Anosov flows whose stable foliations are differentiable. (Flots d’Anosov dont les feuilletages stables sont différentiables.) (French) Zbl 0663.58025

The paper is devoted to Anosov flows on 3-dimensional manifolds with infinitely smooth strong stable and unstable foliations. Two types of such flows are known: suspensions of hyperbolic automorphisms of the torus and geodesic flows on surfaces of constant negative curvature (and also flows related to geodesic flows by finite coverings). The author constructs a new type of such Anosov flows and then proves that his new flows together with old ones exhaust all such Anosov flows. He proves that his new flows differ from the old ones by a rather special change of time. But they are constructed not by a change of time. The construction is unexpected and rather beautiful.
It is well known that the bundle of unit tangent vectors to a surface of constant negative curvature carries an \((\text{SL}(2,{\mathbb R}),\text{SL}(2,{\mathbb R}))\)-structure in the sense of Thurston (the group \(\text{SL}(2,{\mathbb R})\) acts on the space \(\text{SL}(2,{\mathbb R})\) by left translations). The author notes that this structure can be extended in a natural way to an (\(\text{SL}(2,{\mathbb R})\times {\mathbb R}\), \(\text{SL}(2,{\mathbb R}))\)-structure, where \((0,t)\) acts on \(\text{SL}(2,{\mathbb R})\) by right translation by \(\left( \begin{matrix} e^ t\\ 0\end{matrix} \begin{matrix} 0\\ e^{-t}\end{matrix} \right)\). Then he proves that this new structure can be perturbed and any perturbation gives a new Anosov flow.
As an application of his results the author proves the following conjecture of Hurder and Katok. Let \(g\) be \(C^{\infty}\)-smooth Riemannian metric with negative curvature on a surface. If weak stable and unstable foliations of the geodesic flows of \(g\) are \(C^ 2\)-smooth, then the curvature of \(g\) is constant.
Reviewer: N. V. Ivanov


37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37C10 Dynamics induced by flows and semiflows
Full Text: DOI Numdam EuDML


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