Totally geodesic discs in bounded symmetric domains

In this paper, we characterize $C^2$-smooth totally geodesic isometric embeddings $f\colon Ω\toΩ'$ between bounded symmetric domains $Ω$ and $Ω'$ which extend $C^1$-smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if $Ω$ is irreducible, there exist totally geodesic bounded symmetric subdomains $Ω_1$ and $Ω_2$ of $Ω'$ such that $f = (f_1, f_2)$ maps into $Ω_1\times Ω_2\subset Ω$ where $f_1$ is holomorphic and $f_2$ is anti-holomorphic totally geodesic isometric embeddings. If $\text{rank}(Ω')<2\text{rank}(Ω)$, then either $f$ or $\bar f$ is a standard holomorphic embedding.

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