Definition of diffeomorphism functions

Question

I see the definition of Diffeomorphism in Wikipedia homepage, but I don't understand whether "the differential of f (Dfx : Rn → Rn) should be bijective at each point x in U" or "f itself" should be bijective?

Answer 1

To answer the question regarding the what the Wikipedia article say; you need that $Df_{x}$ is bijective for all $x\in U$. This is due to a theorem by Hadamard, and is in essence saying that if $f$ is locally diffeomorphic and $f$, $U$, and $V$ are "nice" then $f$ will in fact be globally diffeomorphic. The answer given by t.b to this question gives a nice exposition and references on this.

That said this is not the definition of a diffeomorphism, but is instead a more advanced characterization of when a local diffeomorphism might in fact be a global diffeomorphism. A map $f:U\rightarrow V$ is a differomorphism if $f$ is bijective and smooth and $f^{-1}$ is also smooth.

Answer 2

I think @JBruce's answer, although elegant and sophisticated, might mislead you. The definition of a diffeomorphism $f\colon M\to N$ is that it is a smooth bijection with a smooth inverse. It now follows that for each $x\in M$, $df_x\colon T_xM\to T_{f(x)}N$ is a vector space isomorphism. Hadamard's Theorem gives some very restrictive conditions under which one can know that a local diffeomorphism is actually a global one.