Let $N$ be a submanifold of a Riemannian manifold $M$. Does a geodesic $\gamma$ in $M$ with image contained in $N$ remain a geodesic in $N$? I'm pretty sure this has to be true true because its locally length minimizing properties.
To prove this can I just say that if $\nabla$ is the connection on $N$ and $\overline{\nabla}$ is the connection on $M$, then $\nabla_{\gamma'}\gamma'=(\overline{\nabla}_{\gamma'}\gamma')^{T}=0$ since $\overline{\nabla}_{\gamma'}\gamma'=0$?