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.

$\phantom{\rule{-0.166667em}{0ex}}$Soit $\left(M\phantom{\rule{-0.166667em}{0ex}},g\right)$, une variété riemannienne compacte simplement connexe de dimension $n\ge 4$, à courbure isotrope positive ou nulle. Nous montrons que pour tout $0, il existe un $\epsilon =\epsilon \left(l,L,n\right)$ qui satisfait la propriété suivante : si la courbure scalaire $s$ de $g$ satisfait

 $l\le s\le L$

et que le tenseur d’Einstein satisfait

 $\left|\mathrm{Ric}\phantom{\rule{0.166667em}{0ex}}-\frac{s}{n}g\right|\le \epsilon$

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 $\left(M,g\right)$, $n\ge 4$, be a compact simply-connected Riemannian $n$-manifold with nonnegative isotropic curvature. Given $0, we prove that there exists $\epsilon =\epsilon \left(l,L,n\right)$ satisfying the following: If the scalar curvature $s$ of $g$ satisfies

 $l\le s\le L$

and the Einstein tensor satisfies

 $\left|\mathrm{Ric}\phantom{\rule{0.166667em}{0ex}}-\frac{s}{n}g\right|\le \epsilon$

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.

DOI : https://doi.org/10.5802/aif.2616
Classification : 53C21
Mots clés : variétés presque-Einstein, courbure isotrope positive ou nulle
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title = {Almost-Einstein manifolds with nonnegative isotropic curvature},
journal = {Annales de l'Institut Fourier},
pages = {2493--2501},
publisher = {Association des Annales de l'institut Fourier},
volume = {60},
number = {7},
year = {2010},
doi = {10.5802/aif.2616},
mrnumber = {2866997},
zbl = {1225.53037},
language = {en},
url = {www.numdam.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. http://www.numdam.org/item/AIF_2010__60_7_2493_0/

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