Diffeomorphisms and the inverse function theorem.

A function $f \colon U \to V$ where $U$ and $V$ are open subsets of $\mathbb{R}^n$ is called a diffeomorphism if it is smooth, invertible and has smooth inverse. If $f$ is a diffeomorphism $f \circ f^{-1} = 1_{ \mathbb{R}^n}$ so it follows from the chain rule that at any point $x \in U$

$$1_{\mathbb{R}^n} = d(1_{\mathbb{R}^n})(x) =d{f^{-1}}(f(x)) \circ df(x)$$

so that $(df(x))^{-1} = d{f^{-1}}(f(x))$. That is, the inverse of the linear map $df(x)$ is the linear map $d{f^{-1}}(f(x))$. Notice that this means that a diffeomorphism necessarily goes from an open subset of $\mathbb{R}^n$ to an open subset of $\mathbb{R}^m$ where $n=m$ so we have lost nothing by putting that in the definition.

It is also useful to have the notion of a local diffeomorphism. We say that $f \colon U \to \mathbb{R}^n$ is a local diffeomorphism at $x \in U$ if there is an open subset $V$ of $\mathbb{R}^n$ containing $x$ such that $f(V)$ is open and $f \colon V \to f(V)$ is a diffeomorphism.

With this notion we have the important inverse function theorem:

Theorem 1.2 (Inverse Function Theorem)   Let $U$ be an open subset of $\mathbb{R}^n$ and $f \colon U \to \mathbb{R}^n$ be a smooth function such that $df(x)$ is invertible. Then $f$ is a local diffeomorphism at $x$ and $d(f^{-1})(f(x)) = (df(x))^{-1}$.

The Lemma proved in the previous section also gives us a characterisation of diffeomorphism:

Lemma 1.2   Let $U$ and $V$ be open subsets of $\mathbb{R}^n$. A bijection $\phi \colon U \to V$ is a diffeomorphism if and only if for every function $f \colon V \to \mathbb{R}$ we have that $f$ is differentiable if and only if $f \circ \phi \colon U \to \mathbb{R}$ is differentiable.

Proof. We just apply Lemma 1.1 to $\phi$ and $\phi^{-1}$. $\qedsymbol$