Two Basic Properties Of Diffeomorphisms

[Tsakanikas Nickos]

Let \( \displaystyle M \) and \( \displaystyle N \) be two differentiable manifolds.

1. If \( \displaystyle f : M \longrightarrow N \) is a diffeomorphism, show that for all \( \displaystyle p \in M \) the differential \( \displaystyle df_{p} : T_{p}(M) \longrightarrow T_{f(p)}(N) \) of \( \displaystyle f \) at \( \displaystyle p \) is a linear isomorphism.
2. Show that \( \displaystyle f : M \longrightarrow N \) is a local diffeomorphism if and only if its differential \( \displaystyle df_{p} : T_{p}(M) \longrightarrow T_{f(p)}(N) \) is a linear isomorphism for all \( \displaystyle p \in M \).