\begin{definition}[Definition:Frame Adapted to Submanifold]
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 $M$'''.
\end{definition}