Holomorphicity of totally geodesic Kobayashi isometry between bounded symmetric domains

Abstract: In this paper, we study the holomorphicity of totally geodesic Kobayashi isometric embeddings between bounded symmetric domains. First we show that for a $C^1$-smooth totally geodesic Kobayashi isometric embedding $f\colon \Omega\to\Omega'$ where $\Omega$, $\Omega'$ are bounded symmetric domains, if $\Omega$ is irreducible and $\text{rank}(\Omega) \geq \text{rank}(\Omega')$ or more generally, $\text{rank}(\Omega) \geq \text{rank}(f_*v)$ for any tangent vector $v$ of $\Omega$, then $f$ is either holomorphic or anti-holomorphic. Secondly we characterize $C^1$ Kobayashi isometries from a reducible bounded symmetric domain to itself.