Vol. 306, No. 1, 2020

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Central splitting of manifolds with no conjugate points

James Dibble

Vol. 306 (2020), No. 1, 95–114
Abstract

Each compact Riemannian manifold with no conjugate points admits a family of functions whose integrals vanish exactly when central Busemann functions split linearly. These functions vanish when all central Busemann functions are sub- or superharmonic. When central Busemann functions are convex or concave, they must be totally geodesic. These yield generalizations of the splitting theorems of O’Sullivan and Eberlein for manifolds with no focal points and, respectively, nonpositive curvature.

Keywords
no conjugate points, no focal points, Busemann function, center theorem
Mathematical Subject Classification 2010
Primary: 53C20, 53C24
Secondary: 53C22
Milestones
Received: 30 July 2018
Revised: 18 September 2019
Accepted: 2 January 2020
Published: 14 June 2020
Authors
James Dibble
Department of Mathematics
University of Iowa
Iowa City, IA
United States
Department of Mathematics and Statistics
University of Southern Maine
Portland, ME
United States