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Flots d’Anosov à distributions stable et instable différentiables. (Anosov flows with smooth stable and unstable distributions). (French) Zbl 0754.58027
This article announces a rigidity result for Anosov flows. With some minor restrictions, this result shows that the stable and unstable distributions of such a flow are \(C^ \infty\) only when, basically, the flow is the geodesic flow of a locally symmetric manifold. An Anosov flow has an associated canonical 1-form which is 0 in the stable and unstable directions and which is 1 in the direction of the flow. The authors restrict attention to the case where this form is a contact form, which is for instance always the case for geodesic flows of negatively curved manifolds. The main result is that, given an Anosov flow with contact canonical 1-form on a compact manifold \(M\), its stable and unstable distributions are \(C^ \infty\) if and only if \(M\) is diffeomorphic to the unit tangent bundle of a negatively curved locally symmetric orbifold by a \(C^ \infty\) diffeomorphism sending the Anosov flow to the geodesic flow. A detailed proof of this result has since been published in J. Am. Math. Soc. 5, 33-78 (1992).

MSC:
37D99 Dynamical systems with hyperbolic behavior
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
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