On the smooth rigidity of almost-Einstein manifolds with nonnegative isotropic curvature

TL;DR

This paper proves that compact, simply-connected manifolds with nonnegative isotropic curvature and scalar curvature bounds are diffeomorphic to symmetric spaces if they are close to Einstein metrics, extending Brendle's rigidity results.

Contribution

It establishes a smooth rigidity result for almost-Einstein manifolds with nonnegative isotropic curvature, providing conditions under which they are diffeomorphic to symmetric spaces.

Findings

Manifolds with scalar curvature bounds and small Einstein tensor deviation are diffeomorphic to symmetric spaces.

The result extends Brendle's isometric rigidity to a smooth diffeomorphism context.

Existence of epsilon depending on curvature bounds and dimension for the rigidity to hold.

Abstract

Let $(M^n,g)$, $n \ge 4$, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given $0<l\le L$, we prove that there exists $\eps = \eps (l,L,n)$ satisfying the following: If the scalar curvature $s$ of $g$ satisfies $$ l \le s \le L $$ and the Einstein tensor satisfies $$ | Ric - \frac {s}{n}g | \le \eps$$ then $M$ is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result of S. Brendle that a compact Einstein manifold with nonnegative isotropic curvature is isometric to a locally symmetric space.