# Curvature and topology

I am studying Riemannian Geometry and I came across various Theorems which give conditions on the topology of a manifold given conditions on curvature, and vice-versa. Just to mention a few of them: Gauss-Bonnet (the version I know works only for surfaces), Hadamard (about negative sectional curvature), Bonnet-Myers (lower bound on Ricci curvature implies compactness), Preissman's theorem on compact manifolds with negative sectional curvature...

Unfortunately, I fail to see the "big picture". In particular, I am interested in the following questions:

• Is there a general heuristic by which I can encompass many of these assertions? In particular, to me it's hard to memorize things just as a bunch of facts.

• Would it be possible to make a list, as concise and at the same time as complete as possible, about the mutual interaction of curvature and topology?

• @LeeMosher, I definitely agree with you. On the other hand, when I see something like this: library.msri.org/books/Book30/files/abresch.pdf I get a little lost, since I have little of an idea where to start, how to organize ideas. For instance, what is the geometrical meaning of $\delta$-pinching? I understand the definition, but I don't have an intuition for it... – snefs Mar 4 '16 at 22:32