Differential Geometry: Proceedings of the International Symposium held at Peñíscola, Spain, October 3–10, 1982

200 Pages·1984·2.06 MB·English

by  Stig I. Andersson (auth.)

Lecture Notes in
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Edited by A. M. Naveira
Springer-Verlag
Berlin Heidelberg New York Tokyo 1984
Editor
Antonio M. Naveira
Departamento de Geometria y Topologia Facultad de Matem~ticas
Burjasot, Valencia, Spain
AMS Subject Classifications (1980): 22E, 53A, 53B, 53C, 57R, 57S, 58A,
58 B, 58 C, 58 G, 18 E
ISBN 3-540-12882-4 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-387-12882-4 Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Main entry Data. under title: Differential geometry.
(Lecture notes in mathematics; 1045) nI English and French.
Proceedings of the Symposium on Differential Geometry. .1 Geometry, Differential-Congresses.
.I A.M. Naveira, (Antonio 1940-. Martfnez), .II Symposium on Differential Geometry (1982:
Spain) PeSfscola, III. Series: Lecture notes in mathematics 1045. (Springer-Verlag);
QA3.L28 .on 1045 [QA641] 510s [516.3'6] 83-20457
ISBN 0-387-12882-4 ).S.U(
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PREFACE
The present volume includes the texts of all lectures given at a
Symposium on Differential Geometry which was held at Pe~fscola, Spain,
from October 3 to i0, 1982. The Symposium was attended by some forty
mathematicians from all over the world.
There have been five International Symposia on Differential Geometry
in Spain during the last twenty years. One of them took place in Sala-
manca (1979), and the remaining four in Santiago de Compostela (1962,
1967,1972,1978). Of those, the first three ones were organized by Prof.
E. Vidal Abascal and the last one was held in his hommage at his reti-
rement. Our wish when organizing this Symposium has been to continue
with this tradition initiated by Prof. Vidal Abascal to whom we express
our deepest gratitude for his exemplary contribution to the development
of Differential Geometry in Spain. We hope to be able to continue with
this tradition by holding these meetings periodically.
The Organizing Commitee of this Symposium is glad to express its grati-
tude to all those who contributed to the success of the meeting and in
particular to all participants. We also wish to express our sincere
thanks to the Ministerio de Educaci6n, to the Facultad de Matem~ticas
de Valencia, to the Diputaci6n Provincial de Castell6n for their gene-
rous financial support, and to the Instituto de Administraci6n Local
de Pen~scola for allowing us to use their premises during the
Symposium. The facilities given by Springer-Verlag, and the co-operation
of Prof. S. I. Andersson in our early contacts with the publishers are
also gratefully acknowledged.
Finally we wish to thank F. Marhuenda and J. Monterde for their careful
typing of most of the manuscripts.
A. M. Naveira
President of the Organizing Commitee
LIST OF PARTICIPANTS
E. Abbena A. Gray
U. Torino, Italy U. Maryland, U.S.A.
.S I. Andersson J. Grifone
U. Clausthal, West Germany U. Toulouse, France
M. Asorey R. Langevin
U. Zaragoza, Spain U. Dijon, France
M. A. Baratta A. Lichnerowicz
U. Parma, Italy Coll~ge de France, France
M. Bendala F. Mascar6
U. Sevilla, Spain U. Valencia, Spain
D. Bernard V. Miquel
U. Strasbourg, France U. Valencia, Spain
J. L. Cabrerizo A. Montesdeoca
U. Sevilla, Spain U. La Laguna, Spain
F. Carreras A. Montesinos
U. Valencia, Spain U. Valencia, Spain
.D Chinea A. M. Naveira
U. La Laguna, Spain U. Valencia, Spain
L. A. Cordero M. Nicolau
U. Santiago, Spain U. Aut. Barcelona, Spain
C. Curr~s-Bosch G. B. Rizza
U. Barcelona, Spain U. Parma, Italy
S. Donnini A. H. Rocamora
U. Parma, Italy U. Pol. Valencia, Spain
F. J. Echarte C. Romero
U. Sevilla, Spain U. Valencia, Spain, and
Southampton, United Kingdom
J. Etayo
U. Madrid, Spain A. Ros
U. Granada, Spain
M. Fernandez
U. Santiago, Spain M. Sekizawa
U. Tokyo, Japan
M. Fernandez-AndrOs
U.zSevilla, Spain R. Sivera
U. Valencia, Spain
A. Ferr~ndez
U. Valencia, Spain F. Torres-Lopera
U. Santiago, Spain
E. Fossas
U. Barcelona, Spain L. Vanhecke
U. Leuven, Belgium
.S Garbiero
U. Torino, Italy F. Varela
U. Murcia, Spain
O. Gil Medrano
U. Valencia, Spain J. L. Viviente
U. Zaragoza, Spain.
J. Girbau
U. Aut. Barcelona, Spain
TABLE OF CONTENTS
S.I. ANDERSSON. Pseudodifferential operators and characteristic
classes for non-abelian cohomology ......................... I
M. ASOREY. Euclidean Yang-Mills flows in the orbit space ......... 11
D. BERNARD. Congruence, contact et rep~res de Frenet ............. 21
C. CURRAS-BOSCH. Killing vector fields and complex structures .... 36
J.J. ETAYO. Derivations in the tangent bundle .................... 43
J. GIRBAU. Some examples of deformations of the transversely
holomorphic foliations ..................................... 53
J. GONZALO et F. VARELA. Sur certaines expressions globales d'une
forme de contact ........................................... 63
J. GRIFONE et F. HASSAN. Connexions singuli~res et classe de
Maslow ..................................................... 71
F. HASSAN. See J. GRIFONE.
A. KUMPERA. Sur la cohomologie des syst~mes d'equatiens diff6-
rentielles et des pseudogroupes de Lie ..................... 84
R. LANGEVIN. Energies et g6om6trie int6grale ..................... 95
A. LICHNEROWICZ. Geometry and cohomologies associated with a
contact manifold ........................................... 104
J.F.T. LOPERA. A note on semisimple flat homogeneous spaces ...... 117
F. MASCARO. Some results on Diffg(~ n) ........................... 125
A. MONTESINOS. Some integral invariants of plane fields on
riemannian manifolds ....................................... 134
A.M. NAVEIRA. A Schur-like Lemma for the NK-manifolds of
constant type ............................................... 142
M. NICOLAU and M. REVENTOS. Compact Hausdorff foliations ......... 147
A. REVENTOS. See M. NICOLAU
G.B. RIZZA. Nijenhuis tensor field and weakly K~hler manifolds... 154
C. ROMERO. Sphere stratifications and the Gauss map .............. 164
A. ROS. Spectral geometry of submanifolds in the complex
projective space ........................................... 182
ILIV
F. TRICERRI and L. VANHECKE. Self-dual and anti-self-dual
homogeneous structures .................................... 186
L. VANHECKE. See F. TRICERRI.
F. VARELA. See J. GONZALO.
PSEUDODIFFERENTIAL OPERATORS AND CHARACTERISTIC
CLASSES FOR NON-ABELIAN COHOMOLOGY
Stig I. Andersson
Institut f~r Theoretische Physik
der Technischen Univ. Clausthal
D-3392 Clausthal-Zellerfeld, FRG.
0. Introduction
The object of study in this work is the interplay between analytic
properties of pseudodifferential operators (psdo) on vector bundles and
the geometry of the vector bundles themselves.
Let X be a connected smooth n-manifold and EI------~ X, 2 E ) X and E ) X
h P~ P
vector bundles of dimensions ml, 2 m and m respectively. By S (El) , S (E 2
and S(E) we denote the smooth sections and PDiffk(Ei,E2 ) (Diffk(Ei,E ) 2
stands for the psdo (partial differential operators, pdo) of order k,
mapping S(EI) ~ S (E2) .
Given P~ PDiffk(Ei,E2) , by a lifting process we shall attach a connection
operator Q: S (El®E) > S (E2®E) . Furthermore, by a procedure analogous to
the Bott-Chern-Weil construction, we shall develop a naturally associated
theory of characteristic classes, based however on a non-abelian
cohomology theory.
Modulo problems in the relevant homological algebra, forcing us to
consider only the one and two dimensional cohomology sets, this provides
a bridge between analytic properties of P and the vector bundle geometry.
The construction extends the one of Asada'~AS i) for local operators.
i. Asada Connections for Diffk(Ei,E2 )
To provide motivation for our later construction, we shall briefly
review the essential steps in the construction of an Asada connection
to a given pdo.
P ~ Diffk(Ei,E ) 2 being a local object, taking restrictions to open sets
is a transitive operation, hence the commutative diagram ( U open set)
S(E1 ) P % S(E2)
S (EI~ U) ) S (E2~ U)
P~U
Let (Ui,~i) be a locally trivializing atlas for E1and 2 E on X and write
Pi: =P~U i. A local trivialization for I E Pl ~ X is then a VB-equivalence
i
F.
-I i (U )x ~ ml
Pl (Ui) > ~i i
i U ) ~i(Ui )
-I
where F~ : Pl (U.) ) U x C ml ~ ~i(Ui)x C ml defines the
1 l
(~i x Id)
fiber chart.
For the sections we obtain the induced isomorphisms
Z 1 -1 2
i : S (Pl (Ui) )~ (~i(Ui)~ ml ; Zi : S (P~I (Ui) ) S[~" (~i (Ui))]m2
(and analogously for S the sections with compact support). So locally
c
P induces the operator p : (U~) ml ) (Ui) m2 , Ui:=~i(U i) C Rn
by
(Z)-i Pi Zi
For two charts Uiand 3 U • with U lj : =Ui~ U. 3 ~ ~ , we thus obtain the
following commutative diagram, describing the compatibility condition
on U
13
ln(s[> ]
m,,..,
.__~ SI (UI)] I m Pi
izT
;t,,
tij S(E ) I
p,
J
with the transition functions -}t . :: Z .I (zl) j -I ' t~ lj :: io 2 z (z2)j -I for
3 l l
l E and 2 E respectively.
Define the (index-preserving) lifting ~ of P by ~i:S (EI®E~Ui)----) S (E2®E~U i)
such that-~'k(~): (~k(P)®Id E (~'k(P) = symbol of P etc) . Explicitly in
a chart, writing Pi a i , D~ (~ a multi-index) we define the
lifting to be
Pi:= ~--'~ (a~, i ~ Id E) D i-
i,,~|.~ k
In general the compatibility requirements are violated though for the
lifting, i.e.
~i ~j ~j j ij
where T[ .:=I tl .... ~e , T~ . :: . . t~ . . .@e , (e : transition function for E)
lj 13 13 lj z3 13 lJ
are the transition functions for EI®E and E2®E respectively.
Definition: Q ={Q) with Qi ~ Diffr (EI®E~Ui,E2~E~Ui) for r < k, is an
Asada E-connection of P iff(P + Q) ~ Diffk(EI®E,E2®E) i.e.
, , 1 = T .2 o (~'. + Q~) on uij.
(~i + Qi )° Tij lj 3
The obstruction Wij: : Q' i1 T - T ij lj .2 Q~ can be simply computed. Putting
Q~ k~.,~= ~-I q~,i i'~D ' we obtain Wij =~ e~ D i~ where
e~ = ~ (a~ , ~ i Id E) tij®eij - i D (tij@eij) =
= ~ (~) D~'T I - 2 T (Jac . ~) with Jac..= the Jacobian
f~|~(~k-I q°~'i " ij ij q~ ,j iJ ' lJ
for the change of coordinates.
On the symbol level, ~'k_i (Wij) : ep ~ .
Assuming 0 = ~k_l (W) = . ...... = (~'k_j ; (W) there exists an E-connection
of P of order ~ k-(j+2) iff ~" (W) : 0.
k-(j+l)
Here ~'k_j (W) = {~k_j (Wij ~ and we call ~'k_ (j+l) (W) the ~bstruction
of order k-(j+2) .
Computing, we obtain explicitly; assuming 0 = ~k_l (Wij)=....= ~-k_s+l (Wij)
~'k-s (W' )~ = - ~ (ao(,i ~iNldE ) ij
l] ~p|=k-s (k- '~l41~,.
:= - s M .. (e..), defining the order s differential operator s M . .
13 13 13
locally.
Example: s=l, (~'k_l (Wij)=- ~ [ ~. (a~+ I ~@i i ~ ---~---~ )] T! .
ipi:k-1
i:i i' ~x(i)l ~3
: : - i M1(e )(t~ ij j~IdE)"
The Asada construction is now completed by the following series of
observations;
i 0. ~-k_s(Wij) is a smooth section in the bundle G k_s~Uij where
*T(S-ks~)2E,IE(moH .)E,E(moH@))X(
~k_s :=
20" ~'k (W.. is independent of Q for order (Q)~ k-s.
-s 13
30 • ~'k_s (W) = 0 in S(~k_s)~ there exists an order k-(s+I)
E-connection of P.
40 "
~'k_s(Wlj) is a special 1-cocycle defines a Mittag-Leffler or
Cousin I-distribution) since
(~'k_s(Wij)T1 jr + T~ lj~'k-s(Wjr) ~--k - s(Wir ) on Uijr::U i U. 3 r U ,
and
1 T
T~i~-k-s (Wij) ji = ~--k_s (Wj ) on Uij.
It is now natural to view s M = [M~j (')] as a differential operator
(of order s) on the sheaf S(Hom(E,E) ) with the image sheaf
Ran(M C ) s S (X, ~k_s) . In particular, 0-k_s (Wij) ~ Ran(M~.) .13
Being a l-cocycle, ~-k_s(Wij) determines an obstruction class,