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

Feres, Renato and Katok, Anatole (1990) Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature. Ergodic Theory And Dynamical Systems, 10 (4). pp. 657-670. ISSN 1469-4417. https://resolver.caltech.edu/CaltechAUTHORS:20170408-163655442

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Abstract

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 χ_μ > 0 such that the Lyapunov exponents are −2χ_μ, −χ_μ, 0, χ_μ, and 2χ_μ, 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.


Item Type:Article
Related URLs:
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http://dx.doi.org/10.1017/S0143385700005836DOIArticle
Additional Information:© Cambridge University Press 1990. (Received 13 December 1988)
Issue or Number:4
Record Number:CaltechAUTHORS:20170408-163655442
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170408-163655442
Official Citation:Feres, R., & Katok, A. (1990). Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature. Ergodic Theory and Dynamical Systems, 10(4), 657-670. doi:10.1017/S0143385700005836
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:76219
Collection:CaltechAUTHORS
Deposited By: 1Science Import
Deposited On:08 Mar 2018 22:02
Last Modified:03 Oct 2019 16:58

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