  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.
