Existence of Adapted Orthonormal Frames

Theorem

Let $\struct {\tilde M, \tilde g}$ be a Riemannian manifold without boundary.

Let $M \subseteq \tilde M$ be an embedded smooth submanifold with or without boundary.

Let $p \in M$ be a point.

Then there exists a neighborhood $\tilde U$ of $p$ in $\tilde M$ and a smooth orthonormal frame for $\tilde M$ on $\tilde U$ 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. Methods for Constructing Riemannian Metrics