I know that this is going to be slightly vague, but it's itching me: why is lower boundedness of the Ricci curvature required by some of the very important theorems in Riemannian geometry? The uniqueness theorem for the heat kernel, Bishop's and Gromov's volume comparison theorems - these are just the first that come to mind. What is so bad about having lower unbounded Ricci curvature, and why isn't upper unboundedness a problem at all? Why not the weaker assumption of lower boundedness of the? scalar curvature? I would be happy to be given some (intuitive) justifications, because no book on Riemannian geometry tackles this issue. Thank you.
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1$\begingroup$ Way too vague for me. $\endgroup$– Anton PetruninCommented Mar 27, 2015 at 19:20
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$\begingroup$ Agreed (and admitted as such from the very first words). Possibly because my intuition of the subject is vague. I just don't get why this hypothesis shows up as fundamental in so many results. What is so special about those manifolds not exhibiting it? $\endgroup$– Alex M.Commented Mar 27, 2015 at 19:22
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3$\begingroup$ Take a look at Gromov's paper Sign and geometric meaning of curvature. It has been 15 years since I read it, but I think there is something relevant there. $\endgroup$– Ben McKayCommented Mar 27, 2015 at 21:21
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1$\begingroup$ @AlexM. it simply impossible to give an answer here, one has to write 20 pages or so and yes Gromov's "Sign and geometric meaning of curvature" is a good sours. $\endgroup$– Anton PetruninCommented Mar 28, 2015 at 3:11
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