I am searching for applications of the Inverse Function Theorem for smooth maps:
Inverse Function Theorem. Let $E\subset \mathbb{R}^n$ be an open subset and let $f\colon E \rightarrow \mathbb{R}^n$ be a $\mathscr{C}^\infty$ map. If for some $a\in U$, $Df_a$ is invertible, then there exists $U,V$ open sets such that $a\in U\subset E$, $V=f(U)$ and $f$ has a $\mathscr{C}^\infty$ inverse $g\colon V \rightarrow U$.
My aim is to gather a list of such applications in one place (here). To be more specific, I would like to know what kind of useful results one can prove using the theorem. Like
- Existence of "normal forms" for smooth maps with some property;
- Examples of interesting classes of maps whose inverses fall in the same class (the holomorphic ones, for instance);
- Existence of "nice" local coordinates in Differential Geometry;
...
The reason I am asking this question is the difficulty I am having to find such applications in literature.
One application that I would like to point out is to prove the holomorphic inverse function theorem.
Since $f\colon E\subset \mathbb{C} \rightarrow \mathbb{C}$ being holomorphic boils down to $f$ being differentiable and $Df_xJ = JDf_x$ for some linear operator $J$ such that $J^2 = -Id$ and for every $x\in E$ (Cauchy- Riemann equations). In particular, $f$ is $\mathscr{C}^\infty$ (analytic, in fact) by Goursat's theorem. Moreover $Df_a \neq 0$ implies $Df_a$ invertible and $f$ has a local inverse whose derivative also commutes with $J$ because $$Df^{-1}_y = \left[ Df_{f^{-1}(y)}\right]^{-1}.$$
Thank you in advance!