Geometry of diffeomorphism 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?
 A: Diffeomorphisms preserve the smooth structure of the manifold. If the transition maps of a manifold are just homeomorphisms instead of diffeomorphisms, then the manifold is just a topological manifold rather than a smooth one. If I have a homeomorphism between open sets of $\mathbb{R}^n$, it is a diffeomorphism iff it is smooth in the calculus sense. 
I'm not sure if this helps at all. If not, can you clarify what background you are coming with?
A: Ad 1: Consider a map $f\colon\ \Omega\to\Omega'$ which is only a homeomorphism or even a $C^1$ diffeomorphism, and assume $f(p)=q$. When $f$ is only a homeomorphism, a small $\epsilon$-neighborhood $U_\epsilon$ of $p$ is mapped homeomorphically onto a certain  neighborhood $V$ of $q$ of pretty arbitrary shape. When $f$ is a diffeomorphism then the increment $f(p+X)-f(p)$ for small $|X|$ is in first approximation a linear function of $X$; therefore $V=f(U_\epsilon)$ will look like an ellipsoid.
Ad 2: A homeomorphism maps curves onto curves, and when two curves meet at some point $p$ then their images will meet at $f(p)$, and that's it. When $f$ is a diffeomorphism it makes sense to look at the tangent direction (resp. at the velocity vector, when time is involved) of such curves. When they intersect transversally at $p$, then their images will intersect transversally at $f(p)$, and if they are mutually tangent at $p$ then their images will be tangent also.
Ad 3: The effect is that you can do differential geometry on the manifold $M$ even if you don't have a single coordinate system that covers all of $M$ (as in the case of an $\Omega\subset{\mathbb R}^n$). In particular you can study the orbits of differential equations $\dot x= X(x)$ where $X(\cdot)$ is a vector field on $M$. Thanks to the transformation rules for tangent vectors not only the direction of $X$ but also its "size" has an invariant meaning.
