|XXXIX Workshop on Geometric Methods in Physics||28.06-4.07.2020|
|IX School on Geometry and Physics||22-26.06.2020|
Some new Ambrose-type theorems via Bakry-Emery Ricci curvature
One of the most fundamental topics in Riemannian geometry is to investigate the relation between topology and geometric structure on Riemannian manifolds. To give nice compactness criteria for complete Riemannian manifolds is one of the most natural and interesting problems in Riemannian geometry. The celebrated theorem of S. B. Myers (Duke Math. J. 24 (1957), 345–348) guarantees the compactness of complete Riemannian manifolds under some positive lower bound on the Ricci curvature. This theorem has been widely generalized in various directions by many authors. The first generalization was given by Ambrose (Duke Math. J. 8 (1941), 401–404), where the positive lower bound on the Ricci curvature was replaced with an integral condition of the Ricci curvature along some geodesics. Both Myers- and Ambrose-type theorems are closely related to General Relativity. In this talk I would like to present some new Ambrose-type theorems via Bakry-Emery Ricci curvature. Our Ambrose-type theorems extend previous Ambrose-type theorems obtained by P. Mastrolia, M. Rimoldi, and G. Veronelli.
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