I'm having trouble showing the following implication:
Let $M$ be a Riemannian manifold, let $L\subset M$ be a submanifold such that the following holds:
- If $\gamma: I \to M$ is a geodesic s.t. $\gamma(0) \in L$, $\dot{\gamma}(0) \in T_{\gamma(0)}L$, then there is an $\epsilon >0$ such that for $|t| < \epsilon$ we have $\gamma(t) \in L$.
Then for any smooth curve $\eta: I \to L$ we have
$P_\eta (s,t) T_{\eta(t)}L = T_{\eta(s)}L $
where $P_\eta (s,t)$ denotes parallel transport along $\eta$.
I don't really have an approach to this one. The differential equations satisfied by a geodesic looks sort of like the ones satisfied by the parallel transport of a vector, but I didn't get anywhere with this.
Any pointers would be greatly appreciated. Thanks in advance!
S.L.