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In this paper we introduce two new notions of sectional curvature forRiemannian manifolds with density. Under both notions of curvature we classifythe constant curvature manifolds. We also prove generalizations of the theoremsof Cartan-Hadamard, Synge, and Bonnet-Myers as well as a generalization of the(non-smooth) 1/4-pinched sphere theorem. The main idea is to modify the radialcurvature equation and second variation formula and then apply the techniquesof classical Riemannian geometry to these new equations.

Sectional curvature for Riemannian manifolds with density