L A Cordero (Elitor) University of Santiago.de Compostela
Differential geometry
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L A Cordero (Elitor) University of Santiago.de Compostela
Differential geometry
Pitman Advanced Publishing Program LONDON MELBOURNE
PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts
02050
PIThIAN PUBLISHING LIMITED 128 Long Acre, London WC2E 9AN Associated Companies Pitman Publishing Pty Ltd. Melbourne Pitman Publishing New Zealand Ltd. Wellington Copp Clark Pitman, Toronto
© 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
All tights reserved. No part of this publication may be reproduced. stored in a retrieval system, or transmitted, in any form or by means, electronic, mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publishers This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers.
Reproduced and printed by photolithography in Great Britain by BiddIes Ltd. Guildiord
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
-
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 + 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 + BT1, (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
=
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, 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
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 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
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
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
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
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
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
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