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Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature

Published online by Cambridge University Press:  19 September 2008

Renato Feres
Affiliation:
Mathematics 253–37, California Institute of Technology, Pasadena, California 91125, USA
Anatole Katok
Affiliation:
Mathematics 253–37, California Institute of Technology, Pasadena, California 91125, USA
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Abstract

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We consider Anosov flows on a 5-dimensional smooth manifold V that possesses an invariant symplectic form (transverse to the flow) and a smooth invariant probability measure λ. Our main technical result is the following: If the Anosov foliations are C∞, then either (1) the manifold is a transversely locally symmetric space, i.e. there is a flow-invariant C∞ affine connection ∇ on V such that ∇R ≡ 0, where R is the curvature tensor of ∇, and the torsion tensor T only has nonzero component along the flow direction, or (2) its Oseledec decomposition extends to a C∞ splitting of TV (defined everywhere on V) and for any invariant ergodic measure μ, there exists χμ > 0 such that the Lyapunov exponents are −2χμ, −χμ, 0, χμ, and 2χμ, μ-almost everywhere.

As an application, we prove: Given a closed three-dimensional manifold of negative curvature, assume the horospheric foliations of its geodesic flow are C∞. Then, this flow is C∞ conjugate to the geodesic flow on a manifold of constant negative curvature.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[A]Anosov, D. V.. Geodesic flows on closed Riemann manifolds with negative curvature. Proc. Steklov Institute of Mathematics, n. 90 (1967).Google Scholar
[B]Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 130.CrossRefGoogle Scholar
[F]Feres, R.. PhD. Thesis, California Institute of Technology, 1989.Google Scholar
[F-K]Feres, R. & Katok, A.. Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows. Ergod. Th. & Dynam. Sys. 9 (1989), 427432.CrossRefGoogle Scholar
[Fl-K]Flaminio, L. & Katok, A.. Rigidity of symplectic Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
[H]Helgason, S.. Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York (1978).Google Scholar
[K]Kanai, M.. Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations. Ergod. Th. & Dynam. Sys. 8 (2) (1988), 215240.CrossRefGoogle Scholar
[KN]Kobayashi, S. & Nomizu, K.. Foundations of Differential Geometry. Vol. I, John Wiley & Sons, New York (1963).Google Scholar