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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
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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

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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