Definition:Vector Normal to Smooth Submanifold

Definition

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

Let $M \subseteq \tilde M$ be a smooth submanifold with or without boundary in $M$.

Let $p \in M$ be a point.

Let $T_p \tilde M$ and $T_p M$ be the tangent spaces of $\tilde M$ and $M$ at $p$ respectively.

Let $v \in T_p \tilde M$ and $w \in T_p M$ be vectors.

Let $\innerprod \cdot \cdot$ be the inner product induced by the Riemannian metric $\tilde g$.

Suppose:

$\forall w \in T_p M : \innerprod v w = 0$

Then $v$ is said to be normal to $M$ (at $p$).

Sources

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