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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 los angeles 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.

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 court cases, contributions conceal lots of the fields studied by way of Marcel Berger in Differential Geometry, specifically : holonomy (Bryant), curvature [positive sectional curvature (Grove), destructive sectional curvature (Abresch and Schroeder, Ballmann and Ledrappier), unfavorable 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 comparable topics [Alexandrov areas (Shiohama and Tanaka, Yamaguchi), elastica (Koiso), subriemannian geometry (Valère and Pelletier)]. Authors are more often than not 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 quick evaluation of contemporary 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 : not easy and tender sphere theorems
T. YAMAGUCHI : A convergence theorem for Alexandrov spaces
J. LOKHAMP : Curvature h-principles
G. ROBERT : Pinching theorems below necessary 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, remarkable 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
D. GROMOLL : confident 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 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

Some of them were released in other places, specifically :

Conjugacy and stress for manifolds with a parallel vector field
J. Differential Geom. 39 (1994), 659-680.
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 additional to this quantity.

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Example text

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 ) . 1. 14) where 0,i (Y, Z) := Pi BJ\I Pi Y , Pi Z BJ,I 1,i (Y, Z) := Pi BJ\I Pi Y , (1l−Pi )Z BJ,I + Pi BJ\I (1l−Pi )Y , Pi Z 2,i (Y, Z) := Pi BJ\I (1l−Pi )Y , (1l−Pi )Z . ) ∧ pξi ΨJ,I,i := β(η, xi ) ξi , Pi (1l + GI ) G−1 (1l − Pi )BJ,I ( . , . ) ∧ pbi +η 2 h (xi ) vi , Pi (1l + GI ) G−1 (1l − Pi )BJ,I ( . , . ) ∧ pξi . 6 ) BJ\I ∧ (1l + GI )Bˆi −1/2 = −xi (1+ xi ) 1/2 Ψ0J,I,i + Ψ1J,I,i + Ψ2J,I,i + ΨJ,I,i .

And R0# = R0 (. ). , . 4) R# (X, Y ; Z, W ) 1 # = R0 (GX, Y ; Z, W )+R0#(X, GY ; Z, W )+R0#(X, Y ; GZ, W )+R0#(X, Y ; Z, GW ) 4 1 2 2 2 2 − D{X,W } g(Y, Z) − D{Y,W } g(X, Z) − D{X,Z} g(Y, W ) + D{Y,Z} g(X, W ) 2 − B(X, W ) , G−1 B(Y, Z) − B(Y, W ) , G−1 B(X, Z) 2 where D{X,W } g(Y, Z) := 1 2 2 2 DX,W g(Y, Z) + DW,X g(Y, Z) is symmetrical in the ar- guments X and W as well as in Y and Z. 4) comes as a sum of three tensors which separately obey all the algebraic symmetries of a curvature tensor.

The method is ˆ in question by a sequence E (µ) of planes whose to approximate the plane E ⊂ TpˆM ˆ We then compute the limit footpoints pˆµ lie in Ω. K(E) = lim K(E (µ) ) . 8 (ii). 2. Proposition. — Let η > 0 be sufficently small13 . Then, at a given point pˆ ∈ SˆI , the curvature K of the metric g vanishes on (i) any plane E ⊂ TpˆF I , and on ˆi|pˆ orthogonally in Li |p . (ii) any plane E which intersects E 13 cf. 9. ´ ` 1 SEMINAIRES & CONGRES ANALYTIC MANIFOLDS OF NONPOSITIVE CURVATURE 49 ˆ , g) is nonBy continuity we know already that the sectional curvature of (M positive.

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