I was wondering the following. Suppose we have some ambient manifold $B$, and we have a totally geodesic surface (meaning second fundamental form vanishes) $M \subset B$.
I know from a proposition that due to $M$ being totally geodesic, the geodesics of $M$ are also geodesics in $B$.
Now also suppose that $B$ is a complete Riemannian manifold. I want to know the following: Suppose we have an immersion $F: M \rightarrow B$. If we let $p \in M$ be a point, is it true in general that $F(M) = \exp_p (T_p M)$? If so, how can we prove this? Or is it trivial?