×

zbMATH — the first resource for mathematics

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

MSC:
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37C10 Dynamics induced by flows and semiflows
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] D. V. ANOSOV , Geodesic Flows on Compact Riemannian Manifolds of Negative Curvature (Proc. Steklov. Math. Inst. A.M.S. Translations, 1969 ). · Zbl 0135.40401
[2] V. ARNOLD , Chapitres supplémentaires de la théorie des équations différentielles ordinaires , Mir, Moscou, 1980 . MR 83a:34003 | Zbl 0455.34001 · Zbl 0455.34001
[3] R. BOWEN , Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Math., n^\circ 470, 1975 , Springer). MR 56 #1364 | Zbl 0308.28010 · Zbl 0308.28010
[4] K. BURNS et A. KATOK , en collaboration avec W. BALLMAN , M. BRIN , P. ELERBEIN et R. OSSERMAN , Manifolds with Non Positive Curvature (Ergod. and Dynam. Syst., vol. 5, 1985 , p. 307-317). Zbl 0572.58019 · Zbl 0572.58019
[5] J. FELDMAN et D. ORNSTEIN , Semirigidity of Horocycle Flows Over Compact Surfaces of Variable Negative Curvature , preprint. · Zbl 0633.58024
[6] D. FRIED , Transitive Anosov Flows and Pseudo-Anosov Maps (Topology, vol. 22, n^\circ 3, 1983 , p. 299-303). MR 84j:58095 | Zbl 0516.58035 · Zbl 0516.58035
[7] E. GHYS , Flots d’Anosov sur les 3-variétés fibrés en cercle [Ergod. Th. and Dynam. Sys., (4), 1984 , p. 67-80]. MR 86b:58098 | Zbl 0527.58030 · Zbl 0527.58030
[8] S. GOODMAN , Dehn Surgery on Anosov Flows, Geometric Dynamics (Lecture Notes in Math., Springer, n^\circ 1007, p. 300-307). MR 1691596 | Zbl 0532.58021 · Zbl 0532.58021
[9] A. HAEFLIGER , Groupoïdes d’holonomie et classifiants (Astérisque, vol. 116, 1984 , p. 70-97). MR 86c:57026a | Zbl 0562.57012 · Zbl 0562.57012
[10] HANDEL et W. THRUSTON , Anosov Flows on New 3-Manifolds (Inv. Math., vol. 59, 1980 , p. 95-103). MR 81i:58032 | Zbl 0435.58019 · Zbl 0435.58019
[11] J. HEMPEL , 3-Manifolds (Annals of Mathematics Studies, n^\circ 86, Princeton University Press, 1976 ). MR 54 #3702 | Zbl 0345.57001 · Zbl 0345.57001
[12] M. HIRSCH , C. PUGH et M. SHUB , Invariant Manifolds (Lecture Notes in Math., n^\circ 583, 1977 , Springer). MR 58 #18595 | Zbl 0355.58009 · Zbl 0355.58009
[13] S. HURDER et A. KATOK , Differentiability, Rigidity and Godbillon-Vey Classes for Anosov Flows , Preprint. · Zbl 0725.58034
[14] Y. MITSUMATSU , A Relation Between the Topological Invariance of the Godbillon-Vey Class and the Differentiability of Anosov Foliations (Advanced Studies in Pure Math., vol. 5, 1985 ). MR 88a:57050 | Zbl 0653.57018 · Zbl 0653.57018
[15] P. ORLIK , Seifert Manifolds (Lecture Notes in Math., n^\circ 291, Springer-Verlag, 1972 ). MR 54 #13950 | Zbl 0263.57001 · Zbl 0263.57001
[16] J. PLANTE , Anosov Flows, Transversely Affine Foliations and a Conjecture of Verjovsky [J. London. Math. Soc., (2), 23, 1981 , n^\circ 2, p. 359-362]. MR 82g:58069 | Zbl 0465.58020 · Zbl 0465.58020
[17] F. RAYMOND et T. VASQUEZ , 3-Manifolds Whose Universal Coverings Are Lie Groups (Topology and its Applications, vol. 12, 1981 , p. 161-179). MR 82i:57011 | Zbl 0468.57009 · Zbl 0468.57009
[18] W. THURSTON , The Geometry and Topology of 3-Manifolds , chap. 4 and 5, Princeton Lectures Notes.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.