Let $M$ be a Riemannian manifold and $N$ an embedded submanifold of $M$. Now I have a geodesic $c \colon (-a,a) \to N$ with $a>0$ and $c(0)=p$ in $N$, s.t. $c'(0)$ is orthogonal to $T_pN$.
Is the parallel transport of $c'(0)$ along $c$ still orthogonal to $T_qN$?
Parallel transport is an isometry, so I think the question is equivalent to whether or not the tangent space of $N$ at $c(0)$ is mapped to the tangent space at $c(t)$ by the parallel transport. Unfortunately I haven't an answer to this.
Regards.