Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T12:47:37.519Z Has data issue: false hasContentIssue false
  • English
  • Français

When is an Anosov flow geodesic?

Published online by Cambridge University Press:  19 September 2008

Leon W. Green
Leon W. Green
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X, H+, H− be vector fields tangent, respectively, to an Anosov flow and its expanding and contracting foliations in a compact three-dimensional manifold, with γ, α+, α− one forms dual to them. If α+([H+, H−]) = α−([H+, H−]) and γ([H+, H−]) = α−([X, H−]) − α+([X, H+]), then the manifold has the structure of the unit tangent bundle of a Riemannian orbifold with geodesic flow field X.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 2 , June 1992 , pp. 227 - 232
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[B]Besse, A. L.. Manifolds all of whose Geodesies are Closed. Ergebnisse, Band 93. (Springer: Berlin-Heidelberg-New York, 1978).CrossRefGoogle Scholar
[BFL]Benoist, Y., Foulon, P. & Labourie, F.. Flots d' Anosov à distributions stables et instable différentiables. Preprint.Google Scholar
[CFF]Croke, C., Fathi, A. & Feldman, J.. The marked length-spectrum of a surface of nonpositive curvature. Preprint.Google Scholar
[E]Eberlein, P.. When is a geodesic flow Anosov? I, II. J. Diff. Geom. 8 (1973), 437–463. 565–577.Google Scholar
[FO]Feldman, J. & Ornstein, D.. Semi-rigidity of horocycle flows over surfaces of variable negative curvature. Ergod. Th. & Dynam. Sys. 7 (1987), 49–72.CrossRefGoogle Scholar
[G1]Ghys, E.. Flots d' Anosov sur les 3-variétés fibrées en cercles. Ergod. Th. & Dynam. Sys. 4 (1984), 67–80.CrossRefGoogle Scholar
[G2]Ghys, E.. Flots d' Anosov dont les feuilletages stables sont différentiables. Ann. Sci. École. Norm. Sup. 20 (1987), 251–270.CrossRefGoogle Scholar
[K]Klingenberg, W.. Riemannian manifolds with geodesic flow of Anosov type. Ann. Math. 99 (2) (1974), 1–13.CrossRefGoogle Scholar
[KN]Kobayashi, S. & Nomizu, K.. Foundations of Differential Geometry. Vol. I. (John Wiley: New York, 1963).Google Scholar
[LS]Livshits, A. N. & Sinai, Ja. G.. On invariant measures compatible with the smooth structure for transitive U-systems. Dokl. Akad. Nauk SSSR 207 (1972), 1039–1041; (Translated in Sov. Math. Dokl. 13 (1972), 1656–1659.Google Scholar
[P]Plante, J. F.. Anosov flows. Amer. J. Math. XCIV (1972), 729–754.CrossRefGoogle Scholar
[T]Thurston, W.. The Geometry and Topology of 3-manifolds. Lecture Notes. Princeton University (1979).Google Scholar