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