Can a diffeomorphism between connected smooth manifolds be both orientation perserving and reversing? i.e. preserving orientation at one point, but revsersing at another point?
I'm reading a book on smooth manifolds where it defines a function as preserving orientation if it preserves orientation for every point in the domain. Similarly for a function reversing orientation if it reverses orientation for every point.
I wonder if there would be functions both orientation preserving and reversing, at different points in the domain. Say the manifold is connected since there're simple counterexamples when it's disconnected.
Hint: Given a diffeomorphism $f: X\to Y$ of oriented manifolds (connected or not), define two subsets:
$P(f)\subset X$, consisting of points where $f$ preserves orientation.
$R(f) \subset X$, consisting of points where $f$ reverses orientation.
Now prove that both subsets $P(f), R(f)$ are open in $X$. Then think about the case when $X$ is connected.