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I hope this finds you all well.

Could someone give me an example of a local diffeomorphism from $\mathbb{R}^p$ to $\mathbb{R}^p$ (function of class say $C^k$ with an invertible differential map in each point) that is not a diffeomorphism..

in the real line (1 dim case) that would mean a function with a continuous non null derivative on an open $V$ of $\mathbb{R}$ that is not bijective which does not make sense thus any local diffeomorphism on the real line is a diffeo..

Could one give me a counterexample in a higher dimension?

Thanks

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Take the complex exponential $e^z$ as a function from $R^2$ to $R^2$. For every horizontal strip [iy+ i(y+2Pi) ) (i.e., including iy, but not i(y+2Pi)) there is an inverse--a logz --, but there is no global inverse. –  gary May 15 '11 at 1:37
    
Thank you a lot gary –  El Moro May 15 '11 at 1:40
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there is no inverse around zero –  yoyo May 15 '11 at 2:27
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Correction: that should be the strip [iy, i(y+2Pi)) –  gary May 15 '11 at 2:38
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@yoyo: I assume you mean the target zero , right? Otherwise, the inverse function guarantees the existence of a local diffeomorphism at each point, including 0 in the domain, since d/dz($e^z$)=1 at z=0; same for all other points' or using the $R^2$ version of the inverse function theorem. –  gary May 15 '11 at 22:36

1 Answer 1

consider $f:\mathbb{R}^2\rightarrow \mathbb{R}^2, f(x,y)=(e^x cosy,e^x siny)$

$Df(x,y)$ is always invertible because $det (Df(x,y))=e^{2x}$ but clearly f is not one to one.it is periodic with period $2\pi$

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