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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 l. a. plupart des sujets abordés par Marcel Berger en Géométrie Différentielle et plus précisément : l'holonomie (Bryant), los angeles courbure [courbure sectionnelle confident (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 was once prepared on the CIRM in Luminy (France) in honour of Marcel Berger. In those lawsuits, contributions disguise lots of the fields studied through Marcel Berger in Differential Geometry, specifically : holonomy (Bryant), curvature [positive sectional curvature (Grove), unfavourable sectional curvature (Abresch and Schroeder, Ballmann and Ledrappier), unfavourable 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 often geometers who labored with Marcel Berger at it slow, and likewise a few more youthful ones. a few papers (Bryant, Colin, Grove...) comprise a short evaluate of contemporary 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 : demanding and smooth sphere theorems
T. YAMAGUCHI : A convergence theorem for Alexandrov spaces
J. LOKHAMP : Curvature h-principles
G. ROBERT : Pinching theorems below fundamental 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 made up our minds via their geodesic flows

Jeudi sixteen juillet 1992

R. BRYANT : Classical, unheard of 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 fresh 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 usually are not reproduced in those notes,
namely these by means of Christopher CROKE, Detlef GROMOLL, Jerry KAZDAN, Ren´e
MICHEL and Gilles ROBERT.

Some of them were released in other places, particularly :

CROKE, KLEINER :
Conjugacy and stress 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 in a position to come to the desk Ronde. He sought after however to offer 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

1 ≈ η→0 πη length(RP ) thin sup E⊂T Mη On the other hand, diam M, g(η) is uniformly bounded for 0 < η ≤ 1. In fact, the expression for R# shows directly5 that for sufficiently small values of ˆi over the stratum Si of the divisor which is η the sectional curvature of any plane E spanned by the unit normal vector of Sˆi and the tangent vector of the fibration Sˆi → Si is approximately −η −2 . Moreover, the region where the sectional curvature gets large in absolute value concentrates more and more along the preimage of the divisor.

So, with our conventions, the curvature R operator of Hn is negative definite. 3. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1996 26 U. ABRESCH V. SCHROEDER Recall that in our context g0 = . , . denotes the standard hyperbolic metric. 3). We shall find it convenient to introduce for each j ∈ J some more objects related to the very special nature of ⊂ Kj . Since the corresponding group ϑj (R/2πZ) of rotations has a subspace Hn−2 j Hn of codimension 2 as its fixed point set, it is clear that the endomorphism field DKj has rank 2 at every point p ∈ Hn .

Moreover, the map g → B is a linear differential operator of 1st order. So the third term in the expression for R# depends quadratically on the 1st derivatives of g, and the factor G−1 in each pairing resembles a common denominator, which depends pointwise linearly on g. 4). 4), which we may think of as an essentially quadratic interaction term in an otherwise linear context. ˆ Λ2 T Ω → Λ2 T Ω or rather for Our sign conventions for the curvature operator R: ˆ # : Λ2 T Ω × Λ2 T Ω → R are explained by the equation its associated bilinear form R ˆ # (X ∧ Y, Z ∧ W ) = −R# (X, Y ; Z, W ) .

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