Equivalence of statements of orientation of a manifold

Question

(From lectures) the following statements are equivalent and, if they hold, we say the manifold $M$ has an orienttation:

a) $\exists$ a nowhere vanishing top form $\alpha$ on $M$

b)There exist a system of coordinate neighborhoods covering $M$ and such that the Jacobian matrix corresponding to the change of corrdinates where two patches overlap, has positive determinant on every overlap.

c) The space of alternating top forms $\Lambda ^N T^* M$, is isomorphic to $M \times \mathbb{R}$

I am struggling to convince myself that $a) \implies b)$, and I think I am missing something really obvious.

If $\alpha = a(x_i)dx_1 \wedge ... \wedge dx_n$ in one set of coordinates, then under a coordinate transformation

$\alpha = a(x_i) det(\frac{\partial x_i}{\partial x'_j}) dx'_1 \wedge ... \wedge dx'_n$.

But if the determinant changes sign for at least one overlap of coordinate neighborhoods for every manifold covering , I don't see why this means the top form must vanish. Since we put the charts onto the manifold, why can the "observer" not merely say that their choice of coordinates is such that the one form is positive in one choice, and negative in another?

Answer 1

It looks like you are trying a proof by contradiction.

Instead of that, why not try a direct proof, like this:

• Assume that $\alpha$ is a nowhere vanishing top form on $M$,
• Use $\alpha$ to construct an appropriate system of coordinate neighborhoods covering $M$, and prove that the system so constructed does satisfy the conditions in b).

Rather than continuing with a complete proof, I'll just end with a hint for the construction.

To construct the desired system, consider an arbitrary coordinate neighborhood $$(x_1(p),x_2(p),...,x_m(p)), \quad u \in U \subset M $$ How might one use $\alpha$ to decide whether or not to admit this neighborhood into the desired system of coordinate neighborhoods satisfying (b)?