Almost-Einstein manifolds with nonnegative isotropic curvature

Seshadri, Harish

Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2493-2501.

$$Soit $(M,g)$, une variété riemannienne compacte simplement connexe de dimension $n\ge 4$, à courbure isotrope positive ou nulle. Nous montrons que pour tout $0<l<L$, il existe un $\epsilon =\epsilon (l,L,n)$ 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

$$\left|\mathrm{Ric}-\frac{s}{n}g\right|\le \epsilon$$

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

$$l\le s\le L$$

and the Einstein tensor satisfies

$$\left|\mathrm{Ric}-\frac{s}{n}g\right|\le \epsilon$$

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.

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