On tori without conjugate points

Christopher B. Croke, Bruce Kleiner

Research output: Contribution to journalArticle

Abstract

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
Volume120
Issue number1
DOIs
StatePublished - Dec 1995

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Conjugate points
Foliation
Torus
Lipschitz
Unit tangent vector
Gradient vector
Metric
Universal Cover
Asymptote
Negative Curvature
Tangent Bundle
Sectional Curvature
Riemannian Metric
Riemannian Manifold
Vector Field
If and only if

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On tori without conjugate points. / Croke, Christopher B.; Kleiner, Bruce.

In: Inventiones Mathematicae, Vol. 120, No. 1, 12.1995, p. 241-257.

Research output: Contribution to journalArticle

Croke, Christopher B. ; Kleiner, Bruce. / On tori without conjugate points. In: Inventiones Mathematicae. 1995 ; Vol. 120, No. 1. pp. 241-257.
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