The weighted connection and sectional curvature for manifolds with density

TL;DR

This paper investigates sectional curvature bounds in manifolds with density using a weighted connection, introducing new comparison tools and extending classical theorems to the weighted setting.

Contribution

It develops a weighted Rauch comparison theorem and a modified convexity notion, generalizing key geometric theorems for manifolds with density.

Findings

Generalized Preissman and Byers theorems for negative curvature

Extended quarter-pinched sphere theorem to weighted manifolds

Improved Cheeger's finiteness theorem for spaces with positive weighted sectional curvature

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.