Read e-book online Actes de la Table ronde de geometrie differentielle: En PDF

By Arthur L. Besse (Ed.)

ISBN-10: 2856290477

ISBN-13: 9782856290477

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 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 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 equipped on the CIRM in Luminy (France) in honour of Marcel Berger. In those complaints, contributions disguise many of the fields studied through Marcel Berger in Differential Geometry, specifically : holonomy (Bryant), curvature [positive sectional curvature (Grove), unfavorable sectional curvature (Abresch and Schroeder, Ballmann and Ledrappier), detrimental 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 topics [Alexandrov areas (Shiohama and Tanaka, Yamaguchi), elastica (Koiso), subriemannian geometry (Valère and Pelletier)]. Authors are normally 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 assessment 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 : challenging and gentle sphere theorems
T. YAMAGUCHI : A convergence theorem for Alexandrov spaces
J. LOKHAMP : Curvature h-principles
G. ROBERT : Pinching theorems less than 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, 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 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 features 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 somewhere else, specifically :

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 used to be invited to offer 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 extra to this quantity.

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

Example text

Thus, it is sufficient to work with the finite piece g0 + gI and verify that ˆˆ ˆˆ I ∩ Ω the following identity holds on U ˆˆ i ,K ˆˆ i = xi +η 2 h(xi ) δi i . (g0 + gI ) K 1 2 1 1 1 2 Here, we have made use of the fact that by the local geometry of the intersection Ki1 , Ki2 = xi1 δi1 i2 on UI , provided that i1 , i2 ∈ I . To finish the proof, we recall that xi1 vanishes identically on SI for any i1 ∈ I and take into account that h(0) = 1 by hypothesis. The next ingredient into our evaluation of the sectional curvatures along SˆI ˆˆ I whose restrictions to pr−1 (SˆI ) generate the line are analytic vector fields vˆi on U I ν ˆˆ I and not bundles Li , i ∈ I.

2 x−2 j h(xj ) . , Kj Kj , . Moreover, for any subset J ⊂ J we let gJ := plies that the corresponding series GJ := j∈J j∈J . gj . t. g0 = , . In particular, the symmetric endomorphism G := 1l + GJ is dual to the metric g from the Theorem. 4) g = g0 + gJ = g0 + gI + gJ\I = g0 + gj . 7. 1 for the actual convergence estimates. 5 cf. 27) ´ ` 1 SEMINAIRES & CONGRES 17 ANALYTIC MANIFOLDS OF NONPOSITIVE CURVATURE ˆI in our covering of M ˆ separately. Note that (ii) We handle each open set U πI∗ (g) = πI∗ (g0 + gI ) + πI∗ (gJ\I ) .

After expressing Kj and DKj in terms of vj and ξj , we just need to collect terms appropriately, using just the definitions of pj , pξj , and the ∧ –product. 2. Lemma. 7. 12) Bj (Y, Z) := −η 2 xj ´ ` 1 SEMINAIRES & CONGRES −1/2 +η 2 xj (1+xj ) 1/2 (1+xj ) 1/2 h (xj ) pξj (Y, Z) vj b h (xj ) − x−1 j h(xj ) pj (Y, Z) ξj . j∈J Bj 29 ANALYTIC MANIFOLDS OF NONPOSITIVE CURVATURE Proof. 3). 12). 13) Again B1 ∧l B2 )(X, Y ; Z, W ) 1 l B1 (X, W ), B2(Y, Z) − l B1 (Y, W ), B2(X, Z) := 2 − l B1 (X, Z), B2(Y, W ) + l B1 (Y, Z), B2(X, W ) B2 has all the algebraic symmetries of a curvature tensor.

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Actes de la Table ronde de geometrie differentielle: En l'honneur de Marcel Berger by Arthur L. Besse (Ed.)


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