Hi people,
I'm interested in results, such as the Gauß-Bonnet theorem, Fàry-Milnor theorem or classification theorems for manifolds, which give topological properties from geometric considerations. Can anyone recommend some good texts? In particular I'd like to see a nice proof of Fàry-Milnor and of the theorem of turning tangents (total curvature of an imbedded plane curve is $2\pi$).
thanks
About the Fary–Milnor theorem. Milnor's original proof is already very nice (see here). I also very much like this proof by Alexander & Bishop (see also a version of this proof in my book).
A nice topic to read about is Chern-Weil theory. This is the generalisation of Gauss-Bonnet to higher dimensions and to vector bundles other than the tangent bundle. Put very briefly, topological invariants of a vector bundle over a manifold (its characteristic classes - certain classes in the cohomology of the base) can be computed using the curvature tensor of any choice of connection in the bundle.
The prototype is Gauss-Bonnet in which, as you know, the Euler characteristic of a (compact orientable) surface is equal to a fixed constant times the integral of the scalar curvature of any Riemannian metric on the surface.
This topic being quite large, I cannot insist enough to recommand you to take a look to Marcel Berger's Panoramic view of Riemannian geometry. The Bonnet-Myers theorem, the sphere theorems (for the recent development on this one, I think the web page of Simon Brendle contains a survey) are two celebrated examples of the topological consequences of geometric properties in the setting or Riemannian geometry.
I think you should read something about the Ricci flow and Perelmann s work (for 3mfs), or Seiberg Witten/Yang-Mills theory (for 4-mfs). These theories give you very deep results in topology. But the hole theory is geometric.
You might want to look up some things about index theorems (particularly Atiyah-Singer). They tend to relate topological and geometric data, so you can put geometric data in and topological data out.
Here is an example where topological objects are constructed from geometrical data through representation theory. Let G/P be a flag variety of a complex Lie group G. Let G0 be a real form of G, and D be an open orbit of G0 in G/P. The Dolbeault cohomology spaces H^n(D, L) of line bundles over D carry irreducible representations of G0 which can be constructed from geometrical data of the orbit. Here is a review article on the subject. When G0 is compact, this construction reduces to the famous Bott-Borel-Weil theorem.
