# Applications of Inverse Function Theorem

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 $$Df^{-1}_y = \left[ Df_{f^{-1}(y)}\right]^{-1}.$$