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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), 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 l. a. 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 available 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 geared up on the CIRM in Luminy (France) in honour of Marcel Berger. In those court cases, contributions conceal lots of the fields studied by means of Marcel Berger in Differential Geometry, particularly : 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 in most cases geometers who labored with Marcel Berger at it slow, and in addition a few more youthful ones. a few papers (Bryant, Colin, Grove...) comprise a short evaluate of modern ends up in 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 : difficult and gentle sphere theorems
T. YAMAGUCHI : A convergence theorem for Alexandrov spaces
J. LOKHAMP : Curvature h-principles
G. ROBERT : Pinching theorems below quintessential 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 by means of their geodesic flows

Jeudi sixteen juillet 1992

R. BRYANT : Classical, unprecedented 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 services 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 haven't 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 via Christopher CROKE, Detlef GROMOLL, Jerry KAZDAN, Ren´e
MICHEL and Gilles ROBERT.

Some of them were released somewhere else, particularly :

CROKE, KLEINER :
Conjugacy and tension 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 offer a conversation, had
not been in a position to come to the desk Ronde. He sought after however to provide a
contribution to Marcel Berger. it's been extra 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

Proof. 2 ) shows that X , KjC |z KjC |z , Y 2 = z , e2j C − z , e1j C C e1j , X z , e2j e1j , Y C + z , e1j e1j , X C e2j , Y C C 2 e2j , X C e2j , Y C + e2j , X C e1j , Y C X , e2j C C C . 2 we have eµj , z 2 C C 2 2 z , ej C X , e2j 2 C z , e1j +2 √ 2 + 1 + 2a(a + 2) eµj , z0 √ 1 ≤ 1 + 2a(a + 2) 1 + 2xj (z0 ) . 6) yield eµj , X 2 C ≤ 1 + 2 eµj , z0 2 X ,X ≤ 1 + 2xj (z0 ) X , X h . 8). ´ ` 1 SEMINAIRES & CONGRES 2 C h C 2 . ANALYTIC MANIFOLDS OF NONPOSITIVE CURVATURE 23 Now, we collect the information.

5) it is clear that πI−1 {p} is a totally geodesic product torus in WIU × (R/2πZ) #I equipped with the metric πI∗ g0 + gI . If η is sufficiently small, then the function x → x + η 2 h(x), x ≥ 0, takes its absolute minimum precisely at x = 0. Hence, for these values of η all closed geodesics of the torus are absolutely minimizing elements in their homotopy classes in WIU × (R/2πZ) metric πI∗ g0 + gI #I . In order to pass from the partial to πI∗ (g), we add a positive semidefinite term which vanishes on the torus.

1 (i). ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1996 44 U. ABRESCH V. SCHROEDER Proof. 7 (ii) that β(η, xi1 ) β(η, xi2 ) ξi1 , LI ξi2 +η 2 h (xi1 ) β(η, xi2 ) vi1 , LI ξi2 + η 2 β(η, xi1 ) h (xi2 ) ξi1 , LI vi2 + η 4 h (xi1 ) h (xi2 ) vi1 , LI vi2 1/2 1/2 ≤ η 2 (1 + 2c4 η 2 )2 cˆ12 (η)xi1 xi2 (1 + xi1 ) −1/2 (1 + xi2 ) −1/2 hence the claim. 13) ∧ LI ∧ pξi vanish identically and that pbi ˆi = B ∧ pbi = −pi 1 + x−1 β(η, xi )2 ξi , LI ξi pi i ∧ ∧ pi pi . 2 (i), we can finish the proof setting cˆ14 := c0 (1 + c4 η)(1 + c4 η 2 ) and cˆ15 := c0 c4 (1 + η + c4 η)(1 + c4 η 2 ) .

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