Abstract: In Riemannian geometry, a maximal rigidity on an n-manifold M of Ricci curvature bounded below by (n - 1)H is a statement that a geometric or topological quantity of M is bounded above by that of an n-manifold of constant sectional curvature H, and \=" implies that M is isometric to an n-space form. For instance, the maximal diameter rigidity (Myers, Cheng) and the maximal volume rigidity (Bishop) for H = 1, the maximal first Betti number rigidity for H = 0(Bochner, Anderson, Ye), and the maximal volume entropy rigidity for H = -1(Ledrappier-Wang). In this talk, we will survey some recent advances in Metric Riemannian geometry in establishing quantitative maximal rigidities, or in extending (quantitative) maximal rigidities to some singular metric spaces (Alexandrov spaces RCD*-spaces).
