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.

  • 3
    $\begingroup$ No list of facts is going to give you a big picture, in my opinion. But what might teach you a big picture is a few well chosen facts --- I like your list so far --- together with a deep understanding of their proofs. $\endgroup$ – Lee Mosher Mar 4 '16 at 3:26
  • $\begingroup$ @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... $\endgroup$ – snefs Mar 4 '16 at 22:32
  • $\begingroup$ Maybe I should rephrase, is it possible to make a rough classification of such statements in categories, according to the intuition behind them and general heuristic ideas? $\endgroup$ – snefs Mar 4 '16 at 22:33
  • $\begingroup$ Looking at the link in your comment just above, I can perhaps understand your question a bit better. $\endgroup$ – Lee Mosher Mar 4 '16 at 23:03
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    $\begingroup$ Still, though, I think it would be better to formulate a narrower question. Pick one particular theorem: maybe the "Sphere Theorem" which is the first theorem in that link (if you like positive curvature); or the Hadamard theorem (if you like nonpositive curvature). Try to understand its proof. Once you have some understanding, try to formulate your "big picture" question in that narrower context. $\endgroup$ – Lee Mosher Mar 4 '16 at 23:07

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