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Let ${\Omega }_1$ and ${\Omega }_2$ be $\mathrm{strongly}\mathrm{pseudoconvex}$ domains in ${ℂ}^n$ and $f:{\Omega }_1\to {\Omega }_2$ an isometry for the Kobayashi or Carathéodory metrics. Suppose that $f$ extends as a $C^1$ map to ${\overline{\Omega }}_1$. We then prove that ${f|}_{\partial {\Omega }_1}:\partial {\Omega }_1\to \partial {\Omega }_2$ is a CR or anti-CR diffeomorphism. It follows that ${\Omega }_1$ and ${\Omega }_2$ must be biholomorphic or anti-biholomorphic.

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.

We discuss the notion of isotropic curvature of a Riemannian manifold and relations between the sign of this curvature and the geometry and topology of the manifold.