Definition:Normal Vector Field along Submanifold

Definition

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 $M$.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics