Almost-Einstein manifolds with nonnegative isotropic curvature  [ Variétés presque Einstein à courbure isotrope positive ou nulle ]
Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2493-2501.

Soit (M,g), une variété riemannienne compacte simplement connexe de dimension n4, à courbure isotrope positive ou nulle. Nous montrons que pour tout 0<l<L, il existe un ε=ε(l,L,n) qui satisfait la propriété suivante : si la courbure scalaire s de g satisfait

lsL

et que le tenseur d’Einstein satisfait

Ric -sngε

alors M est difféomorphe à un espace symétrique de type compact.

Ceci est lié au résultat de S. Brendle sur la rigidité métrique des variétés d’Einstein à courbure isotrope positive ou nulle.

Let (M,g), n4, be a compact simply-connected Riemannian n-manifold with nonnegative isotropic curvature. Given 0<lL, we prove that there exists ε=ε(l,L,n) satisfying the following: If the scalar curvature s of g satisfies

lsL

and the Einstein tensor satisfies

Ric -sngε

then M is diffeomorphic to a symmetric space of compact type.

This is related to the result of S. Brendle on the metric rigidity of Einstein manifolds with nonnegative isotropic curvature.

Reçu le : 2009-07-03
Accepté le : 2010-04-27
DOI : https://doi.org/10.5802/aif.2616
Classification : 53C21
Mots clés: variétés presque-Einstein, courbure isotrope positive ou nulle
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     author = {Seshadri, Harish},
     title = {Almost-Einstein manifolds with nonnegative isotropic curvature},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {7},
     year = {2010},
     pages = {2493-2501},
     doi = {10.5802/aif.2616},
     zbl = {1225.53037},
     mrnumber = {2866997},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2010__60_7_2493_0/}
}
Seshadri, Harish. Almost-Einstein manifolds with nonnegative isotropic curvature. Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2493-2501. doi : 10.5802/aif.2616. https://aif.centre-mersenne.org/item/AIF_2010__60_7_2493_0/

[1] Brendle, Simon Einstein manifolds with nonnegative isotropic curvature are locally symmetric (to appear in Duke Mathematical Journal) | MR 2573825 | Zbl 1189.53042

[2] Brendle, Simon; Schoen, Richard Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc., Tome 22 (2009) no. 1, pp. 287-307 | Article | MR 2449060 | Zbl pre05859406

[3] Koiso, Norihito Rigidity and stability of Einstein metrics—the case of compact symmetric spaces, Osaka J. Math., Tome 17 (1980) no. 1, pp. 51-73 | MR 558319 | Zbl 0426.53037

[4] Micallef, Mario J.; Moore, John Douglas Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2), Tome 127 (1988) no. 1, pp. 199-227 | Article | MR 924677 | Zbl 0661.53027

[5] Micallef, Mario J.; Wang, McKenzie Y. Metrics with nonnegative isotropic curvature, Duke Math. J., Tome 72 (1993) no. 3, pp. 649-672 | Article | MR 1253619 | Zbl 0804.53058

[6] Petersen, Peter; Tao, Terence Classification of almost quarter-pinched manifolds, Proc. Amer. Math. Soc., Tome 137 (2009) no. 7, pp. 2437-2440 | Article | MR 2495279 | Zbl 1168.53020

[7] Petrunin, A.; Tuschmann, W. Diffeomorphism finiteness, positive pinching, and second homotopy, Geom. Funct. Anal., Tome 9 (1999) no. 4, pp. 736-774 | Article | MR 1719602 | Zbl 0941.53026

[8] Seshadri, H. Manifolds with nonnegative isotropic curvature (To appear in Communications in Analysis and Geometry, http://www.math.iisc.ernet.in/~harish/papers/pic-cag.pdf) | MR 2601346

[9] Topping, Peter Lectures on the Ricci flow, London Mathematical Society Lecture Note Series, Tome 325, Cambridge University Press, Cambridge, 2006 | MR 2265040 | Zbl 1105.58013