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Abstract
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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.
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Keywords
no conjugate points, no focal points, Busemann function,
center theorem
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Mathematical Subject Classification 2010
Primary: 53C20, 53C24
Secondary: 53C22
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Milestones
Received: 30 July 2018
Revised: 18 September 2019
Accepted: 2 January 2020
Published: 14 June 2020
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