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.

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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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