Ricci curvature
Here is a short list of global results concerning manifolds with positive Ricci curvature; see also classical theorems of Riemannian geometry. Briefly, positive Ricci curvature of a Riemannian manifold has strong topological consequences, while (for dimension at least 3), negative Ricci curvature has "no" topological implications. (The Ricci curvature is said to be positive if the Ricci curvature function is positive on the set of non-zero tangent vectors .) Some results are also known for pseudo-Riemannian manifolds.