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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!

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  • $\begingroup$ You might want to add the big-list tag :) $\endgroup$ – Severin Schraven Aug 28 at 20:10
  • $\begingroup$ @SeverinSchraven Thanks! $\endgroup$ – Alan Muniz Aug 28 at 20:35
  • $\begingroup$ Well, you can use it to prove the implicit function theorem, which has some nice applications, for example when proving necessary conditions for extrema with side constraints. $\endgroup$ – PhoemueX Aug 29 at 12:59
  • $\begingroup$ @PhoemueX Thanks for the comment. This is a nice application of the implicit function theorem present in every standard text book. I wonder why it is difficult to find other applications... Digging a bit one can find a proof of Picard's ODE theorem using the implicit function theorem (for banach spaces). But it seems to too fancy to me. $\endgroup$ – Alan Muniz Aug 29 at 13:07

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