Abstract:
Two basic rigidity theorems of nonnegative Ricci curvature manifolds are the maximum diameter theorem and the splitting theorem. Each of them asserts if some geometric quantity (diameter) is as large as possible then the metric has to be a warped product of a particular type. Cheeger-Colding 1996 and Colding 1996 showed that if some quantity is almost maximal then the almost rigidity result hold in the sense that the given space is close to the model space under Gromov-Hausdorff distance. I will report these results starting with some basic materials.