  In this paper we introduce two new notions of sectional curvature for
Riemannian manifolds with density. Under both notions of curvature we classify
the constant curvature manifolds. We also prove generalizations of the theorems
of 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 radial
curvature equation and second variation formula and then apply the techniques
of classical Riemannian geometry to these new equations.
