Local characteristic of holomorphic automorphism on classical domains (Q1352475)

Let \(M\) be one of the four types of classical matrix domains. It is shown that if \(f\) is a holomorphic self-map of \(M\) which is an isometry at one point for either the infinitesimal Kobayashi metric or the infinitesimal Carathéodory metric, then \(f\) is an automorphism. The proofs are elementary: one reduces to the case where \(f(0)=0\) and applies Cartan's theorem using the fact that in all cases the indicatrix of both the Carathéodory and Kobayashi metrics at \(0\) coincides with the domain itself.