4
$\begingroup$

I just have some general questions about diffeomorphisms:

1) How can one geometrically interpret a diffeomorphism between two open sets in $\mathbb{R^{n}}$?

2) Typically morphisms preserve some type of structure. Beyond preserving the topology as a homeomorphism, what does a diffeomorphism preserve (if anything)?

3) What effect does the requirement that the transition maps of a smooth manifold be diffeomorphisms have on the geomotry of the manifold?

  • 0
    The structure preserved by diffeos is the *smooth structure*, i.e. that of a manifold, as Oliver's answer points out. The thing is, "open sets in $\mathbb{R}^n$" are such particular examples of manifolds that this fact is a bit obscured. As it is quite often the case, to understand this concept it is useful to *generalize* the situation a little bit, IMHO.2012-01-24

2 Answers 2