Sectional curvature for Riemannian manifolds with density

2013/11/01 by William Wylie, Wylie, William · 1 citation
Mathematics · #53C25 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C25

paper · pdf · doi:10.48550/arxiv.1311.0267

19 pages, The expositiion of the paper has been shortened by a few pages and some of the arguments streamlined at the suggestion of the referee. Final version, to appear in Geometriae Dedicata

arxiv created 2015/01/24 · arxiv updated 2015/01/27

Abstract

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.

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