The Weighted Connection and Sectional Curvature for Manifolds With Density
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In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion-free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger’s finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry.
KeywordsComparison geometry Sectional curvature Manifold with density Jacobi fields Sphere theorem
Mathematics Subject Classification53C20
This work was partially supported by NSF Grant DMS-1440140 while Lee Kennard and William Wylie were in residence at MSRI in Berkeley, California, during the Spring 2016 semester. Lee Kennard was partially supported by NSF Grant DMS-1622541. William Wylie was supported by a grant from the Simons Foundation (#355608, William Wylie) and a grant from the National Science Foundation (DMS-1654034). Dmytro Yeroshkin was partially supported by a grant from the College of Science and Engineering at Idaho State University. We would like to thank the referee for a thorough reading of the paper and many suggestions that improve the readability of the text.
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