Holomorphicity of totally geodesic Kobayashi isometry between bounded symmetric domains

In this paper, we study the holomorphicity of totally geodesic Kobayashi isometric embeddings between bounded symmetric domains. First, we show that for a (Formula presented.) -smooth totally geodesic Kobayashi isometric embedding (Formula presented.) where (Formula presented.), (Formula presented.) are bounded symmetric domains, if (Formula presented.) is irreducible and (Formula presented.) or more generally, (Formula presented.) for any tangent vector (Formula presented.) of (Formula presented.), then (Formula presented.) is either holomorphic or anti-holomorphic. Second, we characterize (Formula presented.) Kobayashi isometries from a reducible bounded symmetric domain to itself.