manuscripta mathematica

, Volume 51, Issue 1–3, pp 145–161

Deux remarques sur les flots riemanniens

  • Pierre Molino
  • Vlad Sergiescu
Article

Abstract

Let M be a connected oriented closed n-manifold. A riemannian flow\(\mathfrak{F}\) on M is an oriented one dimensional foliation which admits a bundle-like metric.

We give a caracterization of isometric flows as riemannian flows whose basic cohomology Hbn−1(M,\(\mathfrak{F}\)) is non trivial in degree (n−1). A second caracterization involves the triviality of the central sheaf.

We show also that\(\mathfrak{F}\) has a section if and only if Hbn−1(M,\(\mathfrak{F}\)) has a non trivial image in Hn−1(M).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographie

  1. [1]
    P. Caron-Y. Carrière “Flots transversalement de Lie Rn, flots transversalement de Lie minimaux” C. R. Ac. Sc. Paris, 291 (1980), série A, pp. 477–478Google Scholar
  2. [2]
    Y. Carrière “Flots riemanniens”-Journées sur les structures transverses, Toulouse 1982-Asterisque (1984)Google Scholar
  3. [3]
    El Kacimi-G. Hector-V. Sergiescu “La cohomologie basique d'un feuilletage riemannien est de dimension finie”. A paraître dans Mathematishe ZeitschriftGoogle Scholar
  4. [4]
    E. Ghys “Classification des feuilletages totalement géodésiques de codimension un”. Comentari Math. Helv. 58, 4 (1983), pp. 543–572Google Scholar
  5. [5]
    E. Ghys “Feuilletages riemanniens sur les variétés simplement connexes”. Ann. Inst. Fourrier (à paraître)Google Scholar
  6. [6]
    H. Gluck “Dynamical behaviour of geodesic flows” In Lectures Notes no 819Google Scholar
  7. [7]
    A. Haefliger “Some remarks on foliations with minimal leaves”. Journal Diff. Geom., 15 (1980), 269–284Google Scholar
  8. [8]
    F. Kamber, P. Tondeur “Duality for riemannian foliations”. Proc. Symp. Pure Math 40 (1983)Google Scholar
  9. [9]
    P. Molino “Géométrie globale des feuilletages riemanniens”. Proc. Kon. Ned, Akad, Al, 85 (1982), pp. 45–76Google Scholar
  10. [10]
    P. Molino “Flots riemanniens et flots isométriques”. dans “Séminaire de Géométrie différentielle 1982–1983”, MontpellierGoogle Scholar
  11. [11]
    M. Pierrot “Groupe des automorphismes d'un flot linéaire sur le tore” dans “Séminaire de Géométrie Différentielle 1983–1984”. MontpellierGoogle Scholar
  12. [12]
    B. Reinhart “Foliated manifolds with bundle-like metrics”. Ann. of Maths, 69 (1959), pp. 119–132Google Scholar
  13. [13]
    V. Sergiescu “Cohomologie basique et dualité des feuilletages riemanniens” preprint Lille 1984Google Scholar
  14. [14]
    D. Sullivan “Cycles for the dynamical study of foliated manifolds” Inv. Math. 36 (1976) pp. 225–255Google Scholar
  15. [15]
    D. B. A. Epstein “Transversaly hyperbolic 1-dimensional foliations”. Journées sur les structures transverses, Toulouse 1982-Asterisque (1984)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Pierre Molino
    • 1
  • Vlad Sergiescu
    • 2
  1. 1.Université Montpellier IIMontpellier CedexFrance
  2. 2.Université Lille IVilleneuve d'Ascq CedexFrance

Personalised recommendations