By Arthur L. Besse (Ed.)

Résumé :
En juillet 1992, une desk Ronde de Géométrie Différentielle s'est tenue au CIRM de Luminy en l'honneur de Marcel Berger. Les conférences qui sont reproduites dans ces Actes recouvrent los angeles plupart des sujets abordés par Marcel Berger en Géométrie Différentielle et plus précisément : l'holonomie (Bryant), l. a. courbure [courbure sectionnelle optimistic (Grove), courbure sectionnelle négative (Abresch et Schroeder, Ballmann et Ledrappier), courbure de Ricci négative (Lohkamp), courbure scalaire (Delanoë, Hebey et Vaugon), courbure totale (Shioya)], le spectre du laplacien (Anné, Colin de Verdière, Matheus, Pesce), les inégalités isopérimétriques et les systoles (Calabi, Carron, Gromov), ainsi que quelques sujets annexes [espaces d'Alexandrov (Shiohama et Tanaka, Yamaguchi), elastica (Koiso), géométrie sous-riemannienne (Valère et Pelletier)]. Les auteurs sont pour los angeles plupart des géomètres confirmés, dont plusieurs ont travaillé avec Marcel Berger, mais aussi quelques jeunes. Plusieurs articles (Bryant, Colin, Grove...) contiennent une présentation synthétique des résultats récents dans le domaine concerné, pour mieux les rendre obtainable à un public de non-spécialistes.

Abstract:
Proceedings of the around desk in Differential Geometry in honour of Marcel Berger
July 1992, a around desk in Differential Geometry used to be prepared on the CIRM in Luminy (France) in honour of Marcel Berger. In those court cases, contributions hide lots of the fields studied by means of Marcel Berger in Differential Geometry, specifically : holonomy (Bryant), curvature [positive sectional curvature (Grove), unfavorable sectional curvature (Abresch and Schroeder, Ballmann and Ledrappier), destructive Ricci curvature (Lohkamp), scalar curvature (Delanoë, Hebey and Vaugon), overall curvature (Shioya)], spectrum of the Laplacian (Anné, Colin de Verdière, Matheus, Pesce), isoperimetric and isosystolic inequalities (Calabi, Carron, Gromov), including a few similar matters [Alexandrov areas (Shiohama and Tanaka, Yamaguchi), elastica (Koiso), subriemannian geometry (Valère and Pelletier)]. Authors are typically geometers who labored with Marcel Berger at a while, and likewise a few more youthful ones. a few papers (Bryant, Colin, Grove...) comprise a short overview of modern leads to their specific fields, with the non-experts in brain.

1. time table of the Mathematical talks given on the around Table

Lundi thirteen juillet 1992

K. GROVE : challenging and tender sphere theorems
T. YAMAGUCHI : A convergence theorem for Alexandrov spaces
J. LOKHAMP : Curvature h-principles
G. ROBERT : Pinching theorems below imperative speculation for curvature

Mardi 14 juillet 1992

Y. COLIN DE VERDIERE : Spectre et topologie
H. PESCE : Isospectral nilmanifolds
F. MATHEUS : Circle packings and conformal approximation
R. MICHEL : From warmth equation to Hamilton-Jacobi equation
C. ANNE : Formes diff´erentielles sur les vari´et´es avec des anses fines
G. CARRON : In´egalit´e isop´erim´etrique de Faber-Krahn

Mercredi 15 juillet 1992

E. CALABI : in the direction of extremal metrics for isosystolic inequality for closed orientable
surfaces with genus > 1
M. GROMOV : Isosystols
Ch. CROKE : Which Riemannian manifolds are decided via their geodesic flows

Jeudi sixteen juillet 1992

R. BRYANT : Classical, unparalleled and unique holonomies : a standing report
T. SHIOYA : habit of maximal geodesics in Riemannian planes
L. VALERE-BOUCHE : Geodesics in subriemannian singular geometry and control
theory
D. GROMOLL : optimistic Ricci curvature : a few contemporary developements
Ph. DELANOE : Ni’s thesis revisited
E. HEBEY : From the Yamabe challenge to the equivariant Yamabe problem
Vendredi 17 juillet 1992
W. BALLMANN : Brownian movement, Harmonic capabilities and Martin boundary
U. ABRESCH : Graph manifolds, ends of negatively curved areas and the hyperbolic
120-cell space
N. KOISO : Elastica
Jerry KAZDAN : Why a few differential equations don't have any solutions
J. P. BOURGUIGNON : challenge session

2. at the contributions

Among the above pointed out meetings, 5 aren't reproduced in those notes,
namely these via Christopher CROKE, Detlef GROMOLL, Jerry KAZDAN, Ren´e
MICHEL and Gilles ROBERT.

Some of them were released in other places, specifically :

CROKE, KLEINER :
Conjugacy and pressure for manifolds with a parallel vector field
J. Differential Geom. 39 (1994), 659-680.
LE COUTURIER, ROBERT :
Lp pinching and the geometry of compact Riemannian manifolds
Comment. Math. Helvetici sixty nine (1994), 249-271.
On the opposite hand, Professor SHIOHAMA, who was once invited to provide a conversation, had
not been capable of come to the desk Ronde. He sought after however to provide a
contribution to Marcel Berger. it's been further to this quantity.

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Extra info for Actes de la Table ronde de geometrie differentielle: En l'honneur de Marcel Berger

Example text

9. 10. Remark. 10 that supp(h) ⊂ [0, sinh2 1 2 d0 ]. 18 ) − 2η 2 h (xi ) g0 i∈I − ∧ pξi − 2η 2 (1 + xi ) h (xi ) pi ∧ pi i∈I 4 η h (xi1 ) h (xi2 ) pξi1 ∧ pξi2 . 8 at all. 4. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1996 6. 5. 1. Proposition. 9. Then, ˆ ε such that for 0 < η < η2 for any ε > 0, there exists a constant η2 = η2 n, h, d0 , N, the following estimates hold on each domain Ω ∩ UI (i) −ε g0 (ii) −ε g0 ∧ ∧ g 0 + ΦI g 0 + ΦI ≤ ≤ BI ∧ G−1 −(1l+GI )−1 BI BJ\I ∧ G−1 BI ≤ ε g 0 ∧ g 0 + ΦI ≤ ε g 0 ∧ g 0 + ΦI .

Hn−2 with ai = 0. Then, pˆ ∈ π −1 (Hn−2 i1 is ), and this submanifold splits as T × D with an s–dimensional torus factor. The vector w is tangent to Ts , and, since w ¯ is ˆi , 1 ≤ k ≤ s, it follows that w ¯ is tangent to D. 3 perpendicular to E k tells us that all the zero curvatures come from the totally geodesic product manifolds of the type π −1 (Hn−2 ∩ . . ∩ Hn−2 i1 is ). 2 by standard polarisation formulae a fairly precise structural statement about the curvature operator along each singular stratum SˆI .

V¯is ) is totally geodesic and splits off a local torus–factor of dimension s. 3 we show that all zero curvatures of the metric g come from these product submanifolds. π −1 In the second part of this chapter, we show that the existence of the submanifolds (V¯i ∩ . . ∩ V¯i ) follows from algebraic properties of the fundamental group, and 1 s so do the basic geometric properties of V¯i1 , . . , V¯is . More precisely, if M ∗ is another compact, real analytic, Riemannian manifold with K ≤ 0 and π1 (M ∗ ) ∼ = π1 (M ), then we find in M ∗ similar totally geodesic product submanifolds.

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