A local diffeomorphism of Euclidean space that is not a diffeomorphism

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?

Solution 1:

Consider $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ defined by $f(x,y)=(e^x \cos y,e^x \sin y)$

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

In higher dimensions, one can use $f(x)=(e^{x_1}\cos x_2, e^{x_1}\sin x_2, x_3, \dots, x_n)$.