\begin{definition}[Definition:Normal Vector Field along Submanifold]
Let $\struct {\tilde M, \tilde g}$ be a Riemannian manifold.
Let $M$ be a smooth submanifold of $\tilde M$:
:$M \subseteq \tilde M$
Let $p \in M$ be a point.
Let $N_p M$ be the normal space of $M$ at $p$.
Let $N$ be a section of the ambient tangent bundle $\valueat{T \tilde M} M$.
Suppose for all $p \in M$ we have that $N_p \in N_p M$.
Then $N$ is called a '''normal vector field along $M$'''.
\end{definition}