Totally geodesic discs in bounded symmetric domains

Abstract: In this paper, we characterize $C^2$-smooth totally geodesic isometric embeddings $f\colon \Omega\to\Omega'$ 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.