Let $U\subset\mathbb{R}^n$, $V\subset\mathbb{R}^m$ and a bijection
$f:U\to V$ is a diffeomorphism if $f$ and $f^{-1}$ are differentiable.
I would like to know the intuitive meaning of two open sets being diffeomorphic.
For example, if two spaces are homeomorphic, these spaces share the same topological properties. And we have a clear intuitive idea of two spaces being homeomorphs, like the classic relationship between a donut and a mug.
Is there a similar intuitive idea for diffeomorphism?
Edit: The proprieties that homeomorphism preserves is, for example, if one of them is compact, then the other is as well; if one of them is connected, then the other is as well; if one of them is Hausdorff, then the other is as well; their homotopy and homology groups will coincide.
So, what are the properties preserved by diffeomorphism?
I quoted homeomorphism, to indicate what I meant by an intuitive idea.
Best Answer
Diffeomorphisms are precisely the isomorphisms in the category of differentiable manifolds.
So it might be helpful to think about diffeomorphisms between differentiable manifolds in exactly the same way as you would think about homeomorphisms between topological spaces.
Since we consider spaces that are isomorphic to be "essentially the same" or "indistinguishable", this is what we have in mind in terms of differentiable manifolds whenever we talk about them being diffeomorphic.