Totally geodesic discs in bounded symmetric domains

Abstract

In this paper, we characterize $C^2$-smooth totally geodesic isometric embeddings $f:\Omega \rightarrow \Omega '$ between bounded symmetric domains $\Omega$ and $\Omega '$ which extend $C^1$-smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if $\Omega$ is irreducible, there exist totally geodesic bounded symmetric subdomains $\Omega _1$ and $\Omega _2$ of $\Omega '$ such that $f = (f_1, f_2)$ maps into $\Omega _1\times \Omega _2\subset \Omega$ where $f_1$ is holomorphic and $f_2$ is anti-holomorphic totally geodesic isometric embeddings. If $\text {rank}(\Omega ')<2\text {rank}(\Omega )$, then either f or ${\bar{f}}$ is a standard holomorphic embedding.