@article{c952de9eba7e433e88b6dd00961c01e2,
title = "The Weighted Connection and Sectional Curvature for Manifolds With Density",
abstract = "In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion-free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger{\textquoteright}s finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry.",
keywords = "Comparison geometry, Jacobi fields, Manifold with density, Sectional curvature, Sphere theorem",
author = "Lee Kennard and William Wylie and Dmytro Yeroshkin",
note = "Funding Information: Acknowledgements This work was partially supported by NSF Grant DMS-1440140 while Lee Kennard and William Wylie were in residence at MSRI in Berkeley, California, during the Spring 2016 semester. Lee Kennard was partially supported by NSF Grant DMS-1622541. William Wylie was supported by a grant from the Simons Foundation (#355608, William Wylie) and a grant from the National Science Foundation (DMS-1654034). Dmytro Yeroshkin was partially supported by a grant from the College of Science and Engineering at Idaho State University. We would like to thank the referee for a thorough reading of the paper and many suggestions that improve the readability of the text. Publisher Copyright: {\textcopyright} 2018, Mathematica Josephina, Inc.",
year = "2019",
month = jan,
day = "15",
doi = "10.1007/s12220-018-0025-3",
language = "English (US)",
volume = "29",
pages = "957--1001",
journal = "Journal of Geometric Analysis",
issn = "1050-6926",
publisher = "Springer New York",
number = "1",
}