Orientation of a Submanifold, given an Orientation of Ambient Manifold

[WWGD]

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.

 

[lavinia]

I think the condition you are looking for is that the normal bundle to the submanifold,S,is orientable(not necessarily trivial). Then S will be orientable if M is.

At each point of S choose a positively orientated basis for the normal bundle. Extend this basis with vectors tangent to S to a positively oriented basis of M. These tangent vector will give you a positively oriented basis for the tangent space of S.

If the normal bundle to S is trivial, then one gets a set of linearly independent normal vector fields along S. Choose some ordering of these vector fields and contract the orientation form of M with them. This will give you an orientation form for S.

One sees from this that if S is not orientable then its normal bundle can not be orientable.

Form an Algebraic Topology point of view, if S is non-orientable then its first Stiefel-Whitney class is not zero. But since M is orientable the Whitney sum of the tangent and normal bundles to S must have zero first Stiefel Whitney class. This can only happen of the first Stiefel Whitey class of the normal bundle equals the first Stiefel Whitney class of S ( by the Whitney sum formula.)