Immersions that are not diffeomorphisms from open sets of $\mathbb{R}^n$ to itself

Question

Let’s say I have an immersion $f:\Omega \to \mathbb{R}^n$, with $\Omega$ an open set of $\mathbb{R}^n$. It is a local diffeomorphism (because its differential is injective and thus bijective), so it is an open map. If I restrict the image and consider it to be not $\mathbb{R}^n$, but $f(\Omega)$, it would also be surjective. The only way it could not be a diffeomorphism is to not be injective, so I was wondering: are there counterexamples? Maybe that work on all dimensions (I don’t see how to give a counterexample in dimension 1, but it might be a special case where there are actually no counterexamples, while in greater dimensions there are)

Answer 1

The fundamental counterexample which shows up repeatedly as you proceed is the exponential mapping $f\colon \Bbb C\to\Bbb C-\{0\}$, $f(z)=e^z$. (In real coordinates, this is, of course, $f(x,y) = (e^x\cos y,e^x\sin y)$.) Each point has infinitely many preimages, but the map is a local diffeomorphism at every point of the domain.