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 $M$.

Proof

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Sources

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