Sectional curvature for Riemannian manifolds with density
Differential Geometry
2015-01-27 v3
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.
Keywords
Cite
@article{arxiv.1311.0267,
title = {Sectional curvature for Riemannian manifolds with density},
author = {William Wylie},
journal= {arXiv preprint arXiv:1311.0267},
year = {2015}
}
Comments
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