The weighted connection and sectional curvature for manifolds with density
Differential Geometry
2017-07-19 v1
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.
Keywords
Cite
@article{arxiv.1707.05376,
title = {The weighted connection and sectional curvature for manifolds with density},
author = {Lee Kennard and William Wylie and Dmytro Yeroshkin},
journal= {arXiv preprint arXiv:1707.05376},
year = {2017}
}