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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 $2\pi$).


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There is a whole host of theorems about smooth homotopy from differential topology that we can lift to results about topological manifolds by proving that continuous maps can be approximated by smooth ones. For instance, this is one of the easiest ways to prove Brouwer's fixed point theorem. –  Harry Gindi Mar 25 '10 at 2:23
By the way, The Fàry-Milnor theorem should always be mentioned along a beautiful result of Milnor which proves a conjecture of Borsuk that the total curvature of a knotted closed curve in $\mathbb R^2$ is at least $4\pi$, so that the knottedness implies curviness... –  Mariano Suárez-Alvarez Mar 25 '10 at 5:15
in $\mathbb R^3$, I mean... –  Mariano Suárez-Alvarez Mar 25 '10 at 5:17
There's a lot of these kinds of theorems. For example, given a knot in $\mathbb R^3$ you can compute the $z^2$-coefficient of the Alexander-polynomial of the knot (a topological invariant) from the straight lines that intersect the knot in $4$ points (a geometric invariant). –  Ryan Budney Mar 25 '10 at 5:21
@Mariano: I'm confused. What is the difference between the Fary-Milnor theorem and the beautiful result of Milnor that you mention? They same one and the same to me. –  Andrea Ferretti May 19 '10 at 20:40

6 Answers 6

up vote 4 down vote accepted

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).

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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.

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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.

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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.

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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.

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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.

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Thanks for all the helpful responses –  kangdon Mar 26 '10 at 3:51

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