# Generalizations of Inverse Function Theorem

A beautiful exercise in Guilleman and Pollack asks us to show the following generalization of the Inverse Function Theorem:

Suppose $f: M \to N$ is a map of smooth manifolds, and $Z$ is a compact submanifold of $M$ such that $\left. f\right|_Z$ is injective, and $f_*$ is an isomorphism at each point of $Z$. Then $f$ maps an open neighborhood of $Z$ diffeomorphically onto an open neighborhood of $f(Z)$.

At the risk of asking a slippery question, is this the "most general" version of the IFT, or is there one more general yet?

## 1 Answer

Eric, even in Guillemin & Pollack you'll find a more general version. Look at Exercise 14 on p. 56. It removes the compactness hypothesis on $Z$.

There are also interesting questions to ask along the lines of this: If $f\colon\Bbb R^n\to\Bbb R^n$ (replace with manifolds if you wish) is a local diffeomorphism at each point, what condition(s) are sufficient to guarantee that $f$ is a global diffeomorphism?

• Orthogonally, one can generalize into the infinite dimensional setting. – user98602 Aug 22 '15 at 17:04