Title: The weighted connection and sectional curvature for manifolds with density,
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.