1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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?

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.