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Say S is a submanifold of an ambient, oriented manifold M; M is embedded in some R^k;

let ## w_m ## be an orientation form for M.

I'm trying to see under what conditions I can orient S , by contracting ## w_m ## , i.e., by

using the interior product with the "right" vector field X. Clearly this is not always possible, since

not every submanifold is orientable, e.g., even-dimensional projective spaces embedded in

odd-dimensional projective spaces. Just curious as to what vector field we can use, and how

to determine when this can or cannot be done, i.e., how this process could fail when S is not

orientable.

Thanks.

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# Orientation of a Submanifold, given an Orientation of Ambient Manifold

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