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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