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Let $(M^n,g)$, $n \ge 4$, be a compact simply-connected Riemannian manifoldwith nonnegative isotropic curvature. Given $0<l\le L$, we prove that thereexists $\eps = \eps (l,L,n)$ 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 spaceof compact type.  This is a smooth analogue of the result of S. Brendle that a compact Einsteinmanifold with nonnegative isotropic curvature is isometric to a locallysymmetric space.

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