On tori without conjugate points

Christopher B. Croke, Bruce Kleiner

Research output: Contribution to journalArticlepeer-review


In this paper we consider Riemannian metrics without conjugate points on an n-torus. Recent work of J. Heber established that the gradient vector fields of Busemann functions on the universal cover of such a manifold induce a natural foliation (akin to the weak stable foliation for a Riemannian manifold with negative sectional curvature) on the unit tangent bundle. The main result in the paper is that the metric is flat if this foliation is Lipschitz. We also prove that this foliation is Lipschitz if and only if the metric has bounded asymptotes. This confirms a conjecture of E. Hopf in this case.

Original languageEnglish (US)
Pages (from-to)241-257
Number of pages17
JournalInventiones Mathematicae
Issue number1
StatePublished - Dec 1995

ASJC Scopus subject areas

  • General Mathematics


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