well,that's pretty much the question. I'd like to know if somebody could point me out if there's any geometrical implications following an upper bound on the Ricci curvature on a riemannian manifold. Specifically, let $(M^n,g)$ be a n-dimensional rimannian manifold with $\text{Ric} \leq k $. For $k > 0, k=0, k<0$ what else can we say about the geometry of $M^n$?

I mean, I know that negative ricci curvature gives has no topological obstructions (Lohkamp) and also that it is known that it's isometry group is finite, but nothing else. Any help is greatly appreciated, thanks!

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    $\begingroup$ All useful theorems seem to involve lower Ricci curvature bounds (and there are many such results), not upper bounds (apart from dimension 2 of course). $\endgroup$ – Moishe Kohan Aug 28 '14 at 22:07
  • $\begingroup$ yes thanks, this i know a bit. I'm just curious about the existence of other geometrical results given upper bounds apart from that one that i stated. I mean even if they are not so 'useful'. $\endgroup$ – Bruce Wayne Sep 15 '14 at 13:10

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