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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Acknowledgments
The author would like to thank Guofang Wei, Peter Petersen, and Frank Morgan for their encouragement and very helpful discussions and suggestions that improved the draft of this paper.
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The author was supported in part by NSF-DMS Grant 0905527.
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Wylie, W. Sectional curvature for Riemannian manifolds with density. Geom Dedicata 178, 151–169 (2015). https://doi.org/10.1007/s10711-015-0050-3
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DOI: https://doi.org/10.1007/s10711-015-0050-3