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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?

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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. – Bruno Stonek Jan 24 '12 at 16:34

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?

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Don't forget that diffeomorphism means smooth with smooth inverse! – Zhen Lin Jan 24 '12 at 7:34
@Oliver : Well, I am currently reading Smooth Manifolds by John Lee. When I go to the wiki page about diffeomorphisms, I see the image of some sort of conformal map. I presume not all diffeomorphisms of open sets are conformal, but I was wondering if there was some weaker geometrical features that a general diffeomorphism preserves (say between open sets in $\mathbb{R}^3$)? Also, does the smoothness of transition maps in a manifold, say anything about the geometrical smoothness of a a surface in $\mathbb{R}^{3}$? I guess I am looking for some physical intuition of what diffeomorophisms are. – JamesMarshallX Jan 24 '12 at 17:34

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.

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