Ricci curvature
Ricci curvature is now known to have "no" topological implications; has shown that any manifold of dimension greater than two admits a complete Riemannian metric of negative Ricci curvature. In the case of two-dimensional manifolds, negativity of the Ricci curvature is synonymous with negativity of the Gaussian curvature, which has very clear topological implications. There are very few two-dimensional manifolds which fail to admit Riemannian metrics of negative Gaussian curvature.