William Wylie
Abstract: In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. The trace of one of our notions of sectional curvature is the Bakry-Emery Ricci tensor which appears in the gradient Ricci soliton equation, while the trace of the other is the modified Ricci tensor that arises in the conformal Einstein equation. 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.
ArXiv: 1311.0267