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Anasov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature. (English) Zbl 0729.58039
The authors consider an Anosov flow on a 5-dimensional smooth manifold V that possesses an invariant symplectic form transverse to the flow and an invariant smooth probability measure, and whose Anosov foliations are smooth. In this situation, it is possible to construct a flow-invariant affine connection \(\nabla\) on V whose torsion tensor is tangent to the flow and which respects the symplectic form. The main technical result of the article is that, either the local space of orbits of the flow is an affine locally symmetric space for the connection induced by \(\nabla\), or the Osedelec decomposition of the flow is smooth and the Lyapunov exponents of any invariant ergodic measure are of the form -2a, -a, 0, a, 2a for some \(a>0\) depending on the measure. As an application, they prove the following result: If the geodesic flow of a negatively curved closed 3-manifold has smooth horospheric foliations, then this flow is smoothly conjugate to the geodesic flow of a manifold of constant negative curvature.

MSC:
37D99 Dynamical systems with hyperbolic behavior
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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References:
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[3] Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (1978)
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[6] DOI: 10.2307/2373590 · Zbl 0254.58005 · doi:10.2307/2373590
[7] Flaminio, Ergod. Th. & Dynam. Sys. none pp none– (none)
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