Holomorphicity of totally geodesic Kobayashi isometry between bounded symmetric domains

TL;DR

This paper proves that certain smooth isometric embeddings between bounded symmetric domains are necessarily holomorphic or anti-holomorphic, and it characterizes isometries for reducible domains, advancing understanding of their geometric structure.

Contribution

It establishes conditions under which totally geodesic Kobayashi isometries are holomorphic or anti-holomorphic and characterizes isometries for reducible bounded symmetric domains.

Findings

Such embeddings are holomorphic or anti-holomorphic under specified rank conditions.

Characterization of Kobayashi isometries for reducible bounded symmetric domains.

Extension of holomorphicity results to more general geometric settings.

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.