Existence and Uniqueness of Outward-Pointing Normal

Theorem

Let $\struct {M, g}$ be a smooth Riemannian manifold with boundary $\partial M$.

Then the normal bundle to $\partial M$ is a rank-$1$ smooth vector bundle over $\partial M$.

Furthermore, there is a unique smooth outward-pointing unit normal vector field along all of $\partial M$.

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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