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arXiv · 1707.05376

The weighted connection and sectional curvature for manifolds with density

Lee Kennard·William Wylie·Dmytro Yeroshkin
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Abstract

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.

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Lee Kennard, William Wylie, Dmytro Yeroshkin. 2017-07-17. The weighted connection and sectional curvature for manifolds with density. https://arxiv.org/abs/1707.05376

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