Definition:Normal Vector Field along Submanifold

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Let $\struct {\tilde M, \tilde g}$ be a Riemannian manifold.

Let $M : M \subseteq \tilde M$ be a smooth submanifold.

Let $p \in M$ be a base 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 $M$.