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here is my exercise:

Let M be a orientable manifold of dimension m and let N be a submanifold of M of codimension 1.

Show that N is orientable <=> it exists a[tex] X \in \tau_1 (M),[/tex] s.t. [tex]span<X(p)> \oplus T_p N= T_p M \;

\forall p\in N[/tex]

The X is a vector field, i.e. X(p) is an tangent vector at p.

But what is the strategy to proof this claim? Excuse me but, i'm so desperate. This stuff is completely new for me and i don't know how this works.

Can you please help me by this proof? Or do you know at least good literature, where i can read something about this topic?

Thanks

Regards