Diffeomorphisms: Examples and Importance

Can anyone give a specific example of a diffeomorphism and also of composing a function with a diffeomorphism and how this helps mathematics as a whole? In other words, how does this fit into the grand scheme of mathematics?

Thanks

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The entirety of "calculus on manifolds" is based on the fact that the local diffeomorphism between a neighborhood on a manifold with an open set in Euclidean space allows you to do your computations relative to the Cartesian coordinate system. –  Willie Wong Aug 31 '11 at 16:27
@WillieWong Thank you. Could you give a specific example? –  analysisj Aug 31 '11 at 16:54

Diffeomorphisms are constructed between the unit tangent bundle of $S^2$, $SO_3$ and $\mathbb RP^3$ in this thread: The circle bundle of $S^2$ and real projective space

On a more nuts-and-bolts level, diffeomorphisms are the setting where you have a change of variables theorem for multi-variable integrals. http://en.wikipedia.org/wiki/Integration_by_substitution#Substitution_for_multiple_variables

Another place where diffeomorphisms come into the picture is if you have a vector field $\vec v$ defined on an open subset of Euclidean space, it generates a flow which is a 1-parameter family of diffeomorphisms. In a sense the process of "solving" an ODE is the process of going from the vector field to the flow. So for example, this leads to the connection that a divergence-free vector field generates an area/volume/content-preserving flow.

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Here is an example from Riemannian geometry that illustrates how diffeomorphisms can play a critical role in an important result from geometry. In a paper by Kazdan and Warner they show that any smooth function on a closed manifold of dimension 3 or higher can be a scalar curvature of a metric on the manifold, by solving the equation

$Lu=(K \cdot \phi) u^\frac{n+2}{n-2},$

where $\phi$ is a diffeomorphism, $L$ is the conformal Laplacian, $K$ is a constant, and $n$ is the dimension of the manifold. You can not solve

$Lu = f u^\frac{n+2}{n-2}$

for general $f$, in general, so the diffeomorphism $\phi$ play a critical role in the proof.

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+1. Thank you very much. –  analysisj Aug 31 '11 at 18:14