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.