Let $M$ be a riemannian manifold and $N$ be a submanifold of $M$. Let $v$ be a vector field in $N$. Then $v$ can be covariantly differentiated along $\gamma$ resulting in new field $u$. (Here consider Levi-Civita connection)
Because $N$ is embedded in $M$ then $\gamma$ and $v$ can be "carried" in $M$. Let us denote them in $M$ the same way as in $N$. Let's consider Levi-Civita connection in $M$ and differentiate $v$ along $\gamma$ resulting in $w$ field.
Is it true that the orthogonal projection of $w$ in $T_p M$ on $T_p N$ is equal to $u$?