Abstract: The holomorphicity of isometries between complex manifolds is a classical topic in complex differential geometry going back to the work of Siu for strong rigidity of compact K¨ahler manifolds. In this talk, we investigate the properties of totally geodesic isometric embedding $f:\Omega\rightarrow\Omega'$ between two bounded symmetric domains $\Omega$ and $\Omega'$ with respect to their Bergman/Kobayashi metrics. In particular, we discuss the question of holomorphicity of totally geodesic isometric discs $f:\Delta\rightarrow\Omega$ that extend sufficiently smooth up to the boundary. As an application, we give a sufficient condition for a totally geodesic isometric embedding $f:\Omega\rightarrow\Omega'$ to be holomorphic or anti-holomorphic. This is a joint work with Aeryeong Seo.
