Geometry Seminar “Luigi Bianchi”: Lectures given at the Scuola Normale Superiore, 1982

182 Pages·1983·2.287 MB·English-French

by  Graziano Gentili, Simon Salamon, Jean-Pierre Vigué (auth.), Edoardo Vesentini (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and B. Eckmann
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Graziano Gentili
Simon Salamon
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Geometry Seminar igiuL" "ihcnaiB
Lectures given at the Scuola Normate Superiore, 1982
Edited by .E Vesentini
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Berlin Heidelberg New York Tokyo 1983
Authors
Graziano Gentifi
Simon Salamon
Scuola Normale Superiore
Piazza dei Cavalieri 7, 56100 Pisa, Italy
Jean-Pierre Vigue
U.ER. de Mathematiques, Universit6 de Paris VI
4 Place Jussieu, ?5230 Paris Cedex 05, France
Editor
Edoardo Vesentini
Scuola Normale Superiore
Piazza dei Cavalieri ?, 56100 Pisa, Italy
AMS Subject Classifications (1980): 53 B, 53 C, 32 A, 32 C, 32 M, 46 A, 15 K
ISBN 3-540-12719-4 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-387-12719-4 Springer-Verlag New York Heidelberg Berlin Tokyo
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INTRODUCTION
The geometry seminar held during the Winter and Spring of 1982
at the Scuola Normale Superiore consisted mainly of three cycles of
lectures delivered by Graziano Gentili, Simon Salamon and Jean-Pierre
Vigu~.
The texts published here are amplified versions of those lectures.
E.V.
CONTENTS
GRAZIANO GENTILI: DISTANCES ON CONVEX CONES
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Preliminaries and formulation of the problem . . . . . . . . . . . . 3
I. Order and Carath~odory pseudo-distance . . . . . . . . . . . . . 7
+ +
2. Special distances on ~. x ~. . . . . . . . . . . . . . . . . . 13
3. Classification of all special pairs of distances
in dimension greater than one . . . . . . . . . . . . . . . . . 22
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
SIMON SALAMON: TOPICS IN FOUR-DIMENSIONAL RIEMANNIAN GEOMETRY
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
I. Elementary Representation Theory . . . . . . . . . . . . . . . 36
2. Representations of SO(4) . . . . . . . . . . . . . . . . . . . 42
3. Spin Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 49
4. Connections and Curvature . . . . . . . . . . . . . . . . . . . 55
5. Riemannian Curvature . . . . . . . . . . . . . . . . . . . . . 62
6. Almost Hermitian Manifolds . . . . . . . . . . . . . . . . . . 69
7. Representations of U(2) . . . . . . . . . . . . . . . . . . . . 76
8. The Twistor Space . . . . . . . . . . . . . . . . . . . . . . . 82
9. The Normal Bundle . . . . . . . . . . . . . . . . . . . . . . . 90
10. K~hler Twistor Spaces . . . . . . . . . . . . . . . . . . . . . 98
11. Differential Operators and Cohomology . . . . . . . . . . . . 105
12. Conformal Structure . . . . . . . . . . . . . . . . . . . . . 114
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
IV
JEAN-PIERRE VIGU~ : DOMAINES BORNES SYM~TRIQUES
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
I. Automorphismes analytiques d'un domaine born~ d'un
espace de Banach complexe: la topologie de la
convergence uniforme locale . . . . . . . . . . . . . . . . . 127
II. L'alg~bre de Lie des transformations infinit~simales
d'un domaine born~ D . . . . . . . . . . . . . . . . . . . . 133
III. Domaines bombs sym~triques . . . . . . . . . . . . . . . . . 139
IV. Automorphismes analytiques des produits continus de
domaines born~s et domaines born~s sym~triques
irr~ductibles . . . . . . . . . . . . . . . . . . . . . . . . 149
V. Automorphismes analytiques des domaines cercl~s born~s . . . 159
VI. Automorphismes analytiques des domaines born~s et
distances invariantes . . . . . . . . . . . . . . . . . . . . 169
Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . 175
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PREFACE
Kobayashi and Carath~odory-type pseudo-distances on convex cones
in locally convex real vector spaces were defined by E. Vesentini [7 .]
Their most characteristic property is the following:
P) Every linear homomorphism between any two cones is a contraction
with respect to the Kobayashi (Carath~odory) pseudo-distance.
In this work all the distances (on cones of dimension greater than
one) having property P) will be classified.
For definitions and properties of Kobayashi and Carath~odory-type
pseudo-distances (denoted respectively by y and 6 ) we refer to
Vesentini [7] and Franzoni [I .]
PRELIMINARIES AND FORMULATION OF THE PROBLEM
Let R be a real vector space and let V ~ {0} be a convex cone
+
in R, i.e. a subset of R such that v E V = tv E V for all t E]R.,
and that u,v 6 V ~ u + v @ V. We shall always suppose that R
coincides with the vector space Sp(V) = V - V spanned by V, and
shall be concerned mainly with the case in which V is sharp (i.e.
does not contain any entire affine straight line of R) and satisfies
the following condition:
)i If v E V and if Z is any affine straight line in R such
that v 6 £ and that £ A V contains a half-line, then there is
a half-line ~6 C 2 A V containing v in its interior.
V
Condition i) turns out to be equivalent to the following (see Vesenti-
ni 7 [ )] :
i)' For every finite-dimensional subspace S of R such that
S N V ~ {0}, S ~ V is open in the subspace Sp(S %f V) of S
generated by S lf V, with respect to the Euclidean topology.
+
For the sharp convex cone ~. of ~ we shall take as a distance
between any two points x and y the Haar measure (with respect to
+
the multiplicative group ~) of the interval determined by x and y:
~(x,y) = Iiog ~tx -
The distance ~ is invariant with respect to the action of the group
+ + +
GL~R.) = {f e GLaR) : faR.) =~.};
+ +
moreover for every affine endomorphism g of ~ such that g~R.) C ~.
we have
o(g(x),g(y)) < o(x,y) (x,y 6(cid:127)+).
+
By means of the distance o on ~., the Kobayashi-type and Carath@o-
dory-type pseudo-distances can be defined on any convex cone V sati-
sfying condition i) (see Vesentini [7 .)]
The Kobayashi-type__pseudo-distance.
Let 0 w =u, wl,...,w = v be points in V, let x ,...,x ,
n 1 n
yl,...,y n be points in ~, + and let f i ,...,f n be affine functions
of ~ into R mapping ~, + into V, and such that
f (x) = w f (y) = w j( = I ..... n).
j j j-l' j j j
Let us define
Yv(U,V) = inf{o(xl,y )I + ... + O(Xn 'yn )}
taking the infimum over all possible choices of n, Wl,...,Wn_ ,I
Xl,...,x , yl,...,y , f ,...,f . The function v T is a Dseudo-distan-
n n I n
ce for every V and property P) (see the Preface) is fulfilled.
Moreover it turns out that:
KI. The pseudo-distance TV is a distance if, and only if, V is
sharp.
+ +
K2. If R = R] x R] and V =JR. x ]R then
T + +((Xl,X2), (yl,Y2)) = max (~(xl,Yl),o(x2,Y2))
]R.x
~,
or
¥ + + ((Xl,X2),(yl,Y2)) = O(xl,Y )I + o(x2{Y2),
~.x ,RJ
depending on whether the affine line determined by (xl,x )2 and
+ +
(yl,y2) intersects ~, x ~, in a half-line or in a segment.
The Carath~odory-type__pseudo-distance.
Let F(V) be the collection of all real-valued affine functionals