Let $M,N,K$ be oriented smooth manifolds. Let $F,G$ be local diffeomorphisms.
Let $F: M \to N$ be orientation preserving and $G: N \to K$ be orientation reversing. Then is $G \circ F: M \to K$ orientation reversing?
Attempt: Let $p \in M$
Take a positively oriented basis $B$ of $T_p M$. Then $F_{*p} (B)$ is also a positively oriented basis of $T_{F(p)} N$ and $(G \circ F)_{*p}= G_{*, F(p)}(F_{*,p}(B))$ is negatively oriented. The result follows.
Is this correct?
Best Answer
Yes, this is correct. Another way to say this is that if you work in oriented local coordinates, an orientation-preserving diffeomorphism is just one whose Jacobian determinant is positive. So, this amounts to the statement that a matrix with positive determinant composed with a matrix of negative determinant gives a matrix of negative determinant.