Orientability of Submanifolds: A Proof and Strategy

[Canavar]

Hello,

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$$X \in \tau_1 (M),$$ s.t. $$span<X(p)> \oplus T_p N= T_p M \; \forall p\in N$$


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

 

[ystael]

Notation in this subject varies widely. What is $$\tau_1(M)$$? Do you mean to say that $$X$$ is a smooth vector field on $$M$$? (that's what I guess from the problem)

Also, what are you using for a definition of 'orientable'? There are many ways to approach the definition.

 

[Canavar]

Hello,

yes you are right, X is a smooth vector field on M.

We have a few equivalent definitions of "orientable".

A manifold is orientable if

1) det $$d(f \circ g^{-1})>0$$ , $$\forall f,g$$ whereas f, g are coordinate maps of the manifold.

<=>2) we have a non-vanishing differential m-form (if dimM=m)

<=>$$\Lambda^m T^{*}M-\{0\}$$ has two components

 

[ystael]

Suppose $$\omega$$ is a volume form on $$M$$ (that is, a nowhere zero $$m$$-form). Think about how you can use $$\omega$$ to convert a vector field $$X$$ which is everywhere linearly independent from $$TN$$, into either a volume form for $$N$$ or a smooth frame field for $$N$$. (By "frame field" I mean a family of $$m - 1$$ vector fields in $$N$$ which are everywhere linearly independent.)

In order to flip between vector fields and forms, it may help you to construct a Riemannian metric on $$M$$.

The best exposition I know of basic material about manifolds is Volume I of Michael Spivak, A comprehensive introduction to differential geometry. I've seen other people say good things about John Lee, Introduction to smooth manifolds, but I've never read it.

 

[Canavar]

Thank you for your help.

Let $$\omega$$ be our volume form. , $$\omega :M->\Lambda^m M$$ That is $$\omega$$ assigns to each point a alternating tensor.
And we need a smooth vector field X:M->TM.

But i don't know how \omega induces a vector field.

$$\omega(p) \in \Lambda^m (T_p M)$$ This is a tensor in this space. Can you please give me another hint?

(excuse me, but this stuff is completely new for me)

Regards