The Weighted Connection and Sectional Curvature for Manifolds With Density

Lee Kennard, William Wylie, Dmytro Yeroshkin

Research output: Contribution to journalArticle

1 Scopus citations

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’s finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry.

Original languageEnglish (US)
Pages (from-to)1-45
Number of pages45
JournalJournal of Geometric Analysis
DOIs
StateAccepted/In press - Apr 24 2018

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Keywords

  • Comparison geometry
  • Jacobi fields
  • Manifold with density
  • Sectional curvature
  • Sphere theorem

ASJC Scopus subject areas

  • Geometry and Topology

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