Orientation preserving maps and volume forms.

Question

Suppose that M is a (connected) oriented manifold and ω is an orientation form on M.

1. Prove that f is orientation preserving if and only if $f^∗ω$ = g.ω where g is a positive function on M.

[We use the definition of f being orientation preserving to be Df being orientation preserving at the level of tangent space]

1. Now USING PART 1. show that the antipodal map from $S^n$ to $S^n$ is orientation preserving iff n is odd.

There are solutions to part 2 elsewhere on this website but they do NOT seem to use part 1 as a way to prove it. Any help?

Answer 1

As far as I know, the solution of part 2 would utilize the volume form $\omega$ of $S^n$ and prove that $f^*\omega = (-1)^{n+1} \omega$ where $f$ is the antipodal map. As part 1 suggests, $f$ is orientation preserving iff $(-1)^{n+1}$ is positive.