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?
I would appreciate your help.