
C) be the isomorphism of Theorem 4.2.
Then we have
= m(N, F). Proof
defines an element of I(0(u) and we have
{dF(log(pa/Da))} =
/0 ))} -
= _d({log(i.Ia/Da)}).
240
- dF({log(/Du)})
By the definition of tech coboundary operator, it follows that -
=
=
-
-
log(1IB/DB)
= —C
ctB
and hence
•({dflog(1/D)}] = ({CB}]. s-simplex such that the image of is contained In a leaf of F. Let C denote the vector space overR with the basis (a }. Then we have obviously for the boundary operator a, and obtain a be a
Let
chain complex a
(Cs, a):
a
F
E
a
F
cs_I —3 ...
Cs
Suppose that
JaaF U)
0,
a) and (A0'
c A0'
and
e
dF).
then one obtains
JGF
it follows that
From the usual Stokes
Proof
F C0
Rm,
One can show a Stokes type formula for Lemma 5.2
a
-
I
+
10F
But We have d1 wE A2'
s-i,
+
d2w E A1'
S
and hence
=0
for
r(F) j
1,...,i+1.
Therefore we get
241
JF
=
=0
JaF
and the conclusion is shown. Let
0q
be an open c-ball around the origin for a sufficiently small > 0,
number
any differentiable map
the standard s-simplex and
is called differentiable, A cochain E is differentiable with respect to x. These cochains make a
such if
R) satisfies the Mayersequence property for finite open covers of N. One can define a and its cohomology
cochain complex Vietoris
homomorphism
5,dF) +
by
= J F
as
Lemia 5.2 shows that =
=
that Is, A Is a cbchain map. We have a natural isomorphism from Theorem 5.3
to
R) as follows.
If F is a foliation on a compact Hausdorff manifold N, then A.
induces an Isomorphism
*
H°'
FDR
'
H5 (M FD
Since the manifold M is compact, by Lema 4.1, one can find a finite F-simple cover U of M by open sets. In exactly the same way as for the
Proof
differentiable singular cochaln complex, for F-contractible set E we have = 0 for s > 0 and the natural isomorphism By making use of Mayer-Vietoris exact sequences of and and by analogous arguments in the proof (6, Appendix Theorem 3.1] of the isomorphism H*(M, H* (N) one can see that the natural cochain map Induces the Isomorphism R). Theorem 5.4 •we have 242
Let (N, F) be a foliation on a compact Hausdorff manifold.
Then
i]),
x(y1) =
= —m(M,F).
N be a Proof The first equation is obvious by Theorem 3.3. Let c:(Q,1] closed piecewise on a leaf of F. By (9, Lenina 2.2 and Section 33,
one obtains A(2ir(h1)0
1)(c)
2nh1 (c)
=This means by Theorem 5.3, that (2tr(h1)0
=
E
Lenina 5.1 shows the conclusion. References
[1]
Bott, R. Lectures on Characteristic Classes and Foliations, Lecture Notes in Math. 279, Springer—Verlag, Berlin (1972) 1-94. (23 Connes, A. Sur l.a rheorie Non de i'Intégration, Lecture (33
(4]
Notes In Math. 725, Springer-Verlag, Berlin (1979) 19—143. Helgason, S. Differential Geometry and Syninetric Spaoe8, Academic Press, New York (1962). Heitsch, J. and Hurder, S. Secondary classes, Well operators and the
geometry of foliations, J. of Differential Geometrij (to appear). (5) (amber, F.W. and Tondeur, Ph. Foliated Bundles and Characteristic Classes, Lecture Notes in Math. 493, Sprtnger-Verlag, Berlin (1975). [6) Massey, W.S. Singular Homology Theory, Springer-Verlag, Berlin (1980). [7) Nilnor, J.W. and Stasheff, J.D. Characteristic Classes, Ann. of Math. Studies, Princeton Univ. Press, Princeton (1974). [8] Reinhart, 8.L. Harmonic integrals on foliated manifolds, Amer. M2th. 81 (1959) 529-536. [9] Suzuki, H. Modular cohomology class from the viewpoint of characteristic class, in Geometric Methods in Operator Algebras, Proceedings of 1983 U.S.—Japan Seminar, RIMS, Kyoto Univ., Pitrnan (to appear).
243
(10] Vatn, L.
feuilletdes,
VarlEtds
cUO7Z.
.T.
Pos,te, Dekker,
New York
22
(1971) 46-75. (113
Vatsaan,
(12)
Yamagami
I.
Cohomol.ogy
and
(1973). S.
Modular
Clans of Poliation and Takeaakt 'a
Duality, Rfl45-417 Kyoto Univ., Kyoto, (1982). Haruo Suzuki
of Mathematics Hokkaldo University Sapporo 060 Japan
244
I VAISMAN
Lagrangian foliations and characteristic classes This coninunication is a preliminary exposition concerning
order gen-
eralizations of the Maslov class within the framework of the theory of secondary characteristic classes. A full version and complete proofs are expected to appear elsewhere.
The Maslov class appeared as an obstruction to the transversal Ity of a Lagrangian submanifold to a fixed Lagranglan foliation (3], and in
(6] it has been remarked that it is the first of a certain series of secondary characteristic classes. Here, we consider all these classes (using the Chern-Simons-Bott approach) in the most general situation, and we discuss them as transversallty obstructions. Then, we compute the classes considered for a Lagrangian submanifold of a K*hler manifold endowed with a parallel Lagrangian foliation, and we show that they are represented by means of various traces of. the second fundamental form of the Lagrangian submanifold.
This generalizes a result of J.M. Morvan (8]. 1.
REMARKS ON LAGRANGIAN FOLIATIONS
Though this is not our main object, we start with a few remarks about Lagrangian foliations. A pair where V is a 2n-dimenslonal differentiable manifold (we work in the C°°-category), and n is a nondegenerate 2-form is an airaoat ayiwpl.otio manifold, and if = 0 it is a raanifol4. A submanifold H of V is Lagr'angian if dim N = n, and if induces on II the zero form. A (distribution) foliation L0 of V is Lagrangian if it consits of Lagrangian (planes) leaves, and we shall say that the pair (V,L0) is an (abnoat) Lagra— ngian manifold. The typical example of a Lagranglan manifold is given by any cotangent bundle with the foliation defined by Its fibres.
It is a basic fact that all the Lagrangian manifolds.are locally equivalent (10], and this follows from Theorem 1.1
(S. Lie).
every point x c V
Let a neighbourhood
be a Lagrangian manifold. Then, endowed with coordinates 245
= 1,...,n) such that 10 Is given by ,? = const., and
(xU.yU)
dXa
a
The local coordinates of this theorem yield an atlas with transition functions of the local form 1(xB),
=
=
(xB)y'(
z
+
(1.1)
Hence, if Spl(n,R) is the group of the symplectic 2n-matrlces of the form
0) )fl (tAc = Id.,
(A
tAB
tBA)
(1.2),.:
nfl we have
ProposItion 1.2 An almost Lagrangian manifold is a manifold endowed with and the 'manifold 1s Lagrangian 1ff the
is Integrable. This remark allows for the utilization of the theory of G-structures in the study of Lagrangian manifolds. On the other hand, the global equivalence of Lagrangian manifolds is a
difficult open problem, and we should like to indicate amethod of obtaining global Irivariants. In view of (1.1), it makes sense to define, on the Lagrangian manifold + b(xB), eheaf S of germs of the functions f = (V.10), the and it is clear that the cohomology spaces H1(V,S) will be global Lagrangian Invariants. Hopefully, these invariants could be computed as follows. Let 0 be the sheaf of the functions V + R that are constant on the leaves of L0, and let be the sheaf of germs of the projectable cross-sections of the transversal bundle of Then, there is an inclusion i:o S, and an epimorphism a:S
q given by
a where
sg h denotes the "c'—gradient" of the function h, and it is easy to
prove
Proposition 1.3 The sequence 246
a
i
a
(1.4)
is an exact sequence of sheaves. theory yields computation methods for H*(V,o) and Since general
(9), the exact sequence (1.4) might provide the computation of H*(V,s). 2.
SECONDARY CHARACTERISTIC CLASSES
Now, before defining Maslov classes, we need an adequate sketch of the ChernSimons—Bott theory of secondary characteristic classes [2], (1]. Let G be a Lie group, let g be Its Lie algebra, and let 1(G) =
be its Well algebra of the multilinear, syninetric, adg—invarlant functions (or polynomials) g P (5]. Furthermore, let iy:P M be a G-principal bundle, and 0,9',... be connection forms on P with the curvature forms 0,0' In the sequel, we shall sometimes Identify the projectable forms on P with the corresponding forms on M.
Following (1], one takes a connection 0 = the theh, where (th) standard r-symplex, with the curvature on P x and one defines +Kx 1k
(G)
+A2kr(M)
r
0
(A denotes the exterior forms functor) by JAr
(f e
(2.1)
.
This yields
r
0" r
h=O
0" h—i h+1
r
is the Chern—Weil homomorphiem, andthe forms in im A,, represent
Then 4,, 0
0
the principai charaOteri8tic ciaeaee of P. The latter do not depend on theC choice of the connection since (2.2) yields dA
9001
f=A f-A f.
(2.3)
For further necessities let us also note the formula
247
,
I
where
(1—t)e0
=
°t
)dt,
+ to1, and
(2.4)
dt).
=
On the other hand, there are the transgression forms on P [2] (2.5)
19f = J
+ {te,te] (the bracket is in g), and
where Ô = d(to)
= ö
(mod. dt).
These
forms satisfy the basic relation d(Tef) =
(2.6)
Accordingly, we have the following definitions. If f c ker , T f is closed, and (T f] E H - (P,R) are the Chern—Simona 01138898 of (P,o). If A ker f is closed, and f) E H are the f ker , o
aecoridary
001
1
0091
characteristic cla8ses
of
(P,e0,81).
= (sf0 s 1), let • be a connection on We shall say that + is a Mx I. and cP/p = Analytically, one has + ads for some of and a iink = function a:P x I + g, and its curvature is
Furthermore, let be I
Px I
• where
- aSS)
(da + (95,a]
+
=
=
whence for f c
+
+ kf
=
Now, if we denote i5:P =
for forms of P x
(2.7)
A ds,
P
(dcx
+ [05,ci] -
'
I
(2.9)
where h is "fibre Integration" on P x I, and applied to view 0f (2.8) 248
(2.8;
x {s) c P x I, it is well known that one has
hd + dh4
-
A ds.
=
k
Jf
this gives, in
- dcx - [95,a], o(k1)))ds + exact form.(2.10)
Particularly, by taking
°o + s(o1 —es) we obtain
The following relation always holds between Chern-Simons and secondary characteristic classes: Theorem 2.1
8001
f] = [1 f) - [1 f).
(2.11)
00
01
Similarly, if
are pairs of connections and
=
+
(x = 0,1), then (2.8) and (2.9) (on N) yield
ctAds are links of
f =k J f
-
-
-
-kf f
-
(2.12)
eO(k_l))ds + exact form.
-
Now, following D. Lehmann [7], we shall say that the connections 00,01 of P are f-homotopic if there is a link of such that f ker
If this happens, the integrals in (2.10) and (2.12) vanish, and we get Theorem 2.2
If
are f—homotopic connections, respect-
and
ively, then [T f] = [1 ,f], and
f] =
00
,f].
,
e0o1
This theorem clarifies the dependency of Chern-Simons and secondary
characteristic classes on the choice of the connections.
Formulas (2.10) and (2.12) yield easily the following generalizations of the Chern-Simons and Heitsch derivation formulas (2), [4) Remark
a(T0 f)
= kf
- dct5 - [05,a5],
k -1
+ exact form,
(2.13)
1
=k
- dct5 — [05,ct5J,
-
)
—
(2.14)
0
where
=
=
f
-
-
00(k_1))} + exact form,
(The original formulas were for a
0.)
249
3.
GENERALIZED MASLOV CLASSES
rank E = Zn) be a eynrpl4otio ueotop bundi. with Its structure defined by a nondegenerate cross section ci of A2E*. Then, a fibre Let 1T:E + M (dim 14 - rn
basis (ei,...,e2n) is aynrpiectic If it assigns to ci the canonical expression, and these bases yield the Sp(n)-principal bundle the symplectic group).
P1
denotes
It is classical that E admits U(n)-reductions (U(n) is the unitary group), of the structure group defined by fibre complex structure operators J, and any two such structures are homotopically related by a family We shall choose one such reduction, and denote by ir:U3(E) + N the U(n)—principal sub—bundle of S(E) given by the unitary bases Jei,...,Jen) or, in the equivalent complex form, by the bases
— /1
=
(i
(3.1)
1,...,n).
The characteristic classes which we have in mind are then related to the Chern polynomials Ck C I(U(n)) defined for A E u(n) = the unitary Lie algebra by [5] -1
c (A) =
First
(LOCk]
(
\k
tr
AkA.
(3.2)
of all, using a connection o on U (E) we obtain the H
oiaeeee
(M,R), anti a simple homotopy argument shows that they depend
only on the symplectic structure of E (i.e., they do not depend on the choice
of J). of E. Then we can further reduce the structure group of E to the orthogonal group 0(n), and get the 0(n)—principal sub-bundle II of defined by the unitary frames (3.1) such that e1 L0 (1 = 1,...,n). Then, it is classical that c,k_f ker for every 0(n)—connection and, therefore, Furthermore,
assume that we also have a Lagrangian
aub-.bundl.e
00
we obtain Chern-Simons classes
ph (E,L0)
c2 (1 e0h-
J
(E), R).
(3.3)
will be called the bundle ?tzelov cZ.aeeea of (E,L0). Since any two 0(n)-connections are C2h_1 homotopic (7), it foUows from Theorem 2.2 that these classes do not depend on the choice of 00. The classes
250
If 00 is represented by the local equations (that UBO the Einstein eumrdatlon convention)
(3.4) with respect to bases (3.1) in is defined by connection form.on
global
then the
(35)
,
is the inverse matrix of
U(n), and
where
from (2.5) and (3.2) that
I0
, =
are represented by i—c
"
I1r
"
J
(2h—2)1 (2w)
A (ö
It follows
2h-1
I, 0.' A ($,,),' A ...
(3.6)
OL
)12h_l] dt,
are computed as shown for (2.5), and using (3.5). Partlculai-ly, we get
where
18c1
d In
(3.7)
is the lift to L13(E) of -(I/2)m(L), where
and it foflows easily that m(L) is
usual Maslov class on the bundle L(E) of the Lagrangian subspaces
of the fibres of E (3). be one more Lagrangian sub-bundle of E, and let 01 be an 0(n)-connection on defined by the new reduction of the structure group to 0(n) given by L1. Then, we clearly get secondary characteristic classes Now,
=
c2h1]
(3.8)
and these will be called the (generalised) Z&zeiov L1 with respect to 10. Using again the c2hl-homotopy argument, it fc'lows that and, also) the do not depend on the choice of the 0(n)-connections to prove that homotopy of any two adapted complex structures J uh(E,Lo,Ll) do not depend on the choice of a. In order to compute the Maslov classes, we represei.. again 00 by (3.4), •
251
and we represent 01 by similar equations = that we have the similar quantities associated to
let us take some fixed unitary bases (ci) in relations of the form .
=
where primes denote
Instead of L0. Then, We shall have transition
èj.
=
(3.9)
and new connection forms
j
,j
jth
h +
(3.10)
, +
where the matrices 3, 3' are Inverse to y, y', respectively. From (3.10), we can further compute the curvature write down
needed in (2.4),and, accordingly,
(1)h+1
i
(2h—2)I
,11
1)
A
JO
(3.1.1) —
I2
A (oIL A ... A (e'). —
"2
dt,
representatives of the Maslov classes [3), (6). and in view of (3.7), we get Particularly, by taking Cj = = (1/2)m(10,L1) where m(L0,L1) is the usual class of with respect to
thereby
Now, we can obtain some basic properties of the Maslov classes defined above.
L1 are everywhere transversal If the Lagrangian sub-bundles = 0. then all Indeed, in this case we may choose such that = JL0, and we may Je1 (I = 1,...,n), and choose bases such that (see formulas =
Theorem 3.1
(3.1) and (3.9) for notation). The forms of (3.10) will then be related by Consequently, the first factor in (3.11) = = 4, vanishes, and we get the conclusion. Remark
sality of L0, 252
are obstructions to the transverbut it is clear that the conclusion also holds if we assume
Theorem 3.1 shows that
only that
11 can be deformed via Lagrangian sub—bundles tO transversal
bundles L6, Theorem 3.2
ons always hold
For Maslov classes, the fc1lowlnq
(a)
)
(b)
=
(c)
+
)
=
-
I ,L0) ,
+ lih(E,L2,bO)
0.
Indeed, (a) follows from (2.11); (b) follows from either (2.1) or (2.4), and (c) follows from (2.2) or, more precisely, from C2h_1)
=
C2h_1
-
C2h_1 +
= 0 if. L0,'L1 admit a Property (c) above shows that (global) conwion transversal Lagrangian sub-bundle L2. 'Hence, these classes
Remark
are obstructions to the existence of the latter. 4.
MASLOV CLASSES AND THE SECOND FUNDAMENTAL FORM
As seen in the introduction, an important transversallty problem Is that of the transversality between a Lagrangian submanifold maniof a and a L.agrangian foliation L0 of the latter. In this case, fold is a symplectic vector bundle E -* M, 10 = 10'M' L = TM are Lagrangian sub— bundles of E, and we are interested in the transversality of these two sub— bundles.
From Section 3, we know that the ?4aBlov
olassee
lJh(E,Lo,L) provide obstructions to the transversality of M and 10. Generally, the computation%of these classes is difficult, but we can compute them in a particular case where the results are both nice and Namefl', we shall assume that V admits important since It includes V = a compatible Kähler structure (J,g) such that L0 is parallel with respect to the metric g. One can prove that g is then, necessarily, a flat Kahler = metric. Clearly, with the "horizontal8' n-dimensional distribution
10 is of this type, and also, if N is any locally flat Riemannian manifold, the cotangent bundle V = T*N has a natural flat structure (J,g) such that the fibres of T*N are g-parallel. Now, let H be a Lagrangian submanifold of the manifold (V,L0) considered 253
(1 1,...,n) needed for (3.9), (3.10), take the bases above, and let of (3.9) to be etc. to be orthonormal tangent bases of II; then take
-/T
=
Since
(4.1)
is parallel, the
induces a connection of usable in the computation some local equations
connection V
in L0, which extends to a of the Maslov classes, and which
irs
t
have
(4.2) =
(hence, in this case, we do not need the bases
of (3.9) for the comput-
On the other hand, it is also clear that we may take the connection needed In the computation of the Maslov classes to be defined by the connection induced by v on 14. The latter is determined by' the Gauss equations of N, which can be written as =
(4.3)
+
is a normal basis of N. In (4.3), it' is the matrix of the Is a matrix of 1-forms which defines the Induced connection, and has the local equations of N. AccordIngly,
since
£j.
Dc1 a
(4.4)
Now, we obtain from (4.1), (4.2) and (4.3) that
- ,q
-
(4.5)
,
and furtheraoi'e, the curvature needed in (3.11) can be compyted from the Gauss—Codazzi Integrability conditions of (4.3) together with the fact that V has zero curvature.
After this computation, we shall get from (3.11) that the representative forms of the Maslov classes of N and o 01
C1
are given by
(4.6)
which can be seen to be equivalent to the interpretation of J.M. Norvan [8], and 254
1
—
-
0001
54
(2rr) k2
1•••j 2h1
"1
(4.7)
k2h_l
A
...AB.
1%
8i,
are constants given by
where
2h-2 h
=
1=0
(-1) h+i+1 (2h—2)!
2
i
4h-i-3
(2h_2)
(4.8)
1
In other words, the Maslov classes uh(M.Lo) are given by various traces of the second fundamental form, and we have Let V be a KShler manifold endowed with a parallel Lagrangian foliation L0, and let N be a Lagrangian submanifold of V. Then the Maslov classes ph(M,LQ) depend only on the second fundamental form of N in V, and Theorem 4.1
they vinish if N is a totally geodesic submanifold of V (and, moreover, = 0 if M is a minimal submanifold). We may expect to be able to use a similar method of computation for any cotangent bundle V = T*N of a Riemannian manifold N, by replacing V with an adequate metric almost complex connection and by replacing with a second fundamental form. The results (except for wifl be more complicated since they will involve the (non—vanishing) curvatue of References
[1] Bott, R. Lectures on Characteristic Classes and Foliations.
Lect.
Notes In Math. 279, Springer-Verlag, New York (1972) 1-94. [2] Chern, S.S. and Simons, 3. Characteristic forms and geometric invariants, Ann. of M2th. 99 (1974) 48-69. [3] Guillemin, V. and Sternberg, S. Geometric Aeynrptotios, Surveys American Math. Soc. 14, Providence, R.I. (1977). (4] Ifeitsch, J.L. Deformations of secondary characteristic classes, Topology 12 (1973) 381-388. [5] Kobsyashi, S. and Nomizu, K. Foundations of Differential Geometry, Vol. II, Intersci. Publ., New York (1969).
(6] Kamber, F.W. and Tondeur, Ph. Foliated Bundles and Characteristic Classes, Lect. Notes in Math. 493, Springer Verlag, New York (1975). 255
[7) Léhmann, 0.
J—homotopie dans les espaces de connexions et classes
exotiques de Chern-Simons, Conrptea Renthsee de l'Aaad. dee
Sci.,
Paris
275 (1972) A, 835-838.
[8] Morvan, J.M. Ann. Inst. H.
Quelques invariants topologiques en Poincaré 38 (1983) 349—370.
symplectique,
M. Dekker, New York Vaisman, I. Cohcn'ology md Differential (1973). (10) Weinstein, A. Symplectic manifolds and their Lagrangian submanifolds,
[9]
Advances in Math.
8 (1971) 329-346.
Izu Vaisman Department of Mathematics
University of Haifa Israel
256
E VOGT
Examples of circle foliations on open 3-manifolds INTRODUCTION
In [3] D.B.A. Epstein showed that every Cr_foliation F (1 see [7]) of a compact 3-manifold M by circles is
r
for r = 0
to a Seifert fibration on M, i.e. to a foliation which near each leaf is given by
the orbits of a locally free circle action.
For non-compact M the situation is more complicated. In [9, pp. 113-115], G. Reeb produces a F x S1 x S1 with all of codimension 2 on an open subset H of com-
pact such that B1(F) = {x M: x is not locally bounded in x} is not empty. (O,co) is the function assigning to x H the volume of the leaf through x with respect to some Riemannian metric. Reeb assumes n 4, but his formulae also work for n = 3. A slight variant of Reeb's example is the real analytic example of D.S.A. Epstein in [3). Here
B1(F) is the "obstruction" for F being a Seifert fibration, i.e. a foliation F by circles a Seifert fibration iff B1(F) = 0. (A completely analogous result is true for higher dimensional foliations. See [4].) B1(F) is the first set in the (coarse) Epstein hierarchy of bad sets of a foliation with all leaves compact. every ordinal a >
One defines by transfinite induction for
1
Ba =B(F)= a
fl 8
,
if a is a limit ordinal.
:AIB1 is not locally bounded in xl,
if ct-i exists. IEFI = sup
01 is a countable ordinal called the length of the
Epstein hierarchy EF = {B1(F) D B2(F)
. . .1 .
measures how complicated
F is: B a is the obstruction to FIB being a Seifert fibration. a-i This paper is one in a series of three. Of the other two,one is mainly ,expository and is concerned with the structure of the bad sets B. In the
0 if N supports a circle foliation with Epstein hierarchy of finite length. last paper we show that
Our interest in the study of circle foliations on open 3—manifolds stems 257
from the following question raised by D.B.A. Epstein in [5): Can P3 be foljated.by circles? The result above shows that such a foliation must have Infinite Epstein hierarchy. The purpose of our examples is to show the following: (1)
Many interesting spaces which do not admit Seifert fibrations admit circle fibrations, e.g. complements of some wild knots in S3.
(2)
B1(F) may have rather unpleasant topological properties even If B2(F) = 0.
(3)
with Epstein hierarchy of infinite length There are circle (this contradicts the remark on p. 28 of [2) claiming that B2(F) = 0 for any foliation with all leaves compact of codimension 2).
(4)
Describe a possible approach to put a circle foliation on P3. This approach necessitates a countable number of extensions of certain
circle foliations from a solid torus V to a larger solid torus In which V is trivially embedded. We show that the first two extensions can be made. The result is a (very complicated) circle foliation F on an open solid torus with EFI = 2. 2.
VARIANTS OF REEB'S EXAMPLE
Let E' be St x S' x [-1,1) with coordinates (x ,x
(x ,x ) ER2,
Letf:[-1,1)-sRbeaC-map, 1
r
w, such that f1(O) = {O}.
We consider on E' the Pfaffian system
{dt, dx1 + f(t)dcp}. o(f) is non—singular on E = E'
1, t = O}
Therefore ci(f) defines a Cr_foliation F(f) on E. (For f(tj = t, t fO,1), we obtain Epstein's example in [3).) Here is a more geometric description of F(f). Consider the level E5 = n (S1 x S1 x (5)). is saturated, and for s 0 E5 Is a 2Because of dt = 0 on F(f) each torus foliated by the graphs of the maps + c from and completely Integrable.
For s (and therefore f(s)) close to 0, a leaf on (for f(s) > 0) is shown in Figure 1. On E0 the leaves are of the form to P mod 1.
258
xl
Figure I
(x11x2) x S' W(O} for -1 is B1(F(f))1 and 82(F(f))
< I (Figure 2). The union of these 'eaves
<
0.
Eo
Figure 2
259
By piecing together various parts of some F(f) with Seifert fibre spaces one obtains already some interesting circle foliations. We will describe infinite) products of torus knots. of f'Iiations on r. of two or h no s not hard this the total spaces of our examples admits a will not carry any Seifert fibre space structure. Let p,q be relatively prime integers. F(p1q) denotes the foliation of = 1) given by the orbits of the restriction + S3 = {(z1,z2) E C2: t ER z1, z the circle action t ((z ,z ) + to s3 mod 1, on C . 'The standard torus knot kp,q is the orbit through (11,17, induces a Seifert fibration on the complement 11,17) of this action. F in S3. By taking of any invariant neighbourhood of q cR3, with 1/,'7) to be in of S3 we may obtain an q hand Kpq contained'in {1x11 <1). the knot being the x1-axis mR3 {1x11 •
aKpq
x1 • —1
Z'
leaf in ôKp,q
FIgure 3
x1
1
1) whose top Let V be a small tubular cylinder around kpq A {1x1! I around' and bottom are two circular disks of radius r in the planes the Intersections of these planes with the x-axis (see Figure 3). Let Z be the union of Z' with the interior of these disks and consider the 3-manifold 260
with boundary D = {1x11
1)
(Kp,q u Z).
aD consists of two components: aKp,q which is a solid torus with a foliation , and an annulus. Let E c E be the union of levels induced from F p,q + 1, and remove the cylinder E0 fl {x2 < 0} from E . Denote the 0 s resulting space by E*. There exists a tomeomorphism h:E* D mapping E1 leaf preserving onto 3K0 q and having the following property: remenl,er that each circle (x11x2) x S1x {0), 1x11 < 1, x2 > 0 is a leaf of F(f) in E*. Then h - each one of these leaves to a concentric circle around maps for -1 < x these leaves (—1 ,0,0) In the plane x1 = -1 in R , and it maps for x1 < to concentric circles around (1,0,0) in the plane x1 = +1. The remaining I
of E0 are mapped onto the annulus leaves (x11x2) x S1 {0}, - < x1 < 3Z'. (To see that such a homeomorphism exists, note that D is Just a thickened rtorus with an annulus in one of Its boundary tori removed and that In any leaf of F together with a concentric circle around (1,0,0) in {x1 — 1) of than r represents free abel Ian generators for that one can choose f and h in such a way that the H1(D).) It is
foliations Fp,q and h(F(f)) fit together across 3Kp,q to form a C°' circle 1}-Z. and that F(p,q,f) has a smooth foliation F(p,q,f) on Np,q = {1x11 extension to X1
U
a
2
, x22 +
x3 a
r2
r around where the new leaves are simply concentric circles of radius the x1-axis In the planes {x1 = c} with a 1. Now let (p1,q1), (p2,q2),... be any finite or countably infinite sequence of pairs of coprime integers. We can put a C°'-circle foliation on the comin the plement K of the product of the torus knots k p1,q1 k p2,q2 following way: K is diffeomorphic to
N ON
UgN
U
g2 N
p3,q3
U
-1, ar}. and M_ • f(x1,x2,x3):x1 Foliate g1 Mpj,qj by g1(F(p1,q1,f)) and foliate 14_ by concentric circles around the x-axls. If the sequence (p ,q ),... ends with (p ,q ), then also foliate {x1 a 2n—11 x2 + x3 a r} by concentric circles around the
where
,x2,x3)
(x1 +
2,x2,x3)
261
•
The second example in this section is intended to give some indication of the possible unpleasant topological properties of the set B1(F). Let K be a continuum in the closed unit disk D2 of R2 K is supposed to have the following properties: (1) (2)
aD24( is the disjoint union of n > 1 open arcs D2-.K is the union of n+1 disjoint simply connected domains E Fr K U A., 1 = 1,...,n. (Fr denotes the 0 1
1
set theoretic boundary with respect to the topology of?2 .)
Such a Continuum can be constructed and it has rather peculiar topological properties [8], §62, VI, Theorem 11, and §48]. We want to describe an example of a F1, on D2 x S', such that B,(F1,) = 02 (K n 0 ) x S and restrjcted\to B1 is the product foliation. Let t. be Caratheodory's primp end compactificatlon of E ([1], 16)). By(6], Theorem 6.6, there exists a homeomorphism + D such that + 0 is a entheds A1 Onto the open sotLjhe n hemisphere D! of the unit (note that A1 is canonically a sub1
space of E'). i = 1,2,...,n. On FKIEI x St will be the pullback
we letFk be th. product foliation.x x Id of a foliation .Ff On
under
which we will now describe.
Let E* c be the submanifold of the total space £ of the F(f) described above and let c be the circle of radius
origin.
Then
[(021 D2)u
around the .E* under the map
Is
x
+ (Cos 2ir'I', sin 2irV, •-'v , 2-2r) where r [0,1], YE? mod 1, are polar coordinates on D2, • ER mod 1. The coordinates on E* are the ones F:
from abQve.
If g1(s) is the distance of the circle = sfl.c from let f1: [0,1) + (O,co) be a
K with respect to the standard metric of map the following properties: (3)
f1(O) =.O;
(4)
f1 (2—s)
6)
exp(-g2(s)) for
<
f1 is equal to some constant c1
to be F*(F(f1)).
We take
of circles Cs
262
in a neighbourhood of 1. Q2) x
Is the union
--
-
cos 2
+ s)
D2) x S1:0
l}.
They are the fibres of a trivial
D2) which can be smoothly
S1-bundle over
x S'. The choice to define F1 on all of x ld)*F1 define circle foliations on E1xS1 off1 wlll.guarantee that the together with the product foliation on (K U E0) x S1 to which fit define a circle foliation FK. Obviously FKI(D2'..K) x S' is a (trivial)
extended to an S1—bundle over
For the converse inclusion note that a leaf on ({r = s}) x S' has at least length exp(g (s)) if we give x S = D x R mod I the obvious product metric. D
bundle.
3.
Therefore B (F )
K x
ITERATIONS OF REEB'S EXAMPLE
(an example with infinite hierarchy)
In this section we will prove: Proposition For any 0 a u there exists'a C°' circle foliation Fa on a connected open 3-manifold such that IEFaI = Proof
We already know that the statement is true for a = 0,1.
To construct
a foliation of length 2, we first modify our standard example a little. The manifolds E, E', E*, E5, -1 s 1, are defined as in the previous section. We also use the same coordinates for E. This time we want f: [-1,1] + [0,1] to be a C°°—map with = (-1,0]. The forniilae of the previous section define again a foliation F(f) on E such that each of the sets E5, E', E* is saturated. Notice that every leaf of F(f) in = U
tE(O,1] (x1,x2)
Et intersects the annulus A =
x
x (0,1]
for any
at exactly one point. To fix notation we take (x11x2) = (0,1) E with to define A. Let B cA be the open disk of points S
1
be a homeomorphism which is a diffeocorners (see Figure 4). Let h:g + morphism in the complement of the corners and which maps the upper half
circle 0
of the outside boundary of A, i.e. the set 1/2, to the lower half circle In the boundary of D2.
Inverse image of
E with
Let B' be the
under h.
For a subset C of E let F(C) be the union of leaves of F(f) through points x S1 onto of C. Then we define a diffeomorphism H from N = (D24 as follows. For Is the intersection of the leaf N, with the annulus
263
t= 1
Figure 4 (cos
in E.
sin
x S' x
Furthermore, let K:N = (cos
1)
E(01) be the diffeomorphism given by sin 2n'4',
4/3 - 4r/3)
are polar coordinates, 'v €P mod 1. where now (r,'i') U Ce). On We ndw define a foliation F(f(2)) on 112 =
u
we take the restriction of F(f), and on F(B-.B') we take (K o The two foliations fit together. To see this, note that H*(F(f)) is x S1, and K*(F(f)IE(Q,l)) is a foliathe product foliation on N = (D2'4,. tion on N which, by the choice of f, can be extended to a foliation on x S1 by taking the product foliation on S1. = R1, 142 is a Since F(f) Is an S -bundle in a neighbourhood of F(B' U 3—manifold (with boundary). The Epstein hierarchy of F(f(2)) Is given by r1(F(fl) U F(T), B2(F(f(2))) = B1(F(f)), where B1(F(f(2))) • = 0 or 1/2, 0 < t < 1) is a union of two arcs (see Figure 4). This concludes the construction of a circle foliation with Epstein hierby glueing archy of length 2. We note that we can plug the hole F(B') in in a trivially fibred solid torus. But H2(112 U F(B')) Z will remain non-
trivial. 264
process. We obtained To obtain a foliation of length 3 we repeat the and then F(f(2)) by first removing F(S) from our standard example F(f) on adding F(B'.B') with a new foliation which, up to the diffeomorphism K o
with its foliation F(f).. This foliation just another copy of E(Q 1] fits into the foliation on -.F(B) (with the length of the leaves growing to infinity when approaching F(aB)) except for the points on the two leaves of = {(O,i,$,i) E4:$'O or 1/2). through the points This is the reason why we also remove F(C ) to obtain 142• by H ° (1(M2 fl E(01)) So to continue, we will simply replace with its foliation (H e F(f(2)). We obtain a manifold (with boundary) N3 c N2 with a circle foliation F(f(3)). The Epstein hierarchy of F(f(3)) is given as is
B1
B2
81(F(f)) U
H
°
= B1(F(f)) U H e
(1(B1(F(f))),
B3 = B1(F(f)).
Continuing in this way, we obtain a sequence E = N1
N3 D
of 3—manifolds and a sequence of circle foliations F(f), F(f(2)), F(f(3)),..., wittrF(f(i))living on N1 and IEF(f(i)fl = 1. It remains to construct a circle foliation with infinite hierarchy. This is done by simply piecing all the N1, F(f(1)) together. Notice that the annulus Z = {(x1,x2,.,O) E: x2 >0) is contained in all N1, and F(f(i))IZ = F(f)jZ is the trivial S1-bundle with fibres L(x ,x ) = { (x ,x ,+,0):0 1 }, where x + x = 1, x >0. Let V be the closed upper half plane on with the points (2i - 1,0), 1 = 1,2,... removed. Then consider the manifold
y xS1
u
Here u denotes
union"
and each
is attached to V x S1 by identi-
fying Z cM1 with (21-1, 21+1) in V S1 via the diffeomorphism (2i + Our choice of f allows us to extend the foliations F(f(i)) to Yx by simply putting on V x S1 the product foliation. Denote the resulting foliation by F(f(w)). Obviously { (y1,O) E Y: y1 > 2i - 1) x c B1(F(f(w))). Therefore EF(f(w)) = w. I
Remark
1
In our example
= 0.
I do not know whether one can construct a 265
circle foliation F on a 3-manifold with B(F) ,' 0. Remark 2
Note that afl manifolds
Remark 3
Rank H2(M1) = I.
I
I
can be
tn
We can close off in each M1 one boundary compon-
ent by adding a trivially foliated solid torus to the single compact boundary component of M1_1-441. This shows that we may construct circle foliations 1—1. But with F1 on 3—manifolds W1 such that IEF1I = I and rank H2(W1) the above methods the rank of H2(W1) cannot be lowered any further. This is because at each stage we have to remove the annulus corresponding to F(T).
by circles. This In the next section we suggest a program to foliate program necessitates the construction of circle foliations F1 on open solid = 1, = 1,2 Up to now I can only complete this tori with 1
program up to 1 4.
2.
ONE-PARAMETER FAMILIES OF DIFFEOMORPHISMS OF THE OPEN DISK
In this section we construct a rather complicated circle foliation F with IEFI = 2 on an open solid torus. A motivation for this example Is the fact that it is the second storey in the construction of a building with infinitely many storeys which would, if completed, result in a circle foliation ofF3. can be written as the union of an ascending sequence V0 c V1 c V2 c is unknotted and contractible in V1f1 and such that Vç..V1_1 is a closed solid torus minus a closed annulus in its boundary. To be more explicit, let V0 and W1 be as in Figure 5.
of open solid tori such that
V0
Figure 5
is a 3-manifold with boundary A , where A Is a meridianal annulus of W the closed solid torus the closure being taken mR . V0 is an open 266
= A1. V0 Is contractible and unknotted in the n solid, torus such that solid torus V = V0 u W • In particular, V0 and V1 Ere unknotted in
Therefore there exists a homeomorphism h:P - R which Is a diffeomorsimple closed curves on av ) mapping V onto V . Since phism in V0 c Vi, we obtain an ascending sequence V0c V1 = h(V0)cV2 = h (V0)c = •u0 h1(V0) is homeomorphic to R and it is not hard to prove that
Rö.
The results of the third paper in
series show the following: if there
is a circle foliation F on
then lEFt u , and if B(F) = 0 there exists ... of open saturated sets, each U1 a com(F) such that torus), and I11(U1) -'
an ascending sequence U0 c ponent of some R3..8
U1
c
So the simplest possible circle foliation onR3 might be obtained in the following way: start with a circle foliation on V0, extend is the 0-map.
this to a circle foliation on V1, extend this to a circle foliation on V2 (after possibly some deformation of the foliation on V1). Continue this process ad Infinitum. Whether this works I do not know. Below we shall show that there exists a circle foliation with the required properties on V2. (The main difficulty comes from the fact that we are not allowed, as in the examples in the preceding sections, to drill holes to remove the polntt where the tangent spaces to the foliation do not also note that a circle foliation on V1 with the required property - i.e., such that each for 0 j < i is saturated — will have Epstein hierarchy of length at
least i.) To begin with the construction of the circle foliation on V2 we observe that we already have an example with the required properties on V1. For
this we take the foliation F(f), where f:[-i,i] -* [0,1] is a C°°-map with [—1,0], and restrict it to the invariant set E E: -1
0, x2
._i_}
/7
-
We attach to this space, along its boundary E1, a solid torus V with the product foliation in such a way that the leaves of F(f) in (they are the 51 x {0}) are homologous in E to meridians of V. sets {(x11x2)} x The resulting space is an open solid torus which we may identify with where V0 corresponds to {(x,,x,,q,t) E: —i < t < 0, x., > and W corresponds to E*
U
I'
V, where E* denotes now the union of E(0
1)
with
267
E: x2 > .—L}.
This example also suggests a general procedure
deform the given foliation on V11 in such a way that it extends to a foliation of a longitudinal U by identifying A1 then attach E*E V to annulus A1 in for passing from a foliation on V1_1 to one on
with the annulus {(x
The result will be an open
>
E:
/7
1
solid torus V, with a fo1iation meeting our requirements. The hard part is to find the deformation of the foliation on V11. We do this now for i = 2. Since it suffices to deform the foliation only near the boundary of V1, it suffices to consider the thickened—up 2-torus V1
£ E: —1 < t < 0, x2 >
V = E* U
which is diffeomorphic to
x
/7
x S1.
S1
It will be convenient to use angular coordinates e P mod I for the first two coordinates Cx ,x ) of E (i.e., x = cos 2irQ, x = sin 2,rO). For coordinates (r,'v,s), r ,1), y,s ER mod 1. ( ,1) x S x S we are polar coordinates for the annulus C = {(y1,y2): +y2 < 1}ofR. x S x S Our plan is the following. Transport F(f) Irom V1'V to via a diffeomorphism b to have better coordinates for V1.V. Then use a diffeomorphism d of
x S1
to deform the foliation b(F(f)) into a
x S1
circle foliation which has an extension to a longitudinal annulus In (1) xS1 x S1. Choose d to be the identity near 4) x 51 x S' so that the construction can be extended over V. We first describe b:V '..V (1,1) x
It will map each circle (0) x S x {t) "Identically" onto the circle {r(e,t), v(o,t)} x S . Thus it suffices to describe the maps r(O,t) and 'v(o,t). We 'iill do this first and we map the corresponding half-open disk for —1 < t 0. Then < 8 < in to the diffeomorphic set R >r , < < v'7sin2iTV (7.1) x S i such that r(e,O) = and = 8. The map on the /7 sin 2iiO set of points (e,t) E Si x (0,1) is more easily described with the 'help of 6. x S'-..R by a family t > 0, of disjoint simple We fill up •closéd curves such that the following holds: S1
x S1.
1
1
1
1
(1) 268
For each 0
P mod 1, the radius {(r,v):v
0) Intersects each Kt in
4+e
Figure 6
exactly one point (r(e,t),e) such that (e,t) (r(e,t),e) is a diffeoinorphlsm. (2)
x (O,t]
.
x
•.R,
For each fixed t, r(o,t) is constant on the intervals + c e and +c U - c it is (non—strictly) increas- c; for e I ng.
(3)
the set {(r(e,t),O): horizontal line in P. For each t
+c
— c) is a straight
0
(4)
the positively oriented unit tangent vector to Kt in (r(t,o),e). Such a family of circles exists, and it is is fixed. We define r(e,t) by (1) and ,(o,t) = e for 51 x (04). Notice that r(0,1) as t and r(e,t) This finishes the descrtption of b. 1
me di ffeomorphlsm d of (1,1) x S' parameter family of diffeomorpnisms,
x S1
once r(O,t)
(o,t) E 0 for
e
will be defined by a smooth 1x S
-.. t!.1) x
S
•
S
[0,1], 269
such that
• d5 for 0
and
tity
$
lithe Iden-
c6
d5 is
The map d corresponding to
S1.
then defined by d(r9q',s) To motivate the somewhat complicated formulae let us consider another way
Let cz:[O,l] + 10,1] be the restriction of a 7-
to describe Reeb examples.
periodic c°—functlon from R to [0,1] of the form shown in Figure 7. Let a
VI).;
V Figure 7
8:11,1)
[O,o') be a Cm-map such that 81(0)
as r with leaves (r,'p) x and
1. S
,
1
+ 6] for some small 6
Let F be the product foliation on [ ,1) x S x S .1) x S . Let d(r,v,s) . B(r),s).
Then the image d(F) of F under d is a circle foliation on ( ,1) x S x S The tori {r} x' S x S are saturated with respect to d(F), the leaves on {r} x S' x S1 become longer and longer as r goes to 1. d(F) can be extended
to the union of the two annuli {1} x S1 x where the foliation 1on these annuli is the obvious product S1-foliation. The resi4lting foliation will be smooth if 8 grows fast enough (any exponential growth suffices). We would like to do the same, but with the foliation b(F(f)) instead of F. It Is an Instructive exercise to show that with the simple minded 1-parameter families of diffeomorphisms d5(r,'p)
Cr, y + a(s)
•
8(r))
above, there will be no point $ .of • MighbowhOOd U e (I) * S victor field towards idilch
in the fact that, for points
x
S1
S1 such that, in is a vanfshing x
this lies
wIth 0 < the positive unit tangent 'vector to b(F) (after we have chosen an orientation E
S' x
as r approaches 1, while for < 'v < I the for b(F)) will converge to Is just the positive unit tangent vector field to F in . limit will be — the example above and the positive unit vectors of d(F) in (r,'r,s) converge to
If r+1 and if
Since in the same domain the positive unit vectors of dC-F) converge to-h, we find in the neighbourhood of any point in positive unit vectors V11V2 of dob(F(f)) with V1 arbitr-
arily close to
and V2 arbitrarily close to-a. This indicates that wehave to make a special effort to find a deformation which also forces the unit tangent vectors of b(F(f)) at points (r,'V,s) with and into the positive 'if direction as r approaches 1. Let be the restriction of a Z—periodic indicated in Figure 8. Let a(r) be a Cm-vector fie'd on
as
with
Figure 8 1,
a(r) = 0 for r near the vector field
and a(r) = r(1-r) for r
x
S'
consider
x S where (r,v) are the coordinates on and let e5 be the corresponding 1-parameter family of diffeomorphisms. Let u:[O,l] + [0,cxf) be the re'striction of a Z-periodic 0 and with u(O) s. U(s). = s for Then let & = h eCS) be our 1—parameter S family of diffeomorphisms of where
r
•
h(r,'y) z
+e
1—r
)
is equal to 0 for r near to r for r We claim properries. that (dob)(F(f)) has th More specifically we prove: Here
Proposition
For proper choices
the positive unit tangent vectors of
of F(f), in (r,'v,s) will cj!Wei-ge to -
271
If (r,v,s) + (1,,0,s0) with
< s0 <
the differential of 4, on the We first investigate the effect of x St x S' for the points (r,'i',s) with in vector fields Proof
I <$<
and r
and
d(r,'V,s) =
r
Notice that in this domain a(r) = r(1-r) and a(s) = s, Therefore -1)) r s.g('P) ,
+e
-r—e
Differentiating, we obtain
r d
(
-
e
+
—
d
r
r
+
d
(
a
-
r a
fairly explicit description of b(F(f)).
a
is the product flow, and we will choose the orientation of F(f) so that the positive unit tangent vector of b(F(f)) in R x S' is 1.. Furthermore, the 2-tori ç x S c ((2,1) x S sR) x S correspond to the saturated 2-tori Et in 0 < t s 1, and for any > 0 the positive unit tangent vector of b(F(f)) on points (r,'y) with < < will be close to for small t, while on points with < I it will be close to +ô< so It will have the form a
X(r,'p,s) = A(r,'P,s) 272
+ B(r,'(',s)
+
On R x
it
with C(r,y,s) close to +1 or -1 depending on whether 6 < < - or is bounded, the only Since in this region also 'V < + 6 if A(r,'V,s) as r ÷ problem for convergence of to and This hap2ens only for is negative and r is close to 1. + here for the first time we have to specify the x S' x S1. If (r('V,t),v) is the positive unit tangent vector to in is somembat more than the then the of 'V
I
—
I
-fold of the component. For r(V,t) fluctuates between 'V and r1(t) with r (t) + 1 as t + 0. We choose b in such a way that r 1-r1(t) e1 •f(t) small t. (This obviously puts a restraint on f. If we choose f(t) = for small t, the above inequality can be satisfied.) Then the negative contribution to coming from will be neg—. ligible when compared with the contribution to — coming for points (r,v,s) with 'V close to r close to 1, and as always So it remains to analyze for close to 0 and 'p is close to 0, A(r,'p,s) = 0. Since B(r,y,s) > 0 everywhere and g'('v) > k for some k > 0, If 'p is close to 0, as long as 'p stays in + some neighbourhood Qf 0, S and r 1. Finally,. for 'p cone into c, + c) the shapes of the curves play. X(r('p,t),'p,s) will be of the form (r('v,t),q') where N is chosen to make sure that X is a unit vector, and + a('V,t) is a function close to 1. It takes care of. the stretching of the curve x t under the map b. By property (3) of the curves c) x - c, K , the .1. component of ak(r('p,t),'p) will be the (1—r(r,t))-fold of the component, as long as Therefore the contribution from 'p + •
(r(v,t),q')) to
will be
r
1
1
( (1
If £
is
+ (1—r)
2)
.(i
+
small enough, Ir.s.g'(v)I <
sg(i')
l-r I
and so this number rapidly goes to +
So a problem for convergence can develop only when 1sIn not small in comparison with I while (which is —
is
as r + 1.
the
coefficient of in d*(&) up to a bounded factor) is not large In son with 1 (which is the coefficient of in d*(.j)). Here wa Introduce the second restraint on b:1-r1(t) for small t. our choice of 273
f(t)
), this is possible. Then, for
small enough, we have
(q -
g(v)
If we further demand that, near
2
< 0, then
where
r('p,t)
r('r,t)
e
Since, for v small t, we obtain
1
Isin 2r4f
is close to 2irjv -
r
1
and
1
=e
lIt2 for
1
)
.
(const. ehl't2_
)
Therefore, if the component of X(r,'Y,s) cannot be neglected in comparison with the component, i.e., if is not very small, then and components of are neglibly small in comparison with the component.
Up till now we have completely ignored the
But, for r close to 1, the
-component of dob(F(f)).
-component of
and
can be
neglected in comparison with the other components. Therefore the above dis-
cussion shows that the positive unit tangent vector field of dob(F(f)) at as r goes to 1. (r,'v,s), < s < converges (rapidly) to As we have Indicated above, the proposition implies the main result of this section. Corollary There exists a circle foliation F on an open solid torus V having the following properties. B2(F) is an annulus splitting V into two open solid tori V1, U1 Invariant under F, where V1 V is contractible and unknotted. B1(19V1) is again an annulus splitting V1 into two invariant open solid tori V0, U0, and V0 c V1 is contractible and unknotted. References
[1)
Caratheodory, C.
Ober die Begrenzung einfach
Annal.n 73 (1913) 323—370. 274
Gebiete,
Foliations with all
[2]
Edwards, R.D., Millett, K.C. and Sullivan, D.
(3)
leaves compact, Topology 16 (1977) 13—32. Epstein, D.B.A. Periodic flows on 3—manifolds, Ann. of Math.
9S
(1972)
68-82.
[4) Epstein, D.B.A. Foliations with all leaves compact, Ann. Inst. Fourier, Grenoble 26 (1976) 265-282. [5) Schweitzer, P.A. Some Problems in Foliation Theory and Related Areas. Problem 36.
Topology, Foliatione, and Geifand—Fuke
Notes in Mathematics 652. Springer Verlag: Berlin - Heidelberg — New York (1978). Epstein, D.B.A. Prime ends, Proc. London Math. Soc., 3. series, 42 Co homology.
[6)
In Differential
Proceedings, Rio de Janeiro 1976.
Lecture
(1981) 385—414.
[7)
tpstein, D.B.A.
Pointwise periodic homeomorphisms, Proc. London Math.
Soc., 3. series, 42 (1981) (8] Kuratowski, K.
415-460.
Topology, Vol. 2, Academic Press: New York - London
(1968).
topologiques des variétés feuilletées. [9] Reeb, G. Sur certaines Act. Sci. Indust., Hermann: Paris (1952).
Elmar Vogt
I. Mathematisches Institut Arnimallee 3 1
Berlin 33, West Berlin
275
RAWOLAK
Some remarks on V-G-foliations In this short note we present two methods for dealing with some problems for transverse structures of 6-foliations and in particular for V-G—foliations.
As examples of applications of these methods we prove two theorems. We construct the graph 0f a v—G—foliation, and then study the transverse structure on the graph. Its properties allow us to prove the following theorem.
Let F be a transversely complete V-G-foliation. of the.foliation F have the same universal covering space. Theorem A
Then the leaves
In the secend part we give definitions of transverse structures of higher order.. They admit a foliation of the same dimension as the initial one, and its leaves are covering spaces of the leaves of the initial foliation. We show the following. A 6—foliation F
a transversely projectible G—connection if and only if the foliation of the normal bundle of order r admits a transversely projectible Gr_connection. Theorem B
1.
PRELIMINARIES
Let Mbe a connected n—manifold1 N a q-manifold (n B(N,G).
phisms
q) with a G—structure
a S-connection in the G-structure 8CN$). Let (M,F) be a on B(N.G), and defined by a cocylce i.e. and are local automor= S-structure B(N,G) and affine automorphisms of the connection
v.
H bq a sub—bundle of the tangent bundle TM supplementary to the foUatim F. Let 1(H) be the frame bundle of H and denote by B(M,F.G) the
by the 6-foliation F. A G-structure obtained in a transverse 6-structure. We denote by f1 the mapping of defined by f1. Let w be the connection fore of the B(M,F,G)1U1 into Then the forms glue together to define a connection & on L(H), whiCh Is. a connection in the transverse G-structure By we denote the corresponding differential operator on H. this way Is
276
The fundamental form ö of the transverse G—structure B(M,F,G) is defined
as follows: Let p be a transverse frame of >
Then
)
M
>
+ M is the natural mapping, p,:TM + H is the projection of is the inverse of the linear isomorphism defined
where
Th onto H along F, and by the frame p. For any vector
we can define a vector field the fundamental horizontal vector field. We demand that: (1)
on B(M,F,G) called
=
(ii)
is a horizontal vector field;
(iii) for any p E B(M,F,G). where Ft is the supplementary distribution in ker to the lifted foliation ker
=
One can easily
where e is the fundamental form of the G-structure B(N,G). The mapping Is an isomorphism, where r is the horizontal space of the connection1w. Additionally, Is the fundamental horizontal vector field on B(N,G) defined by the vector then
;1(u1)
=
if
.lp
d p
Proposition I leaves of F. =
•
Let S be a
section
of the bundle H on
constant along the
Then
f1X
S, -
where S =
Proof
This foreula is obtained directly from the definltio* via the Christo-
fel
—-
Let cx;(O,t] + M be a curve in a leaf of the foliation F, u(O) = x, ct(1).vy. into Let y be a curve such
Then the curve u defines a mapping T of
277
that y(O) = x, d/dty(O)
Let
v
holono.ny lift of a to y(t).
t
Then
be the curve starting at y(t), the ctt(l) Is a curve at y transverse to F.
the H—component of the vector d/dtctt(1)(O) we assume as the value of I on the vector v. One can easily check that it does not depend on the choice of the curve y. The mapping I is a linear isomorphism, and in its turn defines an Isomorphism be an be an adapted chart at the point x and let adapted chart at the point y such that each plaque is contractible. Let be the transverse submanifold be the transverse submanifold at x and let there exist the unique points at y. For any points u and v and v' respectively, such that the points u and u', v and v' u' belong to the same plaque of respectively. or be a neighbourhood of x in such that, for any point x' of Let
Let
there exists the holononiy lift of the curve a to x. the image by the holonomy mapping of
Let us denote by
be the saturations of
and
in and Us,, respectively.
in
and
Let By
is the holonomy lift of the curve is a curve in the plaque linking u to u', is a curve in the a to U', does not plaque linking v to v'. The holonomy mapping I along the curve denote the curve
where
depend on the choice of the curves in such a way that the mapping a:Ox x
is smooth, where
X
I
-t M
x
:
and
x Ii:
=
I
We can choose curves
and
=
any pair of points (u,v)
For
5v•
T(u') = v'}.
0,,, the curve
defines the mapping
and the mapping
1(0)
T-:
x
is smooth. Lenina
t
=
(p)
and be two transverse maniProof Let x = a(O) and y = ci(l), and let folds at x and y, respectively. Additionally,we assume that the manifold
is contained In some and in some Then the mappings are local diffeomorphisms. The mapping 278
and
f(y) +f(V) •.. °
Is just the composition
for some Indices
Thus the mapping :
... ° g.
is equal to g4
01 The vector fields
vector
B(E).
• and therefore it is an affine mapping.
1k—lk
—
on 1(H) or B(M,F,G) are mapped by f1 onto the Then
••• = =
Thus
2.
GRAPH OF A V-G-FOLIATION
For the convenience of the reader we recall the construction of the graph of a foliation, due to Ch. Ehresmann and later developed by I4.E. Winkeinkemper
(4]. Let x and y be two points of the manifold P4 lying In the se leaf and let a be a piecewise smooth curve linking x to y and contained in the same leaf. We say that two such curves atand Bare equivalent if the holonosny along the is trivial. The space of all such triples (x,y,(a]), where (a] denotes the equivalence class of a In the above equivalence relation, Is called the graph of the follation.F, and we denote it by GR(F). The topology Is introduced in the following way. Let z (x0,y0,(a]) be any point of GR(F). Let a be a representative of (a]. Take an adapted chart ) at x and an adapted chart at y such that $ :Dk X (U x0, and verse submanifold
0q
= y0. By W1 we denote the transx and by W3 the submanifold For any
point x U1, by x. we denote the point of the plaque of x belonging to WI, and for any point y of by Yj the point of the plaque of y belonging to By V1 denote all the points of U1 whose plaques can be linked with a plaque of by a chain of plaques following the curve ci. By denote the set of points of which lie in the end plaques of the above chains. Let 279
defined as follows:
Wza be a subset of
GR(F); x
=
V1. y
e=
51*3*5
respects, are curves linking the points x and y with and ively, in the corresponding plaques, and & is the holonomy lift of a to x1. and the equivalence be The set is well defined as the end point of
where
and s,,. .The sets class of B does not depend on the choices of the curves we take as a sub-base of the topology of GR(F). Wz,a In our case, by means of these sets we can introduce a differentiable
structure (cf. [4]).
In this differentiable structure they are adapted charts for a 2k-dimensional foliation F. 'The foliation F can also be defined as the inverse of the foliation F by the canonical projection or
p2
GR(F) + ii, where
= x and
y. The tangent bundle of the manifold GR(F) is isomorphic to the sum F • F ® where the bundle is given by :
H = {v
TGR(F);
E H};
H and
or, in more detail, let z = (x,y,[a))EGR(F), then any tangent vector is equal to (Xx.Yyi(a]). where Z E lxii' Y,, 1)4. In particular, we can consider ii
as
{X E TGR(F); X = (v,T(v),[a]), V
H).
Let us consider the reduction L0 of the frame bundle L(GR(F)) defined by the decomposition F s ii of the tangent bundle TGR(F). Let at z is given z (x,y,[a]). A frame v = by the following vectors:
(Vl,...,Vk) a frame at x of F,
VI
a frame at y of
=
C Il,
I
1,...,q,
Thus
H,
(Wq))
is a frame at y. given by a curve y
280
H and I
=
a frame at x, and Therefore any vector tangent to 10 is
such that
for any i = 1,...,q.
=
Directly from the definition of the foliation F is a v—G—foliation, and the bundle ii can be considered as the normal bundle of the foliation F. be the connection on H defined by the connection v.
Let
Leoma 2
The napping T
is an affine transformation of the connection
Let a(O) = x and a(1) = y, and let x
Proof
some j.
=
=
=
=
f*(fTf1)*
=
=
01 a
for
U1 for some i and y
Then
T
Thus
.
k—lk
Is an affine transformation of w.
Definition A V-G—foliation F is called tran8veraely horizontal vector fields are complete for any
conrplete
If
fundamental
E
Lefllna3 Letav-G-foliation F be transversely complete. on the graph manifold GR(F) is transversely complete.
Then the foliation
be a fundamental horizontal vector field on B(M,1,G), .and its global flow. Let w (w1,...,w ) be a frame of H at z — (x,y,[cg]). Then each WI, I 1,...,q Is equal to where — dl and is a frame at x of H, and w2 a frame at y. Proof
Let
We put =
where the curve
1+t(a)]) is obtained In the following way.
curve, we can lift it to a leaf curve
at w1
Since
is a leer
B(N,F,G), then we take
which is a leaf curve as the vector fields are Infinitesimal lutnimorphisis of the foliation of the fibre hendle. Next, we project this curve beck to P4 and obtain a leaf curve, which we denote by
We have to show that the
homotopy class of the curve •t(a) does not depend on the choice of the lift &. 281
-The segment •(O,t]&(O) of the flow is a transverse curve to the foliation of B94,F,G). It projects to a transverse curve to the foliation F on the foliation F, the mapping manifold N. Since the flow
B:[O,l] x [O,t]
(s,v) •+
is the holonomy lift of the curve & along •CO,t)&(O), and the projection of onto the manifold M is the holonomy lift of the curve along the curve r. To complete the checking that is well defined, we recall that Then the mapping 1 will commute with the flow of and = indeed wilt be an element of B(M,F,G), i.e.
T,t(w1)
=
Since the foliation
,t(w2). on the manifold GR(F) Is defined by a cocycle the vector
the tangent vector to
is the flow of a fundamental horizmatal vector fte)d, as, of the flow course, this vector is horizontal, since, locafly, the cosmectioe on B(GR(F)j,G) is given by 3.
PROOF OF THEOREM A
Any two points of the manifold N can be joined by a pl.ce.lse smooth curve whose segments are either leaf curves or projected segments of Integral curves of fundamental horizontal vector fields. We would like tà lift these curves to GR(?). As the horizontal bundle we assume the bundle which Is traflsverse to the fibre of the subeersion + N. The fibres
I
are covering spaces of the leaves of the foliation F (cf. (4));. First of all, we lift leaf curves. Let y be a leaf curve. The lift of It cannot be tangent to this curve has to be tangent to the bundle ii * Therefor the tangent vector has •to be N, thus it must be tangent to X, the lift of the curve y to of the form (X,O,(c*)). Since th. point z (x,y,(cz]), where, x = a(O•) is the curve 3 t (y(t),y,
(O,t]]). To lift a transverse curve, we need a more subtle construction. Any such curve y Is a projection of a segment of an integral curve •(O,1](p0) of a fundamental horizontal vector field on the manifold B(Mj,G). Take a 282
of an Integral curve of th. victor field
COPP'i$pofldlflg
In the fibre ftC;) on the manifold B(GR(7),?,G) starting from $ point manifold is tangent to The projection of the curve a E H directly from the and the bundle Il, as Additionally, as and definition of the vector field = =
=
it follows that, indeed, the vector tangent
in the fibre is tangent to H. The choice of to corresponds to the choice of the point in the fibre As we have shown, we can lift horizontally from H to GR(F) any curve of the chosen type. We shall call such curves v—curves. Any v—curve y In N defines a diffiomorphism of
Lenina 4
onto
)). Proof
The horizontal lifts of V-curves depend smoothly on the initial
condition; thus, lifting the curve y to the points of the fibre and taking the end points of the lifts, we define the mapping of into which is a diffeomorphism. Using the same methods as in the proof of [1, Theorem I, p. 239]. we show the following theorem. Theorem 1
Let F
be
Let G be a connected Lie group and let N be a connected manifold. a transversely complete V-G-foliation. Then p1:GR(F) H is a
locally trivial fibre bundle, with the structure group is any point of N.
(x0)),
where
As a corollary of this theorem we get our main theorem as the fibres of are covering spaces of the leaves of the foliation F. 4.
TRANSVERSE STRUCTURES OF FOLIATIONS
We present the definitions of some transverse structures and their basic properties. More details can be found in (6]. Example 1
Transverse (p,r)-velocities (pr_jets).
Let m be a point of the manifold H. mapping of
(M,m) be any local smooth Let f:(R1',O) into N mapping 0 onto m. Let f.g be twosuch mappings and let
x (U,+) be any adapted chart at m such that •:U •(x) = thus •2 is constant along the leaves. We shall also use the notation
283
We
2
say that the mappings f and g are
This Is equivalent to
equlvaldnt If 91v1
for any multi-index v E lvi r, i = 1,...,q. We shall denote the number of such indices by p(r). This equivalence relation does not depend on the choice of an adapted chart at the point m. The equivalence class of a mapping The set of all equivalence classes at a point m we f we denote by and the space
denote by N m
U
NP*r(M,,) by
i.e.
denote the natural projection of into can easily check that for any adapted chart Isomorphic to U x
.
above by
()
By
m
the set
wr let
us
p
f(O). U
mEU
One
is
m
and that the isomorphism is given by the mapping Thus, if we denote the mapping defined
•r, ,r.(
pq.p(V')
the collection of all such
defined by an adapted atlas on P4 defines an atlas on the space N ' (M,F). To see this, one has only to notice that if are two adapted charts for a x x foliated manifold (Mj), the composition is of
the form (f 1(y,x),f,(x)), where y denotes the first n-q coordinates, x the
last q,
x
and
,
is the mapping of Suimning up, we have proved that
whose total space admits a by
x
.
then
Is equal to (f,1r(f)) induced by f.
where
is a locally trivial bundle, q.p(r) + q foliation F" projecting
onto the initial foliation.
If p = q and we take only transverse embeddings of into P4, the above construction gives a bundle called the transverse frame bundle of the foliated manifold (M,F) and is denoted by Lr(N,,). It Is a principal fibre
bundle with the fibre Example 2
The Ljndle of transverse A-points of (Mj)
Let A be an associative algebra over the field R with the unit I. The algebra A is loCal If it is counutative, of finite dimension over R, and If it 0 for a of codimension 1 suCh that some non—negative Integer h. The smallest such h is called the height of A. 284
be the algebra of all formal power series in be the maximal ideal of R[p] of all formal power series without and let constant terms. Let A be a non-trivial ideal of REp] such that R[p]/A is of finite dimension. Then A = R[p)/A is a local algebra with the maximal ideal = Any local algebra is isomorphic to such a local algebra (cf. [2)). Let C(Ilj) be the algebra of germs of smooth functions constant on the leaves of the foliation F at the point m of the manifold H. An algebra + A will be called an A-point of (14,F) near to m homeomorphism Let R(p]
(or an infinitely near transverse point to m of kind A) if c(f) f(m) mod I the set of all A-points of (14,7) for every f E C(M,T). We denote by near to m, and by A(M,F) =
u
Am(M,F).
The mapping Am(M*F)
a-pm C 14 is
denoted by
One can prove that the set A(M,F) admits a differentiable structure such that .UA:A(M,f) + 14 is a fibre bundle over 14 with fibre A, and that there exists a canonically defined foliation TA of the same dimension as the folia-
tion F. 5.
PROOF OF ThEOREM B
Roger's definition of the universal Atlyah—Molino class, and therefore we consider foliations as r-structures. be two smooth First consider a pair of groupoids r1 and 12• Let M1 and and respectively. Assume that two groupoids on manifolds, r1 and To prove Theorem B we shall use Cl.
there exist two homeomorphisms of the groupoids F1 and
>r1
2
(1)
>142
M2
such that
,
= 2
f.i = idM and r1im I c in i, yi(y1) = 2
The need to consider such a pair of groupoids is explained by the follow— be the r-tangent bundle of 14, be a smooth manifold, H= TrM. H1 = Let be a given G-structure on 14. Let r2 be a groupoid of germa of automorphisms of the G-structure P(M,G),let r1 be the groupoid of be the r-prolongetion of the Ggerms of lifts of elements of F2 to 1"M, and tng.
structure P(14,G).
of the group G.
It is a Gr_structure on TrM, where G' is the r-prolongation Let i:M + 1r14 be the zero section. Then the mapping
•
285
defined as follows:
c r1. The mapping
we define as the natural projection.
Of course, these
mappings are homeomorphisms of groupoids.
If F is a G—foliation modelled on B(N,G), then the foliation f is a Gr_ = r-structure for foliationmodelled on B'(N,G). Thus a G-foliation is a a suitable choice of the groupoid r, and then the foliation Fr is a structure. A Let (F1,g1) be a r1-sheaf over M1, and (f2,g2) be a r2—sheaf over over i Is called a (r1,r2)-cohomeomorphism if, for any x N2, v E 1(x), y cs1(y)
cohomeomorphism F of F1 into
F(g1(y)(v)) After long computations one can show the following. Lemea 5
Any (r1,r2)-cohomeomorphlsm F of the sheaves F1 and
induces a
mapping in cohomology: F*:H*(BF1,F1)
)
.
(2)
As the next step of the proof let us consider the following situation. be a covering of Y, and = Let ir:X Y be a continuous surjectlon, be a continuous functor, {i(1(U1)} be a covering of X. Let p r1, such that the F2:Y11 + r2, let F1 be another continuous functor, following diagram Is conunutative F1
Additionally, we assume that there exists a section of iT, that lily We require that the diagram
286
X such
F'
on B(M,G) defines such a pair of functors. We have to take Y = II, U the open covering of the dfming cocycle, r2 F on
X = i(M,F), r1 = rr,
= V1}.
=
A F1-sheaf (F1,g1) defines via a sheaf F F2* r2-sheaf (f21g2) defines via F2 sheaf on Y.
on the space X, and a
For the details see [3).
and any Leiua 6 For any (I'11r2)-cohomomorphlsm F of the sheaves F1 and two cofunctors F1 and F2 such that the diagrams (3) and (4) are conrutative,
the following diagram Is comutative: H*(X,FF1)
(.
H*(Br,,F1) I
F*
*
(5)
V
(
F
Having proved the properties contained in Leninas 5 and 6, we can complete the proof of Theorem B. Let us remark first that as a model G—structure we
can always take a trivial G-structure, but then the manifold N does not need to be connected.
Let N be a manifold and let P(N,G) be a trivial principal G-flbre bundle. Letrbeagroupoid of germs of automorphisms of P(N,G).
The tangent bundle
TP admits a natdral action of the group G; let Q = TP/G, let I be the tangent bundle to N, and let I. be the associated fibre bundle with P with standard I denote the r-sheaves of germs of sections of fibre g = Lie(G). Let I., the fibre bundles L, Q I, respectively. Then the following sequence of sheaves is exact: (6) Qr
the bundle TPr,Gr over IrN, by Tr the bundle fl.rN Let i:N -'. IrN be the zero section. We have the following conmutative diagram Denote by
287
of sheavesand their (r,r")-cohomomorphisms over 1r
o
>
(7) o
Q
—> 1 —4 0
In the proof of the fact that the vertical arrows, defined by the mapping we use the existence and properties of liftings are
of vector fields (ci. [2], (6]). From diagram (7) we get o
(8)
I o+
!2m(lA) +
0
and therefore the following diagram of long exact sequences is coirniiutatlve: o
o
Homrr(T",Q") -t
Homr(T,L)
Homr(T,Q)
4
Horn r(T",T')
HOmr(T,T)
.i.).
_!4
H1(Br,Hom(LLJ).
the sheaf Let us take as the sheaf F1 the sheaf Hom(TrLr) and as Hom(T,L). As the F over i we take the corresponding vertical arrow. Then from diagrams (9) and (5) we get the following coirnnu-
tative diagram, taking into account that the sheaf is equal to the sheaf where and FHom(!,L) to the sheaf p(g") and P(g) are the associated fibre bundles with the standard fibre gr and g, respectively: HI(Brr,Hom(Tr,Lr))
•j,b HOmr(u,T)
+
1
H (Br,Hom(T,L))
>H
According to (3], the Atlyah-Mollno class F" Is equal to
288
(10)
of the lifted foliation
N4[Fr]
ó(Id1r).
Then =
One can easily check, directly front the definition, that a(Id1r) = Id1. Thus b(AM[Fr])
=
A14(F).
Up till now we have considered the foliation ( as a r"-foliation. Let rr be the groupoid of germs of automorphisms of the r-prolongation of the G-structure P(N,G). This groupold contains the groupoid rr as an open subgroupoid. We have also to consider the foliation as a rr-foliation and look at the relations between the Atiyah-Molino classes. It Is not difficult to check that they are equal. This last remark effectively ends the sketch of the proof of Theorem B. References
(1) Hermann, R. A sufficient condition that a mapping of a Riemannian manifold be a fibre bundle, Proc. A.M.S. ii (1960) 236-242. (23 Morlmoto, A. Proiongationa of Geometric Structures, Lecture Notes, Math. Inst. Nagoya Univ. (1969). hontotopiques et cohomologiques en tMorie des (3] Roger, Cl. feuilletages, Univ. Paris XI (1976). Winkelnkemper, H.E. The graph of a foliation, Ann. Giobal An. Geom. I (4] (3)(1983) 51-76. 6(1984)329-341. (5] Wolak, R. On (6] Wolak, R. On transverse structures of foliations, preprint 1984. Robert A. Wolak Krakow
Poland
289
CTJDODSON
Fibrilations and group actions 1.
FIBRED MANIFOLDS
The aim of this paper is to present some results on fibred structures which may be viewed as generalisations of fibre bundles, and to report some joint work with D. Canarutto concerning the stability of frame bundle Incomplete-
for quantization in general relativity.
ness which has
The context is thai of fibred manifolds (or surmersions) the geometry of which, following Ures,nann, has been studied in particular by Llbertnann, MangiarotLi iudugno, Ferraris and Francaviglia. The geometry is quite rich becaLI3e a fibred manifold may be viewed as the least structure needed to suv submersion E
A
fold
Then p has
F Peas an open neighbourhood V and a mani-
maximum rank
with
fibril.
B.
pV x over p. We shall call a OV:V fibred manifolds Is a commuting diagram of smooth
A
fibre-preservin
I
-p B
There is a.naturäl composition of such diagrams, so yielding a category FM.
One reason for studying this category is that it admits puilbacks; in fact, that is a consequence of the following result, which says that every finite diagram in FM has a left limit. The category FM is finitely left complete.
Theorem
Proof
It is sufficient to show that FM
The former Is clear enough. squares:
290
finite products and equalizers.
Consider the equalizer diagram of conhinzting
E2 <
E1
—--———4
p1
82< Our
=
2
B1
1
>
F2
F, say)with
f2(y))
E11f1(y)
}
= {x p1E =
>
B2
candidate left limit object is E —3--> B E = {y
F1
B, E is a smooth submanifold of E1 and p is evidently a smooth
surjection. Take any y
E, then by connutativity in the tangent diagram, =
so p Inherits the submersion property from p1. The morphism required for the universal property can be. obtained from inclusions and compositions. 2.
SHEAF STRUCTURES
A smooth map p:E -, B is a sheaf manifold over B if p Is an open, local diff€'omorphism.
The categories of sheaves on smooth manifolds and sheaf manifolds are
related via a functor which carries sheaf
to sheaves of their
smooth local sections. Proposition open and its In the
p
fibred manifold E —94 B is a sh€. are discrete spaces.
nifold over B if p is
actually equivalent as a given base space, for there the 'i'1ctor, carrying sheaf spaces to of their continuous local ar inverse which carries sheaves to sheaf spaces of germs of :wns. However, this Inverse is not available to us even for fibred topological spaces, because case, sheaves and sheaf
291
the latter need not have discrete fibres, Of course, the fibrils of a fibred manifold constitute a sheaf and this may allow any coniiion algebraic structure to be exploited. Proposition
p
Every fibred manifold E
> ) B determines a sheaf SE of smooth
local sections of p. First we obtain a presheaf cofunctor
Proof
Set
S
:
S(U,E) =
U
(a
E
EVIPa
=
!
fibred every point of E has a sectionable neighbourhood over a sufficiently small base set; so SE is not empty. a
A) of any U A) with
Now take any open cover
a collection
S(U ,E)I cx
1(B).
Suppose that we have
Unu (va,B
A)
Hence Gala
=
8a
a
S
cx
B
Then by functoriality of S
a8i8,
Unu
Unu
aB
PU
aB
U
Unu
aB
a
U8
PIJci
U
B
PU
fl U8
'cx) E(
U8
x e U. It follows that a E S(U,E) and, as required, it satisfies
Define a:U + E
292
:
x+
If
= a.
a=
S determines a functor from the category of B to the category of sheaves over B. 3.
manifolds over
G—FIBRILATIONS
is a fibred manifold E B with a action a over B by a Lie group G acting as a group of fibred manifold automorphisms of E. We shall denote this by G x E _29 E 29.4 B and call it a Logically we might call a fibred manifold a A of group-fibrilations is a cosmnuting diagram of smooth maps with f a Lie group homomorphism:
A group—fibriZation
GxE
>
E
B
,
I
B'
a category, under diagram compositlom. finite prodects and we can use the FM equalizer but In
Again we
Is clOsed under we cannot
obtain a suitable G-fibrilation equalizer except by takhul it with trivial 6, and likewise for puilbacks. Proposition Let 6 x E E >B be a G-fibrilatioe. Given a smooth curve c B and any y0 positive 6 and a smooth curve E:(—6,6] E with pE —
there is soma Moreover
is a,mth.r such curve for all smooth curves-$(—6,4] Proof
Since E 24—4
B
is a fibred manifold, we can
hood V of y0, a fibril
y(O) — 1.
somi open neighbour-
and diffeomorphism
Take 6 > 0 with c(—6,6] c pV, possible since E is a manifold.. pV x
(x,v) •
:
(-6,6]
E
:
v.
Difine
Then,
t 293
is smooth and projects onto the restriction of c to (—s,5). Also, if y.(t) EG is well-defined for acts vertically so then the action automorphism it may leave V.
the same
•The usefulness of this lifting is essentially measured by 6, on a scale from zero to c; the more of the curve that can be lifted the better. It may be of value to take the supremum of over all fibril neighbourhoods of y. Each lifted curvn determines a transport process among fibrils over the the 3ction of G, also to their base curve ad extends, t5
cm G-fibrilations, essentia
Next we same way
a G—fibrilation G x E dimensional distribution on E A cclnnection
r:y
-,
.,
the
C8J.
.
on
c
E
)—>B is a smooth
with TPIH
that is invai'iant under a, namely: H
agY
H
9
Y
(vy€E,vg€G).
We see that a G—flbrilatlon generalises the notion of a G—bundle.
It has
a local product structure that is not locally trivial nor even a fibration, since the fibres need not be homotopic.
Moreover, the action of G is not necessarily transitive nor free. However, locally a G-fibrilation has a sufftclently simple structure to make differential analysis easy through adapted charts. Moreover, It can support the useful geometric notion of an invariant connection. The study of these was begun In (3) where some principal bundle theory was adapted to obtain induced and coinduced connections from group—fibrilatlon morphisms. Jet calculus and connection geometry on fibred manifolds (Cf. (8], (9], (4], [7] for example, and references therein) Is transferable to G—fibrilations which may prove a useful setting for variational problems with group syninetries.
We turn now to a result concerning a very particular type of G-fibrilatlon; namely, a principal 6—bundle.
294
4.
CONNECTION-STABILITY OF BASE SINGULARITIES
It is well known that the notion of geodesic completeness is Inadequate for pseudo-Riemannian manifolds, such as spacetimes, where It has become the due to Schmidt practice to lift the problem to a convenient bundle by a
(10] (cf. also (2] for a detailed accountand survey). General theorems suggest that any theory of gravity is likely to predict physical singularIties In the classical geometry. Recently,.we have proved the iollowing result for manifolds with linear connections. Bundle-incompleteness is stable under perturbations of the connection. The geometric details of the proof are given In (1] and they depend on Nodugno's structure of connections (9]. This is a flbred aenifold JP/G where In case P P(Gj4) is the frame bundle, sections of which are connections r:P + JP invariant under G (cf. Libermann (8]). Our trick is to use a canonical connection (cf. (5]) on the fibred manifold 6P/G x P .JP/G to obtain a bilinear form on its total space. Now, this restricts to become a Riemannian metric on certain submenifolds which have diffeomorphisms to the frame bundle and these become isometries for each choice of connection. Then, If M is bundle-incomplete with respect to one connection, it is also bundle-Incomplete with respect to a nearby connection. Theorem
This theorem has physical significance in that it lends weight to the belief that general relativistic singularities cannot be quantized away.
It
was already known from the work of Gotay and Isenberg (6] that geometric quantization of a massless klein-Gordon scalar field on a positively curved spacetime could not prevent the collapse of the state vector. Our result
and not tied to any particular method of It may also be useful to extend it to the case of nonlinear connections when they can be made to induce suitable metric structures some convenient fibred manifold total space. Is more
References
(1)
On the bundle of principal connections and the stability of b-Incompleteness of manifolds. Math. Camb. Phil. Soc. (1985) 98 (in press). Canarutto, D. and Dodson, C.T.J.
295
[2]
Dodson, C.1.J. (6)
[3]
Space-time edge geometry.
mt.
Theor.
Phys.
17
(1978) 389—504.
Invariant connections on G-fibrilations. Presented at Colloquium on Differential Geometry 26 August — 2 September 1984, Dodson, C.T.J.
Hajduszoboszl o, Hungary.
[4]
Ferraris, M. and Francaviglia, M. On the global structure of Lagrangian and Hamiltonian formalisms in higher order calculus of variations. Proc. Meeting, Geometry and Physics, Florence October 12—15 1982. Ed. M. Modugno, Pitagora Editrice, Bologna (1983) 44-70. Cf. also Fibred
connections and higher order calculus of variations.
Presented at Colloquium on Differential Geometry 26 August - 2 September 1984, Hajduszoboszlo, Hungary.
(5] [63
[7]
GarcIa, P.L. Gauge algebras, curvature and symplectic structure J. Diff. Gecin 12 (1977) 209—227. Gotay, N.J. and Isenberg, J.A. Geometric quantization and gravitational collapse. ffiys. Rev. D22 (1980) 235-260. I. Prolongations of generalized connections Coil. Math. Soc. Janos 31. Differential Geometry, Budapest, Hungary (1979) 317325.
[8] Liberiiiann, P. [9] Mangiarotti, N.
J. Diff. Geom.
8
(1973) 511—539.
Fibred spaces, jet spaces and connections for field theories, in Proc. International Meeting Gecvrietry and Thyewf 12—IS October 1982 ed. H. Modugno, Pitagora Editrice, (1w) 135—165.
ClO]
296
M.
L&. Aw definition of singular points in general relativity
M FERRARIS & M FRANCAVIGLIA
The theory of formal connections and fibred connections in fibred manifolds 1.
INTRODUCTION
In the framework of higher order calculus of variations in a fibred manifold one often encounters fields of objects which may be naturally Y= identified with sections of vector bundles of the kind
T(X) where V and I are standard functors and (p,q,r,s) are non—negative integers.
Objects of this type are called in short '(fields of) fibred tensors°, because of their transformation properties under changes of fibred coordinates In Y. As an example, we can mention Lagrangians, their vertical differentials, momenta, etc.
The local structure of higher order calculus of variations is fairly well understood, both at the L.agrangian and at the Hamiltonian level. However, in many physically interesting situations one needs to deal also with global problems, which only recently have received serious consideration and have Penong the global problems been given a reasonably satisfactory
a nuober of different interprea tations we r.c11 the problem of a correct global definition of the so-called that have
fcrm' (which bas long keen known to exist upiquely for
to exist uniquely also for higher und first ordme field theory, but recently shown to be order mache highly non—unique in the meat general sItuation; (2], (8], (9), (10], (11], (12), [13], [17], (19]). There are of course several tediniques to handle global problems (direct or intrinsic methods based on methods of globalization from local sophisticated tools such as sheaf theory, cohomology, etc.): In the direct dpproach, one of the standard procederes consists in trying to patch together local expressions by showing that their transformation laws may be Interpreted suitable bundle. A major difficulty which as transition functions of arises in applications to higher order calculus of variations is hidden In 297
the wide use of the so-called "formal derivative operator", which unfortunately does not transform fibred tensors Into fibred tensors. More pre-
cisely, If
a
1' 2'"'
fibred tensor t, the formal partial dervatives
'°2'
Accordingly, it Is convenient to replace higher-order (formal) derivatives of fibred tensors with suitably defined "formal covenant derivatives", constructed In such a that they transform again as fibred tensors. For this purpose, one needs first to Introduco suitable global objects which are called "forusi connections" and then use a formal connection to define a "fibred connection" which allows calculation of formal covariant derivatives of any ftbred tensor. A preliminary short discussion of formal connections and fibred connections In fibred manifolds has already been given In (2] and [5] and the purpose of this paper is to provide a more detailed exposition of this subject. Applications to higher order calculus of variations have already been discussed In [2], [3], (5], where the existence was shown, by an explicit construction, of an Infinite family of Polncari-Cartan forms parametrized by a family of are no longer components of a fibred tensor.
"f I bred connections".
In this paper we shall first define the relevant notions in the classical coordinate formalism and then we shall turn to more intrinsic definitions in terths of principal fibrations and exact sequences of vector bundles. Section 2 wIll be devoted to a short discussion of preliminaries and notation; fri SectIon 3 we shall develop the theory of formal connections and formal '(firstorder) covaniant derivatives; Sections 4 and 5 will contain the intrinsic description of these notions. 2.
PRELIMINARIES AND NOTATION
We shall here recall some standard definitions and set the notation which will be used throughout this paper. We assume that the reader is already familiar with differential geometry In fibred manifolds and with the theory 298
of jet-prolongations (details and references may be found in [13] and [18]). All manifold and fibred manifold structures considered here are assumed to over the be smooth in the category of (paracoinpact) topological reals. be a fibred manifold over the Let X be a manifold and let Y = = (Vx(Y),X,nbvy). where manifold X. The vertioal bundle of Y IS Vx(Y) = Ker(Tn) c 1(Y) and the restriction to Vx(Y) of the canonical projection V. If U = (U,Y,v) is a fibred manifold having for basis the total space Y of Y (namely, we have a double fibration over X), then the composition defines a fibred manifold We recall that In this case Vx(U) stands for while stands for Ker(Tv). Whenever
there is no need to specify the basis manifold 0f the fibratlon we shall omit the basis from the notation (writing, for example, V(Y) instead of Vx(Y)). For any quadruple of non-negative integers (p,q,r,s) we shall also set and we define the following family of vector bundles =
where Tr(X) denotes the standard
tensor power of T(X).
The sections of
over V will be called (fielde of) fibred teneore over Y. For any point y £ Y we consider the space consisting of all bases of tie vector space and we form the union = VF(Y)
u yEY
VF (Y).
This space is endowed with a natural manifold structure and it Is fibred over Y by the canonical projection . V. Moreover, there is a canonical action of the linear group GL(nR) (n dim(Y) - dim(X)) onto the fibres which Induces on VF(Y) a natural structure of principal GL(n;R)-
is shortly and It is called the bundle of vertical framea of Y. bundle over Y.
The bundle
by VF[r1]
The k-th order of V (where k is any non-n'egative Integer) is denoted by iktri] Also in this case, whenever there is no danger of confusion we shall omit the Indication Of the basis manifold X. For any pair (r,s) of integers there is a canonical embedding r,s :J r+s I (Y) Jr (JS(V)). For any local section o:X V we denote by 299
ik0.x
the k-th order jet—prolongation of a.
is a further fibred manifold over X, a fibred morphism from the fibred manifold V into the fibred manifold Z is a map F:Y + 2 such that
If Z =
with and any fibred morphism F:Jk(Y) + For any integer r we define the r-th order (holonomic) prolongation P(F) of the
= rj. k>
0,
1
morphism F, by setting
r(F) =
jr(2)
:
(2.1)
.. where JrF) denotes the standard r-th order jet-prolongation of the fibred mornhism F. f:Jk(Y) +R be a smooth map. There exists a unique 1-form Df over
the manifold
Jk41(y)
(11k+I)*(Df)
a
such that the following holds:
d((jko)*(f))
and
—0
for *ny (local) section a:X
Y and any field of vertical vectors here i(.) denotes the Interior product between + vectors and forms. The unique 1-form D? smt$sl4es the properties above is calld the differential of the f. For our later purposes we now turn to ltst coordinate notations which will be used throughout the paper. Consider a flbred manifold Y, with a m = dim(X) and n — dim(Y) — dim(X). If with 1 A •, is a local chart of the manifold X, Its domain Is denoted By a fibred
of Y over Is denoted by = (W;xA,yi) (with x 1,...,m and I = 1,...,n); the local coordinates associated to a fibred chart will be = (W1 ;X1' is a further fibred chart called fibred coordirtatee. If has a non-empty intersection with
whose a
A'A
and
y1' =
are the corresponding transition functions, the following notation will be used to indicate their partial derivatives: a
=
a.A/axX,
(xA,yk)
=
a
and so on. 300
a
with a fibred chart A fibred chart * of Y induces canonically in in J En], with coordinates ,v1) and a fibred chart coordinates is a multi-index with 0 (here k). All charti induced canonically by fibred charts of V will be called natural fibred oharta. For any fibred chart of Y, the formal differential Df is defined over the domain of *k+1 and its local representation is Df = (dxf)dxA, where the coefficients are given by dxf
af/axA
•
(af/ay1).
y1
Here standard multi-index notation has been used:
(2.2)
denotes the multi-index
(0,...,0,1,0,...,0) (with 1'in the x-th position) and sunination of multiindices is defined componentwise. The meaning of Df is clear from the local expressions above. We remark that the partial differential operator is
often called the formal partial derivative with respect to the coordinate xx.
Let us now consider any field t of fibred tensors in local components
in any natural fibred chart. local functions defined by d
ii
having
-
easy but tedious calculations show that the
j1,j2,... ,j
do not transform as the local components of a field of fibred tensors in For example, we have the following transformation law for the formal partial derivatives of the components v1 of a vertical vector field: =
+
+
(2.3)
This localformula Is our starting point toward the definition of formal connections, which will be discussed In the next section.
We finally recall the following well known alternative intrinsic definitions of connections in a principal G-bundle P = (P,M,1T;G), over any manifold 301
M with structure group any Lie group G
(i)
exists a natural action J'(P) x G + J1(P), which is the first prolongation of the canonical (right) action of G onto P There
by
This prolonged action is free and differentiable and it admits a quotient maniftid = J1(P)/G Moreover, it can be shown that the canonical proN is a (surjective) submersion. The fibred manifold jection k ) is an affine bundle, because the affine structure of = Finally, there exists J1(P) over P can be shown to pass to the quotient correspondence between the space of global sections
a canonical of the bundle
and thetspace of all connections of the principal bundle
P (see [7])
(ii)
Given
any G-bundle P, there exists a short exact sequence of vector
bundles and vector bundle morphisms over M, 0
V(P)/G -, T(P)/G
3
0,
w 1(M) -' T(P)/G of this sequence defines a coonection of
and any
P (and
1(M)
versa). CONNECTIONS AND FIBRED CONNECTIONS
of this paper is to define a family of objects which allow replace.entOf the partial formal As
we already announced in the Introduction, the
of the components of any fibwed tensor by local functions which
still ha* a "fibred-tensorial a
Objects of this kind will be "fibred connections' and it turns out that their major ingredient is of a suitably defined affine bundle over which will be "formal connection" over Y. In order to fibred connections derivatives of fibred tensors, be convenient first the notion of formal covariant derivahve of a fibred morphism + V(Y), with k any non-negative integer Derivation of fibred behaviour
tensors vrll then be defined by standard tensorizatloft procedures. us. consider a fibred manifold Y = together with fibred morphiss + V(Y). For any fibred chart * of V1 we introduce a set + of local ssoeth functions and we set by definitions =
302
+
(3 1)
where
a V1
o F are the components of the morphism F with respect to
Let us then require that the local expressions the given fibred chart Easy (3.1) above define the components of a fibred tensor in calculations based on relation (2.3) tell us that the local functions should obey the following transformation laws:
-
(3.2)
-
for any pair (4,,g,) of fibred charts of Y such that Dom(4,) n Dom(qs') i' 0.
It is easy to show that the relations (3.2) are invertible any satisfy the composition property of a cocycle over the manifold 31(Y) with values in the Lie group GL(n2.mR). Accordingly, they define the transition func= tions of a bundle over ,.J1(Y) Itself, which is unique up to isomorphlsms.
It is easily seen from (3.2) that this bundle
can be given a canonical structure of affine bundle over J1(Y); moreover, is affine, whenever is the pull-back over J1(Y) of an affine bundle over the basis X. The bundle will the bundie of fibred manifold Y; being an affine bundle, it admits oonneotiona over oonneotione global sections + e(J'(Y)), which we shall call over
Turning to local coordinate expressions, let us first r.mirk that any fibred chart g, of! induces in a canonical way a natural fibred chart of In such a the affine bundle C(n1], with fibred coordinates natural chart, the local representation of a formal connection over V has then the following expression: (3.3)
>
of the where the functions are defined in the domain given chart and transform according to (3.2). We turn now to define the foz,w1 covariant derivative of a fibred morphism. We consider then any formal connection F over ! and we set, in any
fibred chart of Y, =
+
.
(3.4)
(xA,yi,va,yl,va) are the natural fibred coordinates in J1(V(Y)) Induced by the given chart of Y. It is easily checked that the relations (3.3)
where
303
define in fact a global vector bundle morphism over v:J1(V(Y))
>
which will be called a connection ft).
V(Y) covariant
derivation (associated to the formal
F:i(Y)
v(Y) be any (global) fibred morphism over V. We consider the holonomic prologation p'(F) = J1(F) o and we define fibred morphisms and v(F) by setting Let now
V(F) * VoJ1(F)
—+ v(Y)
(3.5)
>v(v)
(3.6)
and
v(F)
Vep'(F)
Jk+l(y)
These flbred morphisms are respectively called the anhoionai,ic and the
fowirvzi covariant derivative of F with respect to F; they are the
fit into the
unique (global)
V
J'(V(Y))
p'(F)
V(Y)
.
JK(y)
Figure 1
304
diagram, Figure 1.
Consider now a local section v : X • V(Y) and define Its fo1n,zi oov4wiant obtained by as the section of V(Y) derivative (with respect to
setting V(v)
voj1(v)
:
X
(3J)
>
Is a local the composition F : x . For any local section section of V(Y) over X. Therefore we can calculate its formal covariint derivative, which is easily shown to satisfy the following property: v(F In
=
Y,
(3.8)
Is the jet-prolongation jk(0) of a local section relation (3.8) and Figure 1 imply the following:
particular, If
a:X •
v(F)ojl(ak).
v(F e jk())
ak
v(F)
jk+I(a).
(3.9)
We are now in a position to define foriaal covariant derivatives of any field of fibred tensors over V. In fact, let us first remark that standard tensorizatlon procedures allow us to extend the notion of formal covariant
for any pair (p,q). derivative to morphisms from into any bundle On the other hand, whenever a linear connection y is given on X, one may calculate covariant derivatives of any tensor field over X. Accordingly, any pair r consisting of a formal connection F over Y and of a linear connection y over the basis X, will naturally allow us to define formal covarlant derivatives of morphisms from into any bundle of the Any such pair r will be thence called a form fibred connection over V. A standard construction then provides uniquely, for any fibered connection r and any quadruple (p,q,r,s) of non-negative integers, a global vector bundle morphism
(q,s)
(q,s)
>
(q,s+1)
(y)
over V, which will be called the covariant derivation (of fibred tensors of type associated to the fibred connection r. (In the sequel, whenever there is no danger of confusion, the of the type (p,q,r,s,) will be omitted and we shall more simply write v). The local 305.
coordinate expressions for V may be easily from the definitions given above. As an example, the expression of the relevant part of V In the
Isgivenby: +
°V= faa k8 where
and
In
+ A
I
'so' -
faa ba
10)
6(XA)
are the natural fibred coordinates
and
respectIvely,induced by a fibred of Y. The generalization of fornula (3.10) to arbitrary values of the four Integers (p,q,r,s) is analogous to the standard one for covarlant derivatives of arbitrary tensor fields over a manifold and to avoid con(plicited expressions, It will not be reported here. It is now easy to define also the and lioionomio fo1I'mzi chart
ooua2'jtint der.ivativ. a
of any morphism
F:Jk(Y)
conmiutative diagram by the obvious replacements in Figure 1,
which yield the following: VoJ1(F)
(3.11)
:
and
Y(F)—
V. p1(F)
:
yP(y)
(3.12)
In terms of these notions, we have the following intrinsic characterization of the operator v. Let us first remark that the set of all fibred morphisms
.
for all integers (k,p,q,r,s), forms a graded algebra
over thermals. (This algebra Is In fact the pullback over f(Y) of the graded algebra of all fibred tensors over V.) Thee V is uniquely characterized by the following property: FE(Y)
Given any fibred connection the differential operator v defined bj (3.12) is the unique derivation of the graded algebri fl(Y) which satisfies the following properties: Theorem I
(1)
v restricted to functions coincides with the formal derivative
(II) v restricted to vertical vector fields coincides with the V defined by (3.6); 306
D;
operator
(lii)
V restrictfd to "horizontal' tensor fields coincides with the covenant derivation with respect to y;
(iv)
V comnutes with contractions.
The proof of this theorem is straightforward, by recalling that V is local by definition and applying a classical theorem of Wilimore concerning the extension of differential operators on tensor bundles (see, e.g., (13, p. 50). Proof
4.
FORMAL CONNECTIONS THROUGH VERTICAL FRAMES
In this section we shall provIde a first Intridsic definition of the bundle of formal connections, discussing an equivalent construction through suitable quotients of jet-prolongations of the bundle
of vertical frames in 'V. Let us then consider the principal bundle of vertioai framee of the fibred manifold where G = GL(n,P) with n * dlm(Y) - dim(X). We denote by A VF(Y) x G VF(Y) the canonical (right) action G onto VF(Y). If we prolong this action with respect to the projection we obtain a natural right action A,;
4(VF(v))
x G
:
whose quotient manifold defines the bundle = The are in one-to-one correspondence with the linear connections sections of of the vector bundle which will be called the connection8 of Y. Composing with we obtain a further fibred manifold
Although this Is not a principal bundle over X, we may adapt to it the above construction. In fact, there exists a natural right action :
4(VF(Y))
x G
-9
4(VF(Y))
which is obtained by prolonging A with respect to the projection This action Ak is free and differentiable and it admits a manifold having a natural projection over which makes Kx(VF(Y)) It an affine bundle over the manifold J1(Y) itself. Turning to local calculations in natural fibred coordinates, one can easily show that the bundle and the bundle
constructed above admit the same transition 307
functions, so that they are canonically isomorphic as affine bundles over Since the natural composition of functions induces a (natural) eplmorphipm between first jets of functions, there exists a natural epimorphism * J1(VF(Y)Y)
J1(VF(Y);X)
x
x which by restriction defines an epimorphism a from onto 4(VF(v)). It Is not hard to show that this epimorphism is equivariant under the prolonged actions and so that it passes to the quotients and defines uniquely a natural epimorphism which fits into the coninutative diagram, Figure 2
K9
x
(VF(V))
(Y)
C(J1 (Y))
Figure 2
A local coordinate description of the projection & may be given as follows. (W;xA,yl) of V and let us denote by (xA,yl,yl, Let us fix a fibred chart and ,y1 ) respectively the induced fibred coordinates in x
and Kx(VF(Y)).
Then the epimorphism & reads as follows:
(4.1) from which it is immediately seen that & is in fact an affine morphism of affine bundles over J'(Y). 5.
FORMAL CONNECTIONS AS SPLITTINGS OF EXACT SEQUENCES
We give here a further description of formal connections, in terms of splitting of exact sequences of bundles. 308
Let us then consider the exact diagram
of vector bundles and vector bundle morphisms over the manifold V shown in which acts naturally on VF(Y)C i1 Figure 3, where G is the group are natural einbeddings; (with I = 1.2,3,4) are natural projections. and
(VF(Y))/G
V., (VF(Y))/G
1
®
1
T(VF(Y)WG
0
Q
I 0
-ø Vx (Y)
7(Y)
-0
-
0
0
Figure 3
We then define affine bundles C1(Y) = {r1 c2(Y)
{r2
c3(Y) = {r3 c4(Y)
{r4
(C1(Y),Y,c1) Ci = 1,2,3,4) by setting
(T(VF(Y))/G) 1(Y)
(T(vF(Y))/G)
(T(Y))* I Tf3
(T(x))* I
Id1)
°
(Vx(VF(Y))/G) ey(Vx(v))* I iT2
F2
Id2)
= Id3) F4 = id4)
and taking for c1 the natural projections onto Y (here Id1 are abbreviations for the appropriate identity mappings). From these definitions It follows directly that the spaces r(c1) of all global sections r1:Y C1(Y) coincide with the spaces S1(r,) of all splittings of the four exact lines of Figure 3. We remark the following:
The splittings r1:V(Y) Vx(VG(Y))/G of the first short exact column (I.e., the elements of rCc1)) allow definition of covariant derivatives of (1)
309
vertical tensor fields along "vertical directions". For this reason they might be called very vertical connections. Since they have nQ direct relevance to our present purposes they will not be discussed here.
(ii)
The splittings r2:T(Y) + T(VF(Y))/G (I.e., the elements of
coincide with the vertlcal connections over V which have already been defined in Section 4 above.
(iii) The splittings r3:n*(T(X))
1(Y) (i.e., the elements of r(c3))may be
called nonlinear connections (or "generalized connections") over the fibred manifold V. They have been considered by several authors, also in view of
their possible application to physical field theories (see, e.g., [15]).
(iv) The splittings r4:n*(T(x))
T(YF(Y))/G (i.e., the elements of r(c4)) will be called here formal preoonnectione over V. In fact, as we shall see below, although they do not correspond directly to formal connections, it is exactly this row of the diagram which allows us to define formal connections over V. The rest of this section will be devoted to a discussion of this claim. We have the following:
Proposition I There are canonical projections C3(Y) which define affine bundle structures. Proof
C1(Y) and
From the exactness of Figure 3 we have
im(11) = ker(,r4) = ker(ir3 °
•
=
so that a canonical projection
+ C1(Y), may be defined by setting (5.1)
r2
All the bundles and mappings involved are affine and easy calculations show
that also the fibration (C2(Y),C1(Yh4) defines an affine bundle over the manifold C1(Y). Let us then define a mapping
o r4. 310
:
C4(Y) + C3(Y) by setting
(5.2)
see that From the coninutativity of Figure 3 and our definitions above, is well defined and turns out to be an affine surjective submersion, so is an affine bundle over C3(Y). that
We have also the following result: For any fibred manifold Y there exists a canonical epimorphism + c4(Y), defined by
PropositIon 2
j:C2(Y) x JU'211'3) Proof
r2
°
r3.
is a splitting of (2) and
Since
is a splitting of (3), the
composition is well defined and provides us with an injective mapping from into T(VF(Y))/G. Owing to the coninutativity of Figure 1 we have ° ° is a splitting of (4). Suralso iT3 it4. This implies that
jectivity of j is easily shown in local coordinates (see E6]). Finally, we state the following: There exist canonical isomorphisms A:C3(Y) + J1(Y) and
Theorem 2
MC4(Y) + C(J1(v)) of affine bundles over Y, such that Figure 4 is coninutati ye.
Pr1
C2(Y)
C4(Y)
-4
C1(Y)
93(Y)
I
A
-
J'(y)
I Figure 4
Proof
Let us first recall that an equivalent definition of first—order jets
of a fibred manifold Z =
assures the existence of a canonical oneto-one correspondence between global sections + J1(Z) and splittings T(Z) of the canonical exact sequence 311
•
0
V(Z)
1(Z)
>
1(X)
0.
-
Accordingly, there exists a canonical one—to—one correspondence between splittings r3:n*(T(X)) T(Y) and global sections a:Y J1(Y), defines uniquely a canonical ,isomorphism x:C3(Y) J1(Y) of affine bundles over V.
By analogy, from the above definition of C4(Y) one can see that there exists a canonical one-to-one correspondence between splittings. r4:n*(T*(X)) T(VF(Y))/G and global sections a:V 4(VF(V))/G, turn provides a canOnical isomorphism A:C4(V) + e(J'(Y)).
It is imediate to see that it projects Onto A, i.e., the following holds:
Therefore these affine isomorphisms fit into Figure 4 and make its right-hand square comutative. The rest of Figure 4 is coninutative by virtue of Propositions I and 2 above. We are now in a position to explain the terminology "formal preconnectionsTM
we used above to denote the splittings r4 of (4), by showing how they allow one to construct an important sub-family of formal connections over Y. For this purpose, let us first consider the exact coninutative diagram (Figure 5) of vector bundles and vector bundle morphisms over J1(Y), which is obtained by over J1(Y) of the conmiutative Figure 3. DefIne
(VFrfl)/G)
I
'I o
—0 0
I
I, o
o
Figure 5 312
II L
0
then affine bundles over J1(Y) by setting 1
'
1,2,3,4), so that their global sections Jt(Y) + C(Jt(Y)) can be canonically identified to the splittings of the fourexact lines of Figure 5 (numbered as in Figure 3). From the definition of pull-back bundles, it (1
follows that any section r1:J1(Y) + C1(J1(Y)) may be canonically and uniquely
Identified to a function
:
J1(Y)
which satisfies the relation
i.e., such that Figure 6 is coimnutative.
Jl(Y)
C1(Y)
V
Figure 6
We remark that all pull-backs where is any section of are sections of the bundles C1(ri1), although the converse Is not true (i.e., not all sections of are pull—backs). In particular, the short exact sequence 0
0
admits infinitely many splittings
which form a space, say
much larger than the space of pull-backs
of all splittings r4:rI*(T(X))
T(VF(Y))/G.
- Let us now recall that there exists a canonical embedding 313
I
:
a. splitting of thd short exact sequence
which is in
0.
0
This Implies that
all me buncfle C3(J1(Y)) + Jt(Y) admits a canonical
section K3
:
J1(Y)
C3(V) associated to K3 satisfies the
the mapping 1(3 ':
relation (5.3) where x:C3(Y) -* J1(Y) is the canonical isomorphism described in Theorem 2 above.
c.nsequence of (5.3), it follows that there is no section
As
r3:Y
coincides with 1(3..
se
We claim the following:
There are infinitely many splittings r4
Theorem 3
S4(111) which are not
pull—backs and which satisfy the following relation 113
° r4
K3,
namely, they are projected onto the canonical section 1(3.
space of all these splittings r4 c with the space
Moreover, the is in one—to-one correspondence
of all formal connections over V.
Let us first recall that the formal connections over V are by definition the sections of the affine bundle so that they are the only which fit into the coranutative diagram, Figure functions ?:J1(Y) isomor'phism of affine 7. Let us also recall that there exists a bundles over Y, A:C4(V) C(J1(v)), so that all sections r4:J1(Y) C4(J1(Y)) may be uniquely and canonically identified (through A) to all functions Proof
+
314
which fit into the couvautative diagram, Figure 8.
J'(Y)
"i.
(Y))
4Y)
'4 Figure
7
r4 J1(Y)
Cu' (v))
'41 J'(Y)
Figure 8
.
However, there are infinitely many functions ?4:J1(Y) C(J1(Y)) which fit into Figure 8 but do not make also Figure 7 coninutatlve (i.e., which are not formal connections). For example, if r :Y -. C CV) is a splitting of the mapping r4 uniquely associated to its pull-back (ri0)*(r4)eS4(ri ) cannot make Figure 7 and the above Using relation (5.3), recalling the definition of identification, it is easy to see that a splitting r4 is projected onto the canonical section 1(3 (i.e., its satisfies (5.4)) if and only if its associated
function
makes Figure 7 coninutative.
Accordingly, to generate the whole
family of splittings F4 satisfying relation (5.4) amounts to constructing them out of all formal connections, which are infinitely many. Finally, the
fact that all splittings satisfying (5.4) are not pull-backs follows trivially from our remark above that the canonical section K3 is not a pull-back as well. 315
Remark
A formal construction which allows us to generate the whole set of
splittings satisfying relation (5.4) through the existence of a surjective mapping from 52(n1) onto
will
be discussed elsewhere (see [6]), where
we shall also give local coordinate descriptions of all the notions introduced In this paper. References
[1]
Abraham, R.
Pou'ndatione
of
Mechanics, 1st
Edition, W.A. Benjamin,
Reading Mass. (1967).
(2]
Ferraris, 14.
Flbered connections and global
higher order calculus of variations, in Proceedings
forms in of the
Conference
Differential Geometry and its Applications, Rove MeVBto no Morave, Sept. 1983; Vol. II (Applications), D. Krupka ed.: Univerzita Karlova, on
Praga (1984)
[3]
61-91.
Ferraris, 1.1, Francaviglia, M. On the globalization of Lagrangian and Hamiltonlan formalisms in higher order mechanics, in Proceedings of the IUI'AM—ISI!4 Synrpoaium on Modern Developnente in Analytical Mechanics, Torino July 7—11, 1982; S. Benenti, 14. Francavlglia and A. Lichnerowicz eds., Tecnoprint, Bologna (1983) 109—125.
..[4] Ferrarls, II., Francaviglia, 14. On the global structure of Lagrangian -
•
and Hamiltonian formalisms in higher order calculus of variations, in Proceedings of the International Meeting on Geanetry and Phyaice, Florence, October 12—15, 1982; 14. Ilodugno ed.; Pitagora Editrice, Bologna (1983)
[5]
[6]
Ferrarls, N., Francaviglia, 14. Global formalisms In higher order calculus of variations, as [2] above, 93—117. Ferraris, N., Francaviglia, 14. Formal connections in fibered manifolds, in Proceedinge of the Conference on Differential Geometry, Debrecen, 26 Aug.—1
[7)
[8]
Sept. 1984; L. Tamassy and A. Rapcsak eds.; North-Holland
(to appear). Garcfa, P.L. Connections and r-jet fibre bundles, Rend. Sem. Mot. Univ. Padova 4? (1972) 227-242. GarcIa, P.L. The PoincarE-Cartan invariant in the calculus of variations, Symposia Math., 14, Academic Press, London (1979) 219246.
316
43-70.
J. On the geometrical structure of higher order Garcia, P.1., variational calculus, as (3] above, 127—147. The Hamilton Cartan formalism in the (10] Goldschmidt, H., calculus of variations, Ann. That. Fourier (Grenoble) 23 forms, Czech N., KoIa$, I. On the higher order (11]
(9]
Math. J., 33
n. 3 (1983) 467-475. I. A geometrical version of the higher order Hamilton formalism [12] in fibred manifolds, J. Geometry & Phyaiae 1 (2) (1984) 127-137. [13] Krupka, D. Some geometric aspects of variational problems In fibered 'manifolds, Folio Rae. Sci. Nat. UJEP Brunene'ia (Phyeica), 111' (1973) (108)
1-65.
(14] Krupka, D.
Lepagean forms in higher order variational theory, as (3)
above, 197—238.
Mangiarotti, L., Modugno, 14.
Fibered spaces, jet spaces and connections
for field theories, as (4] above, 135-165. (16) Mangiarotti, L., Modugno, 14. New operators on jet spaces, Annalee Rae. Sci. Toulouse (to appear). (17] Muiioz, J.N. Canonical Cartan equations for higher order variational problems, J. Geometry & Physics 1(2) (1984) 1-7. (18] Poamiaret, J.F. Systoma of Partial Differential Equatione and Lie Peeudogroupa, Gordon and Breach, New York (1978). (19] Sternberg, S. Some preliminary remarks on the formal variational calculus of Gel 'fand and Dlkii, in Differential Geometrical Methods in Mathematical Phyiioe II; K. Bleuler, H.R. Petry, A. Reetz eds., Lect. Notes in Maths. 676, Springer, Berlin (1977) 399-407.
N. Ferraris and Pt. Francaviglia Istituto dl Fisica Matematica "J.-Louis Lagrange" Unlversità dl TorinoVia C. Alberto 10, 10123
Italy
317
J GANCARZEWICZ
Horizontal lift of connections to a natural vector bundle 0.
INTRODUCTION
Let 1T:E + N be a vector field and D be a connection in E, that is, D
1(N) x E 3 (X,s)
E
on is a mapping (where 1(M) denotes the module of vector fields of class of E) which satisfies the N and E denotes the module of sections of class following conditions:
Dfx + gys = f D>5 + 9 DySi Dx(S + 5')
+ DxS',
Dx(fS) = x(f) S
+
f
for all vector fields X, V on H, all functions f, g on H and all sections s, s' of E. In Section 1 we recall the basic properties of a connection in a vector bundle E.
In particular, we define the horizontal lift of vector fields from
N to E. In Section 2 we study vector fields on E.
At first, for each section a of the dual vector bundle E* we define a function a on E. This family of functions is very important in the study of vector fields on E because two vector fields X and V on E such that X() = Y() for all a coincide on E (see Proposition 2.1). We prove (Proposition 2.2) that the horizontal lift xD of a vector field X from N toE verifies the formula D
Secondly, we define a vertical lift of sections of E. If s is a section of E then we define a vertical vector field on £ called the vertical lift of s. This vertical lift generalizes the previous definitions due to K.Yano, 318
S. Kobayashi, S. Ishihara (8], (9], (Ii] in the case of tangent bundles and due to K. Yano and E.M, Patterson [12], [13] In the case of cotangent bundles.
Our definition generalizes also the horizontal lift of tensor fields to tensor bundles (see [4]).
The vector field
verifies the, condition (Propos-
ition 2.8) = is the vertical lift of a function for every section a of E*, where fV f from N to E. In Section 2 we define also a vertical vector field (R(X,Y))° on £, where
0 Dy - 0y 0 Dx -
R(X.Y)
is the curvature transformation of D. This vector field satisfies the following condition (ProposItion 2.10) (R(X,Y)) (a)
R(X,Y)c,
for each section a of E*. This vector field generalizes the constructions due to K. Yano, S. Y.obayashi, S. Ishihara and EM. Patterson (8), (9), (10), (11], (12), (13] in the case of taigent and cotangent bundles. Next we study properUes of these vector fields on E. We have the followIng formelas (ProposItions 2.9, 2.13, 2.14): txD.yD]
(X,Y)D + (R(X,Y))° = (Dxs)!'
0
for all vector fields X, V on N and all sections s s' of E. In SectIon 3 we define the horizontal lift of connections of order r to a natural vector bundle 'and we study its properties. Let ir:E N be a natural vector bundle. According to the theorem of R.S. Palais and C.-L. Terng (7], E Is an associated vector jndle to FrN for some r, where denotes the principal fibre bundle of frames of order r. Let r be a connection of order r on N, that Is, r is a connection In F'M. For a such conpection r we define a linear connection on a manifold E called the horizontal lift of F 319
to E.
This connection V satisfies the conditions (Theorem 3.1) =
(vxY)D
x
(Ds)V
5V
X
=
sV
sV
for all vector fields X, Y on M and all sections s, s' of E, where v is the linear part of I' and D is the covarlant derivation of sections of E determined by the connection r (see R. Crittenden (1)). This construction generalizes the horizontal lifts of linear connections to tangent and cotangent bundles (see K. Yano, S Ishihara and E.M. Patterson
[9), (10), (13]) and also the horizontal lifts of linear connections to vector bundles associated with the principal fibre bundle of linear frames (3). Next we study properties of the horizontal lift of connections of order r. Our results generalize the results obtained by K. Yano, S. Ishihara and E.M. Patterson (9], (10]. (13) in the case of tangent and cotangent bundles. The results of this paper can be generalized for an arbitrary natural bundle (no vector bundle). In this case we need another characterization of vertical vector fields on a natural bundle (in the construction of sV the fact that £ Is a vector bundle Is Important). This generalization be published spparately. 1.
PRELIMINARIES: CONNECTIONS IN A VECTOR BUNDLE
Let ,r:E + N be a vector bundle. We denote by E the module of all sections of class C" of E and by X(M) (resp. 11(E)) the module of all vector fields of class C" on N (rasp. on E). A connection In E is a mapping D:X(M)
E 3 (X,s) —> DxS E E
satisfying the following conditions: DfX +
g
+ s') = 320
+ Dxs'i
(1.1)
(1.2)
(1.3)
Dx(fs) = X(f) S + f
for all vectof fields X, V on 14, all functions f g of class
on M and all
sections s, sa of E. Let ,:EIU + U x RN be a trivialization of E and let El,...,EN be the RN.
canonical base of
We consider sections
of EIU defined by (1.4)
a = 1,...,N. p ,...,p are called the adapted sections to the trivialization 4.
If (U,x
functions
is a chart on 14, then there are (uniquely determined) on U such that b
(1.5)
na is the canonical frame associated to (U,x1,...,x").
where
The mapping 0 can be prolonged to a connection in the bundle
denoted also by 0.
This prolongation satisfies the following conditions:
Dx(t 8 t') = Dxt 8 t' + t 9
(1.6)
= X(f),
(1.7) (1.8)
=
for all XE X(M), t
t,
and f
the operator of contraction. Let
be sections of E*JU such that
where
pl(x),PN(x)
is
form the dual are the
base to pl(x),...,pN(x) for every point x of U, where From the conditions (1.6) adapted sections to a trivialization of (1.8) we obtain
t
We use the following convention: the Indexes i, i, n, and the indexes a, b, c,... run from I to N.
k,...
run from 1 to
321
pa
— rIb
(1.9)
on M. for any chart Let y:(a,b) + M be a curve of class and let J(E) be the set of all sections of E defined along y, that is, an element of J(E) is a mapping s:(a,b) • E (of class such that 1TOS = y. For every curve y, a connect-
ion D in E defines a mapping
J(E)
J(E)
called the covarlant derivation along of then for a chart D1s =
{d 5a
d
If s
=
5a
° y) is an element
on N we have
i Sb)
(1.10)
+
o y, i = 1,...,n.
where
P:oj,ositlon 1.1 1T
(y(t0)), to
From (1.10) we have:
If y:(a,b) -, M Is a curve and y is an element of
=
(a,b), then there is one and only one section s
such that
s(t0)
y,
= 0.
(1.11) (1.12)
Let y be a fixed element of E and x = n(y). We denote by the set of all velocity vectors (0), where s:(-c, +c) E is a section along y 1to S satisfying the conditions (1.11) and (1.12) with t = 0. Let •:EIU + U x R be a trivialization and let (U,x ,...,x ) be a chart on N.
Now we can define a chart
on E called an induced chart,
where
x1(y) = x1(ir(y)), =
(1.13)
for all y Let al,....an.s1.....6N be the canonical frame associated to the induced chart. If X = is a velocity vector of y and s is the unique section defined along y satisfying the conditions (1.11) and 322
(1.12), then =
(1.14)
-
This implies that: is a vector subspace of
Proposition 1.2
is the subspace of vertical vectors.
ker
where
and
In
is an isomorphism. particular, d?j&:FY Let X be a vector field on M. Using Proposition 1.2 we can define its
horizontal lift xD(y)
by the formula (1.15)
=
It is easy to verify: Proposition 1.3
If X,Y are vector fields on M, and f, g are functions on M,
then
(fx + where
fV
=f°
=
fV xD and
+ 9V
= gori are
vertical lifts of f and g.
From (1.15) and (1.14) we have (1.16)
- X1(1T(y))
for any induced chart on E. 2.
VECTOR FIELDS ON C
a defines a function
be a section of the. dual vector bundle E*. Let a:M -, on E by the formula .
for every point y of E.
(We observe that R.)
is, is a linear mapping to verify
(2.1)
is an element of that Using an induced chart it is easy
323
(y,
c(y) — where
=
.
2.2
is a function of class
Thus
on E.
We have the
following proposition.
If
Proposition 2.1
o' are sections of E* and f, g are functions on 14,
then =
where
fY
=
is
+
gV
the vertical lift of f.
The proof is trivial. This family of functions
is very important to
the study of vector fields on E because we Proposition 2.2
Let and be vector fields of class C°' on E. for every section a of E*, then =
Proof
It is sufficient to show that the equality X(cj) =
a of E* implies X = 0.
0
If X() =
for every section
Let
=
of X with respect to an induced chart on E.
be the we obtain —1
Ia
From (2.2)
+
a
for all functions aa,a 1,...,N, on U. This_implies that for I = 1,...,n and a = 1,...,N, that is, X = 0.
a 0
and
=0
This proposition signifies that vector fields on E are uniquely determined by their actions on the functions of type a, where a is a section of E*. We have:
ProposItion 2.3 If X is a vector field on 14 and a is a section of E*, then
Proof
324
Let a
a
aa
'!Pom (t.16), (2.2) and (1.9) we have
-
=
ab
t'ia
A vector field on E is called projectable on 14 if there Is a vector field X on N such that =X°
o
X is called projection of and X is uniquely determined by L The set of all projectable vector fields on N is a Lie algebra and the projection mapping Is a Lie algebra homomorphism.
We have the following proposition
(3]. 2.4
Let X and
be vector fields on K and E respectively.
is
projectable on M and X is its projection if and only if, for each function f on H, we have =
(Xf)V,
where fe" — foir is the vertical lift of f. A vector field on E is called vertical if, for each point y of E, is a vertical vector, that is belongs to VIE. A vertical vector field on E is projectable on II and its projection is zero. Thus, by Proposition 2.4, we have (see (3]): Corollary 2.5 Let be a vector field on E. = 0 for each function f on M.
X is vertical if and only if
Corollary 2.6 If X is a vector field on N and f is a function on M, then
(Xf)". Since
=
,i1(,T(y)) is a vector space, there is for each point y of E
a natural isomorphism =
-__-
>
(2.3)
325
If
•
E is a section of E, then we can define a vector field E,
s
by the formula (2.4)
(2.5) sections.
generalizes the definitions of vertical lifts of vector fields to the tangent bundle (K. Yano, S. Kobayashi and S. Ishihara [8], [9], [11]) and vertical lifts of 1—forms to the cotangent bundle (K. Yano and E.M. Patterson [12), [13]). Our definition generalizes also the definition of vertical lifts of tensor introduced by J. Gancarzewicz and N. Rahmani (2]. We have
If s, SS are sections of E and f, g are functions on 14, then
Pr.oposition 2.7
(fs +
gss)V
fV sV s
9V
is a section of E,a is a section of E*and f is a
function on 14, then we have
5V(fV)
= 0,
where a•s is the function on P4 defined by the formula (a•s)(x) Proof
From (2.5).and (2.2) we have 5a
6a(Ob
°a
The second formula is a consequence of Corollary 2.5.
Proposition 2.9 If s, s' are sections of E and X is a vector field on then
5'Vj =
326
=
Proof
Let a be a section of E*.
According to Proposition 2.8 we have
_SIV(SV(;)) = 0.
Thue, by Proposition 2.2, = o. According to Propositions 2.8, 2.3 and Corollary 2.6 we have xD(Sv(;)) - sV(XD()) Dx(O.S) —
Usingithe formula Dx(a.s)
we obtain
=
a•DXS
and hence, by Proposition 2.2, Remark
Propositions 2.7, 2.8 and 2.9 generalize the analogical proposition
shown in (2], (3], [8], [9], (10], [11], [12], [13]. We will introduce a new vertical vector field on E using the following proposition. Prpposition 2.10 Let A:E*
be a vec;
bundle homomorphism
which covers
the identity on 14; that is, the diagram A
E*
,ve and "ie restrictions of
is
one and on'y one vector field a of (*, we have there
.. Proof
=Ae
fibres of E* are Ihen on E such that, for every section
a.
The uniqueness of A° is a
Proposition 2.2.
To prove
the existence of A° we consider a vector field •
327.
+
=
with respect to an induced chart. on EIU with coordinates section of E*, by (2.2) we have
For a
?+
=
ab ,
a
A(p ) =Abp
(2.7)
and hence, using (2.2), we have
then Aou = (Ca =
(2.8)
0a
Thus, if we set =
=
(2.9)
= the equality is verified for every section a Thus we have constructed a vector field on EIU such that = A°a for all a. Using two charts(U,x1) and (U',x1 ) we can construct two vector fields X and X' on EIU and EIU' respectively. For any section a of E*I(U n U') = —
(E*IU) n
we have
——
—
X(o) = Aoa
X'(a)
—
and hence, according to Proposition 2.2, X and X' coincide on EI(U n U'). Thus, using an atlas on N, we can define a (global) vector field A° on E = Aoa. such that This construction generalizes the operation y defined by K. Yano, S. Kobayashi, S. Ishihara and E.M. Patterson [8], [9), [11), [12) in the case of tangent and cotangent bundles and also the lift ( )° defined by J. Gancarzewicz and N. Rahmani [23 in the case of tensor bundles. According to (2.9) we have:
A° is a vertical vector field on E.
Corollary 2.11 N, then A° = 328
yb 6a
If (U,x1) is a chart on
with respect to the induced chart, where From Proposition 2.4 we obtain:
are defined by (2.7).
Corollary 2.12 If f is a function on M, then A0(fV) = 0. We have the following properties of A°.
If A, B:E* E* are vector bundle section of E and X is a vector fie'd on 14, then Proposition 2.13
(XD,A0)
= (DxA)°,
(A*oSf',
ISV,AD3
=
CAD,BO)
= (A,B]°,
where A*:E + E is a homeo.norphism of vector bundles such that transposed mapping of A e B — B o A. Proof
$ is a
Let a be a section of E*. [XD,AD)()
Is the
Using Propositions 2.3 and 2.10 we have
= Dx(A00) —
section of E* Q E. Aoo is obtained from A and a by the tensor product and contraction, thus using (1.6) - (1.8) we have A
Dx(A0a)
a
(DxA)0a
+
or
-
(XD,A0](;)
(DxA)oa
= (DxA)°(). Hence, according to PropositIon 2.2, we obtain the first formula. Using 2.8, 2.10 and Corollary 2.12 we have
[$v,Ao](;)
I C s(Aoa)-A((a.s)) =
((sAoa).s)V. 329
On the other hand, from (2.1) and (2.7), we obtain (Aoa).s = Sa(AO) = 5a
°b
= ab(Aos) = a• (A*os) and hence =
=
that is,
15V AD]
=
The verification of the last formula of our proposition is by analogy. Let X and V be two vector fields on M. R(X.Y)
Dx 0
—
o Dx — Dtxy)
We denote by
(2.10)
E*
R(X,V) is called the curvature transformation of the connection 0. From (1.1) - (1.3) (we have the same formulas for sections of E*) we obtain R(X,Y)(a+ a') = R(X,V)cr + R(X,V)o'
R(X,Y)(fa) = f R(X,Y)o
for all sections a, a' of E* and any function f, and hence, R(X,Y) can be considered as a vector bundle homeomorphism R(X,Y):E* ....-._>
The vector
field (R(X,Y))° is important for the characterization of the vertical component of [XD,YD].
Proposition 2.14 =
We have
If X and V are two vector fields on M, then tx,y3D + (R(X,Y))°,
where R(X,V) is the curvature transformation of 0 defined by (2.10). Proof
Let
be a section of E*.
(2.10) we have 330
Using Propositions 2.3, 2.10 and formula
- YD(xD(.))
DDyO)Dy(Dxo) =
R(X,Y)o +
= [X,Y]0— (a) + (R(X,Y)) (a), and hence, using Proposition 2.2, we obtain our formula. 3.
HORIZONTAL LIFTING OF CONNECTIONS TO A NATURAL VECTOR BUNDLE
If q,:M ÷ N is a local diffeomorphism, then we denote by E the induced mapping. For each point x of M, is an isomorphism, where 111(x) is the = E,(X) and fibre of E. By the theorem of R.S.
Let •:E + M be a natural vector bundle.
number r such that, for all local diffeomorphisms •, q,:M M and every pointx of M, the equality = implies IEX = The smallest number r satisfying this property is called order of E. Let r be the order of E. We suppose that r 1. The vector bundle E is isomorphic to an associated fibre bundle with FrM (see (7], (6]), where FrM
a
is the principal fibre bundle of frames of order r, that is, F"M
q
is a diffeomorphism of a neighbourhood of 0 in into some open subset of N).
x F E the canonical Let F be the standard fibre of E. We denote by mapping for the associated fibre bundle E. Let I' be a connection in the principal fibre bundle FrM (r is called connection of order r on H). r determines a horizontal distribution on E.
If y = 4(p,z) is a point of E, then (3.1)
=
= e(p,z). The connection I' determines the covarlant derivation of sections of associated fibre bundles with FrM. In particular, we have the covariant derivation where
E,
D:x(M) x
E3(X,s)
——>
E
331
df sections of E. It is well known that D satisfies conditions (1.1) - (1.3), that is, using the terminology of Section 1, D is a connection in £ (see
(1]). It is easy to verify that the distribution H defined by (3.1) is the same as the distribution r defined in Section 1 for the connection D. Hence, with the the horizontal lift of vector fields with respect to D
usual horizontal lift of vector fields with respect to the connection r of order r on II. + ESM, s
Using = r, be the natural projection, this projection, for a given connection of order r on ti we can Induce a connection of order s, s r. In particular, the given connection r 'of order r on H induces a linear connection on H called linear part of r. We denote by v the covariant derivation of vector fields with respect to the
Let
linear part of r. The main theorem of this paper Is the following one. Let be a connection of order r on H. If w:E + H is a natural vector bundle of order t, then there is one and only one linear connection on the manifold £ such that Theorem 3.1
(32) x V
D
sV =
(3.3)
=0
(3.4)
sy —0
(3.5)
V Is the for all vector fields X, Y on N and all sections s, S of E, covarlant derivation of vector fields on H with respect to the linear part of r and 0 Isthe covarlant derivation of sections of E with respect to r. To
prove this theorem we need the following leura.
Lemma 3,2
on E.
be a llneas Connection
For a chart (U,x1) on H we denote by Va1
332
Let r be a connection of order r on H and
=
ak
+ r1j
(3.6)
a. + 1'ia
aa
= 1'ab a.
+
rai 6b
+
rab 6c
(3.8)
the Christoffel symbols of v with respect to the induced chart on E.
If
conditions (3.2) - (3.5) are satisfied, then (3.10)
=
+
=
1'jb -
rkb)yb
(3.11)
=0
(3.12)
=
(3.13)
=0
(3.14)
=
(3.15)
1'ab =
(3.16)
1'ab =
(3.17)
where rJk are the
symbols of the linear part of r and rlb are
defined by (1.5). Proof
Let
be the adapted section of E to the induced chart. According to (2.5) we have V
(3.18)
=
Now formulas (3.18), (3.5) and (3.9) imply (3.16) and (3.17). Next from (1.16) we have —
(3.19) 333
and hence, using (3.16), (3.17), (3.8) and (3.4), we obtain (3.14) and (3.15). Using (3.7), (3.19) and (3.18) we can calculate —
v
V
-r.
b
=v o —
y
c
V
a
=
On the other hand, using (1.5) and (2.5) we obtain b
V
'5b
Thus the equality (3.3) implies (3.12) and (3.13).
Finally, using (3.19)
and (3.12) — (3.17) we calculate V
a.
D
=
j aj
äb
a
—
d C
°a
=
=
c
ya
6b
Hence, formulas (3.7) and (3.2) imply (3.10) and (3.11).
The proof of our
is finished. The uniqueness of a linear connection V on E satisfying conditions (3.2) - (3.5) is clear because, according to Lenina 3.2, the Christoffel symbols of V are uniquely determined by the given connection r Proof of Theorem 3.1
of order r. Thus we need to prove only the existence of V. Let (U,x1) be a chart on H. We can define a linear connection v on EtU such that its Christoffel symbols with respect to the induced chart are verifies given by formulas (3.10) - (3.17). This linear connection V on the conditions
334
—
¼Va1
D
3j
—
V
—
D3
I
0
—
V
V
•
(3.20)
0
a1
V
V
Pb
for i,j = 1,...,n and a,b = 1,...,N. Using the propositions of Sections 1 and 2 it is easy to prove that 0
—
x
x
D
(3.21)
S
=
o S
S
for all vector fields X, Y on U and all sections s, s' of EIU. We show only the first formula of (3.21). Let X and V be vector fields on U. If we denote by X1
X
V=
a1,
a1
the coordinates of X and V with respect to the chart (U,x1), then according to Proposition 1.3 we have 3D
x0
=
(yl)V 3D
and hence, using Propositions 1.3, Corollary 2.6 and the first equality of (3.20), we obtain (xi)V {aD((yi)V)
y0 x
D
(y.3)V
+ J
{(ay.J)V
=
= =
{X'
+
(yJ)V
(V
a)Dl
aJ}D
+ y3 Va 1
335
The other formulas of (3.21) are verified by analogy. Tf (U,x') and (U',x' ) are two charts on M, then we can define two linear respectively on EIU and Elu'. From (3.21) we have v and
D_m
D
x
D
x
sV
= v0
= (D
x
x
=o
x0
=
0
for all vector fields X, Y on U fl U, and all sections s, s' of EI(U n U') = (E
)
Hence, by Lefihia 3.11 the linear connections
n
and
coincide on EI(U n Ii'). Using an atlas onM we can define a linear connection on E. This connection verifies the conditions (3.2) — (3.5) and the proof is complete. The linear connection on E verifying conditions (3.2) - (3.5) is called the horizontal lift of r from 14 to E. The following three corollaries are imediate consequences of Theorem 3.1.
Corollary 3.3 (K. Yano, S. Ishihara [10], [9)). If v is a linear connection on M, then there is one and only one linear connection v on TM such that VH V
=
VH
xv
yV
=
(vy)v
xv
for all vector fields X and Y on M, where
xH
is the horizontal lift of X to
TM with respect to V..
Corollary 3 4 (K Yano, E.M. Patterson [13], (9)). If V is a linear connection on M, then there is one and only one linear connection on such that =
x
H
H.
V(DVW
336
(v4)V
H
x V
=0
for all vector fields X, V on N and all 1—forms horizontal lift of X to T*M with respect to V.
on M, where
is the
Corollary 3.5 (J. Gancarzewicz, N. Rahmani [3]). If E is a vector bundle associated to the principal fibre bundle IN of linear frames and V is a linear connection on N, then there is one and only one linear connection on E such that ,H_
—
x
H
H
-
'
V
—
S
x
H
—
x'l
=a
for all vector fields X, V on N and all sections s, s' of E, where the horizontal lift of )C to E with respect to v.
is
Next we will study the torsion tensor and the curvature tensor of the horizontal lift of a connection of order r to any natural vector bundle of order r (r is arbitrary). We have the following properties of these tensors.
ition 3.6 Let E be a vector bundle associated to and let r be a connection of order r on N. If is the horizontal lift of r to E and is the torsion tensor of v, then we have =
(T(x,v))0 — (R(x,V))°
=
=0
for all vector fields X, V on N and all sections s, of E, where T is the torsion tensor of the linear part of v and R(X,Y) is the curvature transformation of r defined by (2.10). Proof
Using Theorem 3.1 and Proposition 2.10 we have =
x = =
0
-
(vxY)° -
V
D
- EX,Y]D -
(T(x,Y))D - (R(X,Y))°.
Next, using Theorem 3.1 and Proposition 2.9 we obtain 337
x
=
- [xDSV]
sV -
= D
(Dxs)V - (Ds)V = 0
=v
- [SVsSV)
S'
S
= 0.
To calculate the curvature tensor of V we need the following lema. Lema 3.7 If V is the horizontal lift of a connection of order r to E and A:E* + is a vector bundle homomorphism, then
A
for every vector field X on M and every section s of E. Using an induced chart, according to Corollary 2.11 and formula (2.5)
Proof we have
'i
.,b
ia
tS
b
)
we obtain
and hence, by
A°
-
=
=0 A°
—
-
=
=0 Now we have
Proposit on 3 8 If V M to d vecto bundle then
338
is the horizontal lift of a
on
with FrM and R is the curvature tensor of v,
(r(X,Y)Z)0 (R(X,Y)s}V =
=0
for all vector fields X, V. Z on M and all sections s, S' is the curvature transformation of r defined by (2.10) and r(x,'. curvature tensor of the linear part of r. Proof
Using Theorem 3.1, Proposition 2.14 and Lemma 3.7 we have 10) -
=
x
=
V
-
D V
X
(vx(vyZ))D —
(X,Y)
-
(R,X,Y))D =
(r(X,Y)Z)D,
D
X
sV) -
V
D 0 [X,YJ
D V
X
(Dy(oxs))" -
y]S)
S
(R(X,Y) )° =
Using Proposition 2.9 we can calculate 0
,
0
0
for all vector fields X, V. Z on M and all sections s, s',
of E.
This
remark finishes the proof of our proposition. From Proposition 3.6 we have:
339
.Progosltion 3.9 Let V be the horizontal lift of a connection r of order r to a vector bundle E associated with FrM. If the linear part of r is without torsion, then Is without torsion If and only if the curvature transformation R(X,Y) is zero for all vector fields X, V on N. From Proposition 3.8 we have:
Proposition 3.10 Let be the horizontal lift of a connection r of order r on N to a vector bundle E associated with FrM. Then the linear connection
= 0) if and only if the curvature transformation R(X,Y) of r defined by (2.10) is zero for all vector fields X and Is without curvature (that is, YonM.
To prove this proposition it is sufficient to observe that if the curvature transformation R(X,Y) of r is zero then the linear part of r Is without curvature. Propositions 3.9 and 3.10 generalize the analogic propositions proved by
K. Yano, S. Ishihara in the case of tangent bundles (10], (9], by K. Yano, E.M. Patterson In the case of cotangent bundles (13], [9) and by .3. Gancarzewlcz, N. Rahmani (3) In the case of a vector bundle associated with the principal fibre bundle of linear frames. References
[1) Crittenden, R.
Covariant differentIation, Quart. J. Math. Oxford (2) F
(1962) 285-298. Gancarzewicz, J. Connections of order r, Ann. Pol. Math. IV (1977) 13
(2]
69—83.
(3)
Gancarzewlcz, .3., and Rahmanl, N.
Relbvement horizontal des connexions
au fibre vectoriel associd avec le fibre principal des lindaires (in press). •
[4] Gancarzewicz, .3., and de type (1,1) au fibre C (5)
horlzontaux des tenseurs N. TN e T*M (in press). Kobayashi • S. and Nomizu, K. Foundations of Differential Geometry,
vol. 1 New York (1963). Nijenhuis, A. Natural bundles and their general properties, Diff.
in honor of (7]
K. Yano, Tokyo
Palais, R.S. and Terng, C.-L. Topology 16 (1978) 271—277.
340
(1972) 317-334.
Natural bundles have a finite order,
Geom.
[8]
Yano, K. and Ishihara, S.
Differential geometry in tangent bundles,
Kodaj Math. Sam. Rep. 18 (1966) 271-292.
(9] Yano, K. and Ishihara, S.
Tangent and Cotangent Bundiea, Marcell Dekker
Inc. New York (1973).
(10) Yano, K. and Ishihara, S. Horizontal lifts of tensor fields and connections to tangent bundles, J. Math, and Mach. 16 (1967) 1015—1030. [11) Yano, K. and Kobayashi, S. Prolongations of tensor fields and connections to tangent bundles, J. Math. Soc. Japan, 19 (1967) 185-198. [12] Yano, K. and Patterson, E.M. Vertical and complete lifts from a mani-
fold to its cotangent bundle, J.
Math. Soc. Japan, 19 (1967) 91—113.
[13] Yano, K. and Patterson, E.M. Horizontal lift from a manifold to its cotangent bundle, J. Math. Soc. Japan, 19 (1967) 185-198. Jacek Gancarzewicz
Uniwersytet ul. Reymonta 4, p.V Krakdw, Poland
341
