Ricci Curvature - Global Geometry and Topology

Global Geometry and Topology

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 Ric(ξ,ξ) is positive on the set of non-zero tangent vectors ξ.) Some results are also known for pseudo-Riemannian manifolds.

1. Myers' theorem states that if the Ricci curvature is bounded from below on a complete Riemannian manifold by, then the manifold has diameter, with equality only if the manifold is isometric to a sphere of a constant curvature k. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental group.
2. The Bishop–Gromov inequality states that if a complete m-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a ball is smaller or equal to the volume of a ball of the same radius in Euclidean m-space. Moreover, if denotes the volume of the ball with center p and radius in the manifold and denotes the volume of the ball of radius R in Euclidean m-space then function is nonincreasing. (The last inequality can be generalized to arbitrary curvature bound and is the key point in the proof of Gromov's compactness theorem.)
3. The Cheeger-Gromoll splitting theorem states that if a complete Riemannian manifold with contains a line, meaning a geodesic γ such that for all, then it is isometric to a product space . Consequently, a complete manifold of positive Ricci curvature can have at most one topological end. The theorem is also true under some additional hypotheses for complete Lorentzian manifolds (of metric signature (+−−...)) with non-negative Ricci tensor (Galloway 2000).

These results show that positive Ricci curvature has strong topological consequences. By contrast, excluding the case of surfaces, negative Ricci curvature is now known to have no topological implications; Lohkamp (1994) has shown that any manifold of dimension greater than two admits a Riemannian metric of negative Ricci curvature. (For surfaces, negative Ricci curvature implies negative sectional curvature; but the point is that this fails rather dramatically in all higher dimensions.)