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

• WWGD
Gold Member
Hi, All:

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.