Necessary and Sufficient Condition for Hypersurface in Oriented Riemannian Manifold to be Orientable
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Theorem
Let $\struct {\tilde M, \tilde g}$ be an oriented Riemannian manifold.
Let $M$ be a hypersurface in $\tilde M$.
Let $g$ be the induced metric on $M$.
Then $M$ is orientable if and only if there exists a unit global normal vector $N$ for $M$.
Proof
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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds