Suppose we have Riemannian manifold $M$ isometrically immersed in a Kähler manifold $\tilde{M}$. Let $\tilde{g}$ be the metric and $\tilde{\nabla}$ the Levi-Civita connection on $\tilde{M}$ , we know that the isometric immersion defines a metric $g$ on $M$ which then has a Levi-Civita connection $\nabla$. Now, the Kähler manifold has an almost complex structure $J$ for which holds that
$$ \tilde{\nabla}_XJY = J\tilde{\nabla}_XY $$
However, does the same hold for $J$ and $\nabla$?
$$ \nabla_XJY = J\nabla_XY $$
I would assume so since $\tilde{\nabla}$ and $\nabla$ are related through the immersion somehow, but I don't quite know how to prove this. And if it does not hold in general, are there conditions for which it does hold? (Totally real, holomorphic, minimal, ...)