$F_{*p} : T_pM \to T_pN$ is an isomorphism of vector spaces for any $p \in M$.

Question

Let $F : M \to N$ be a diffeomorphism between smooth manifolds. Show $F_{*p} : T_pM \to T_{F(p)}N$ is an isomorphism of vector spaces for any $p \in M$.

Here $F_{*p}$ means pushforward/differential of the map $F$.

How to proceed with the problem?

Answer 1

Hint: If $f$ is the identity, $F_{*p}$ is the identity, $(F\circ F^{-1}=Id_N, F^{-1}\circ F=Id_M$ and $d(F\circ G)_p=dF_{G(p)}\circ dG_p$.