Let $X$ be a complex manifold, and let $Y$ be a complex submanifold of $X$.
If $X$ has an hermitian structure(on its tangent bundle), we can consider the Chern connection $\nabla$ on the holomorphic tangent bundle $\mathcal{T}_X$ which is compatible with the hermitian structure.
Then, for $Y$, is it possible to define the second fundamental form with respect to the connection and the hermitian structure?
Also, is it available to say that $Y$ is totally geodesic if its second fundamental form vanishes on $Y$?
I have googled about this thing, but it is hard to find any reference.
One that I am studying is 'Complex Geometry' of Huybrechts. On page 175, there is a definition of the second fundamental form.
It seems possible to define those things, but I am not sure of it, and also I am wondering that is it a useful definition.
Thanks in advance.