Holomorphic mappings into strictly convex domains which are Kobayashi isometries at one point

Abstract

Let $M$ be a complex manifold of dimension $n$ and let $\Omega$ be a domain in ${{\mathbf {C}}^n}$. Let $f:M \to \Omega$ be a holomorphic map which is an isometry for the infinitesimal Kobayashi metric at one point. We given conditions on $M$ and on $\Omega$ which imply that $F$ must be a biholomorphic map.