Existence of Pseudo-Riemannian Adapted Orthonormal Frames

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

Let $M \subseteq \tilde M$ be an embedded pseudo-Riemannian or Riemannian submanifold.

Then for each $p \in M$, there exists a smooth orthonormal frame on a neighborhood of $p \in \tilde M$ that is adapted to $M$.