The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 957–1001

# The Weighted Connection and Sectional Curvature for Manifolds With Density

• Lee Kennard
• William Wylie
• Dmytro Yeroshkin
Article

## 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

Comparison geometry Sectional curvature Manifold with density Jacobi fields Sphere theorem

53C20

## Notes

### Acknowledgements

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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