On proper holomorphic mappings among irreducible bounded symmetric domains of rank at least $2$

Abstract:

We give a characterization for totally geodesic embeddings between higher-rank irreducible bounded symmetric domains in terms of certain totally geodesic rank-1 symmetric subspaces. More explicitly, we prove that for two irreducible bounded symmetric domains $\Omega _1,\Omega _2$ of rank at least 2, a holomorphic map $F:\Omega _1\rightarrow \Omega _2$ is a totally geodesic embedding with respect to the Bergman metrics if $F$ maps the minimal disks of $\Omega _1$ into rank-1 characteristic symmetric subspaces of $\Omega _2$. As a simple corollary, we obtain a much simpler proof for a theorem of Tsai which says that $F$ is totally geodesic if $F$ is proper and $\operatorname {rank}(\Omega _1) = \operatorname {rank}(\Omega _2)$.