TITLE: Topological results from geometry
QUESTION [5 upvotes]: 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

REPLY [5 votes]: 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.