Definition:Frame Adapted to Submanifold

Definition

Let $\struct {\tilde M, \tilde g}$ be an $m$-dimensional Riemannian manifold.

Let $M : M \subseteq \tilde M$ be an $n$-dimensional submanifold.

Let $\tilde U \subseteq \tilde M$ be an open subset.

Let $E = \tuple {E_1, \ldots E_n}$ be a local frame for $\tilde M$ on $\tilde U$.

Let $T_p M$ be the tangent space of $M$ at the point $p \in M$.

Suppose the first $n$ vector fields of $E$ are tangent to $M$:

$\forall p \in \tilde U : E_1, \ldots E_n \in T_p M$

Then $E$ is called the frame adapted to $M$.

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