Géométrie Différentielle: Colloque, Santiago de Compostela Espagne 1972

230 Pages·1974·2.142 MB·French-English

by  Luis A. Cordero (auth.), Enrique Vidal (eds.)

Lecture Notes ni
Mathematics
Edited yb .A Dold, Heidelberg and .B Eckmann, Zerich
392
Geom6trie Diff6rentielle
,euqoiloC ogaitnaS ed aletsopmoC
engapsE 1972
Edite rap Enrique Vidal
Springer-Verlag
Berlin-Heidelberg • New York 1974
euqirnE ladiV
dadisrevinU de de Santiago ,aletsopmoC
Santiago ed Compostela/Espagne
AMS Subject Classifications (1970): 53-XX, 22-XX, 28-XX, 55-XX,
57-XX, 58-XX
ISBN 3-540-06?97-3 Springer-Verlag Berlin • Heidelberg - New York
ISBN 0-38?-06?97-3 Springer-Verlag New York • Heidelberg • Berlin
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© by Springer-Verlag Berlin • Heidelberg 1974. Library of Congress
Catalog Card Number ?4-?908. Printed in Germany.
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Preface
Ce volume contient Iss Conferences prononc~es au Colioque de
G~om~trie Di~rentlelle organis~ par le D~partement de G~om~trie st
Topologle de l'Onlversit@ de Santiago de Compostela (EspaKne) du 18
au 12 octobre 1972. Le oolloQue a b~n~ici@ du support du Minist@re
de l'Education et la Science, du Rectorat de l'Universit~ de Santiago
ainsi qua de la Facult~ des Sciences de carte unlversit~. A tous nous
exprimons notre reconnaissance. Je remercie ~galement tousles parti-
cipants oui ont asslst~ aux discussions, pr~sent~ des conferences de
grand int~r~t et dont la g~n~rosit~ a 9aoillt~ leur d~placement
Santiago. Merci encore ~ tous lee Dro~esseurs du D~partement qui ont
collabor~ avsc enthcusiasme ~ l'organisation du ColloQue; Je voudrais
signaler sp~clalement Qua is pro~esseur A.M. Naveira a oarteg~ avec
moi le plupart des t~ches de l'organisation, et remercier Mme. Peraza
qui a dactylographi~ avec ~rand spin ce manuscrit. Finalement Je
remeroie lee ~diteurs des "Lecture Notes In Mathematics" d'avoir
accspt~ cette publication dams leur s~rie.
Enrique Vidal
Santiago de Ccmpostela 1972
Table des Mati~res
I. STRUCTURES PRESQUE-PRODUIT ET FEUILLETAGES
°] L.A.CORDER0, On p-normal almost-product structures ...... I
2. C.GODBILLON, ~euilletages de Lie . . . . . . . . . . . . . . . 10
3. G.~ECTOR, Actions de groupes de diff~omorphismes de ~0,I). . . 14
4. R.LUTZ, Structures de contact en codimension quelconque 23
5. J.NARTINET, Classes caract~ristiques des syst~mes de Pfaff . 30
6. R.~OUSSU, Sur les classes exotiques des feuilletages ..... 37
7. G.REEB, Apropos de l'~quation (de Painlev~)
y" = 6y 2 + ~ . . . . . . . . . . . . . . . . . . . . . . . . . 43
8. B.L°REINHART, Holonomy invariants for framed foliations • . 47
9. R.ROUSSARIE, Ph~nom~nes de stabilit~ et d'instabilit~ dans les
feuilletages . . . . . . . . . . . . . . . . . . . . . . . . . 53
II. C OURBURE
10. A.M.NAVEIRA - C.FUERTES, Les zeros des op~rateurs courbure
holomorphes non-negatifs . . . . . . . . . . . . . . . . . . 6]
11. D.C.SPENCER, Curvatures associated with differential
operators . . . . . . . . . . . . . . . . . . . . . . . . . 64
III. SOUSVARIETES, CALCUL DES VARIATIONS ET APPLICATIONS ARMONIQUES
12. H.GOLDSCHMIDT, Le formalisme de Hamilton-Cartan en calcul
des variations . . . . . . . . . . . . . . . . . . . . . . . 76
13. J.-C.MITTEAU, Existence des applications harmoniques .... 90
14. R.THOM, Sur le cut Locus d'une vari~t~ plong~e (R~sum~) . . . 97
15. A.G.WALKER, A certain conformal structure . . . . . . . . . . 98
16. T.J.WILLMORE, Minimal conformal immersions . . . . . . . . . ]11
IV. GROUPES ET ALGEBRES DE LIE
]7. A.KUNPERA, Invariants diff~rentiels d'un pseudogroupe de Lie 121
18. A.LICHNEROWICZ, L'alg~bre de Lie des automorphismes
symplectiques . . . . . . . . . . . . . . . . . . . . . . . . 163
VI
19. J.SANCHO SAN ROMAN, Sur des groupes associ@s A certaines
vari~t~s tridimensionelles . . . . . . . . . . . . . . . . . 175
V. MESURES INVARIANTES
20. R.SACKSTEDER, The measures invariant under an expanding map . 179
VI. CONVEXITE
21. J.J.KOHN, Convexity and pseudo-convexity . . . . . . . . . . 195
VII. NOEUDS
22. J.L.VIVIENTE, Sur le module deriv@ d'un homomorphisme .... 203
.I STRUCTURES PRESQUE-PRODUIT ET FEUILLETAGES
ON P-NORMAL ALMOST-PRODUCT STRUCTURES
Luis A. Cordero
NOITCUDORTNI
nA structure, no na n-dimensional differentiable
manifold ,V given by a non-null tensor field f, of cons-
tant rank r, satisfying
f3 + f = 0
is said na f-structure. If n = r the f-structure defines
na almost-complex structure dna r is even; if V is na
orientable manifold dna r = n - I, the f-structure defines
no V na almost-contact structure na r is odd.
If f defines na f-structure no V dna n-1)r, there
exists no V two regular distributions I D dna D 2, corres-
ponding respectively to the projection operators m = f2+I
dna l = _f2; f acts no I D sa the zero tensor mf( = fm = )O
dna induces no 2 D na almost-complex structure. Ishihara ({4})
constructs na almost-complex structure no the total space
DI(V) of the vector bundle I n = (DI(v),HI,v), which is ca-
I
nonically defined from f dna a linear connection *m no n
ehT f-structure f is said normal with respect to *m if that
almost-complex structure is integrable.
Here, ew explain a similar construction starting from
na almost-product structure no the manifold ,V which per-
mits su to introduce the notion of P-normal almost-product
structure; besides, owt examples of manifolds admiting -P
normal almost-product structures dna na aplication to the
theory of foliations are given•
morF won on, all manifolds, tensor fields, dna os forth
are demussa to eb differentiable of class C .®
§I.- P-normal almost-product structure
Let V eb na n-dimensional differentiable man%fold, en-
dewod with na almost-product structure ,H 2 = H I, P dna Q being
the projection operators of the structure, 1 = D mI ,P 2 = D mI ,Q
mid 1 = D p, mid 2 = D q, q+p = n.
1
Let su consider the vector bundle of dimension p, n =
1
{DI(v),nl,v); a local tri?vialltation of n is given sa follows:
let U eb a local coordinate neighborhood of V; no ,U the pro-
jectors P dna Q are expressed yb
p = xd@j~.~p i
Q = xd@j@~q i
being i@ = B/@xi dna (x I ,x )n the coordinate functions no U
, , . ,
For each x~U, ew take n vectors X{ i} defined yb I)
aX = P~i
uX q iB
where X{ } is a basts of ID dna X{ u} is a basis of .~D eW shall
a- i " x
denote I e = yjdx J the dual basis of X{ i} Every point o~{nl)-1(U)
)1 ehT ind~ces nur over the rank sa follows:
1~i,j,k. ... n~
l~a,b,c,... p~
p+lsu,v,w,... n~
is uniquely expressed yb =~ ~aXa; then, the functions (xl,...
1 xn,~ ..... )P~ form a systemo f coordinate functions no (~1)-1(U)
dna , thus, DI(v) sah na structure of (n+p)-dimensional dif-
ferentiable manifold.
yB identifying the tangent space at a point of the fibre
of I n with the fibre itself, the tangent ecaps T (DI(v)) of DI(v)
o
al a point ~ nac eb expressed yb
To(DI(v)) = Tx(V) x 0 F x x = 2 D 0 I D 0 x F
erehw x = nl(~) dna T (V), x 2 F D dna 1 D denote the tangent ecaps
' X X
of ,V the fibre of I D V), the tangent plane belonging to 2 D dna
the tangent plane belonging to D ], respectively. Besides, there
exists a natural identification
j:D 1 ~ F
X X
Let *~ eb a linear connection no n .1 If XsSx(V), ew write
h X its horizontal lift with respect to ~* to each point ~ of
(~I)-I(x). ,woN ew define a linear operator F applied to the tan-
gent ecaps T {DI(v)), at each point ~sDI(v), yb
X(oF )h = 0
hy(sF ) : j(Y)
Z(aF ) = .(j-l(z))h
erehw ZsD~, YsD~, F~Z x dna x = nl(o). It is easily verified that
the operators ° defined F in each tangent space To(DI(v) determine
na f-structure F of rank p2 no the manifold DI(v), i.e. that
3 + F F = .O
If a r b~ era the local stnenopmoc of *m in ,U the tensor field
of type (1,1) defined above fs represented in (x1)-1(U) yb
I j a a pJs~r~ )
-Pari iY + b , j
~ F = (*)
B
_p~ i a
irbP
erehw ~ I = a rib~ b dna ~,B,... nur over eht set {1,...,n,n+1,...,n+p}.
MEROEHT I
I f a differentiable manifold V admits an almost-product
structure H, then there exists an f-structure F of rank 2p in the
I
total space of the vector bundle n ; given a linear connection ~*
on n with local componentsr~b, F = (F) is determined by (~1.
KRAMER I.-
A similar construction is possible no eht total space of
eht vector eldnub 2 = n (D2(V),~2,V).
KRAMER 2.-
meroehT I is similar to that eno of Ishihara ({4}); if eht
almost-product structure is given yb = H I+2f ,2 f being na f-struc-
ture no eht dlof~nam ,V then F is eht almost-complex structure
defined yb Ishihara f% ew tup F (xh):(fX) ,h for hcae .~D~X
REMARK 3.-
1
If 1=q dna the total space DI{V) ~o n s~ na orientable
,dlof%nam ew nac assert the existence of na almost-contact struc-
ture over DI(v).
DEFINIT!ON I
The almost-product structure H is ~aid P-normal with res-
pect to ~* if t~e f-structure F is integrable.
sA it s% well ,nwonk eht f-structure F is integrable if
dna only if the Nijenhuis tensor
)Y,X(N : [FX,FY1- F~.X,FY} F[FX,Y1 + F2[X,Y]
vanishes identically.
Then. the following theorem is proved:
MEROEHT 2
A necessary and sufficient condition for an almost-
product structure be P-normal with respect to ~* is that
the tensor fields i S jk and i S vanish identically and the con-
nection ~* be of zero curvature.
There, the tensor fields !S jk dna i S are defined sa
follows; in a coordinate neighborhood U of V !S is a ten-
' jk
sor field of type (1,2) given by:
i S = (ajya a i b a b a i
jk k - akYj)Pa - (Yjrkb - Ykrjb)Pa
dna i S is na n( 1. @ n1*)-valued tensor field of type (1,0)
given by:
i S i a be
= ebaS
erehw
~S b 1 i_p~ai Ic Ic i
= PaalPb aPI - (Parlb - Pbrla)P c
1. 1
n being the dual of n •
§2.- selpmaxE
elpmaxE 1.-
Let V eb na n-dimensional differentiable manifold dna