parallel transportation on submanifold

parallel transportation on submanifold

$M$ is a Riemannian manifold, $N$ is a submanifold of $M$, not totally
geodesic. Given two points $p,q \in N$, let $\gamma_N$, $\gamma_M$ be the
geodesics connecting $p$, $q$ in $N$ and $M$, respectively. Let $\xi \in
{T_p}N$, $\xi $ orthogonal to ${{\dot \gamma }_N}(p)$. Is $\xi $
orthogonal to ${{\dot \gamma }_M}(p)$? Does the result of parallel
transportation of $\xi$ from $p$ to $q$ in $N$ coincide with that in $M$?
If not, please give counter-examples.