Abstract

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