摘要：

The classical theorem in Riemannian geometry, the Myers’s theorem, says that a compact Riemannian manifold with positive Ricci curvature has finite fundamental group. Another classical theorem, the Bochner theorem, asserts that a compact Riemannian manifold with negative Ricci curvature has finite isometry group. In this talk, I will show how to generalize these two classical theorems to the almost nonnegative Ricci curvature and almost nonpositive Ricci curvature cases respectively. Our main tools are the Atiyah-Singer index theorem and the rigidity theorems for genera. This represents our joint work with Xiaoyang Chen and Jian Ge.