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?