Topological results from geometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:00:51Z http://mathoverflow.net/feeds/question/19258 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19258/topological-results-from-geometry Topological results from geometry kangdon 2010-03-25T01:41:13Z 2010-05-19T20:25:20Z <p>Hi people,</p> <p>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 $2\pi$).</p> <p>thanks</p> http://mathoverflow.net/questions/19258/topological-results-from-geometry/19271#19271 Answer by Igor Pak for Topological results from geometry Igor Pak 2010-03-25T04:11:38Z 2010-03-25T04:11:38Z <p>About the Fary–Milnor theorem. Milnor's original proof is already very nice (see <a href="http://www.jstor.org/stable/1969467" rel="nofollow">here</a>). I also very much like <a href="http://www.jstor.org/stable/119165" rel="nofollow">this proof</a> by Alexander &amp; Bishop (see also a version of this proof in <a href="http://www.math.ucla.edu/~pak/book.htm" rel="nofollow">my book</a>).</p> http://mathoverflow.net/questions/19258/topological-results-from-geometry/19284#19284 Answer by Sebastian for Topological results from geometry Sebastian 2010-03-25T07:22:47Z 2010-03-25T07:22:47Z <p>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.</p> http://mathoverflow.net/questions/19258/topological-results-from-geometry/19293#19293 Answer by Charles Siegel for Topological results from geometry Charles Siegel 2010-03-25T11:11:53Z 2010-03-25T11:11:53Z <p>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.</p> http://mathoverflow.net/questions/19258/topological-results-from-geometry/19298#19298 Answer by Joel Fine for Topological results from geometry Joel Fine 2010-03-25T12:34:26Z 2010-03-25T12:55:25Z <p>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. </p> <p>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.</p> http://mathoverflow.net/questions/19258/topological-results-from-geometry/19302#19302 Answer by David Bar Moshe for Topological results from geometry David Bar Moshe 2010-03-25T13:25:06Z 2010-03-25T13:25:06Z <p>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 <a href="http://cauchy.math.okstate.edu/~zierau/papers/parkcity/Zierau.pdf" rel="nofollow">article</a> on the subject. When G0 is compact, this construction reduces to the famous Bott-Borel-Weil theorem.</p> http://mathoverflow.net/questions/19258/topological-results-from-geometry/25274#25274 Answer by Benoît Kloeckner for Topological results from geometry Benoît Kloeckner 2010-05-19T20:25:20Z 2010-05-19T20:25:20Z <p>This topic being quite large, I cannot insist enough to recommand you to take a look to Marcel Berger's <em>Panoramic view of Riemannian geometry</em>. 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.</p>