manuscripta mathematica

, Volume 66, Issue 1, pp 45–59 | Cite as

Foliated g-structures and riemannian foliations

  • Robert A. Wolak
Article

Abstract

Abstract Using the properties of the commuting sheaf of aG-foliation of finite type we prove that some of theseG-foliations must be Riemannian.

Keywords

Finite Type Basic Manifold Riemannian Foliation Compact Type Local Diffeomorphisms 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Albert, P. Molino,Pseudogroupes de Lie transitifs, Travaux en cours, Herman, Paris 1984MATHGoogle Scholar
  2. [2]
    R.A. Blumenthal, Cartan connections in foliated bundles,Michigan Math. J.,32 (1984), 55–63.MathSciNetGoogle Scholar
  3. [3]
    J.P. Conze, Y. Guivarch, Remarques sur la distalité dans les espaces vectoriels,C. R. Acad. Sc. Paris 278 (1974), 1083–1086MATHMathSciNetGoogle Scholar
  4. [4]
    M. Davis,G-smooth manifolds as collections of fibre bundles,Pacific Math. J. 77 (1978), 315–363MATHGoogle Scholar
  5. [5]
    E. Ghys, Feuilletages riemanniens sur les variétés simplement connèxes,Ann. Inst. Fourier 34 (1984), 203–223.MATHMathSciNetGoogle Scholar
  6. [6]
    H. Gluck, Dynamical behaviour of geodesic fields,Global Theory of Dynamical Systems, Evanston, Springer LN819 (1980), 190–215CrossRefMathSciNetGoogle Scholar
  7. [7]
    A. Haefliger, Pseudogroups of local isometries,Proceedings of Vth International Colloquium on Differential Geometry, Santiago de Compostela, 1984, ed L.A. Cordero, Pitman 1985, 174–197Google Scholar
  8. [8]
    A. Haefliger, Closures of leaves in Riemannian foliations,A fête of topology, Academic Press, Boston, MA, 1988, 3–32Google Scholar
  9. [9]
    A. Haefliger, E. Salem, Pseudogroupes d'holonomie des feuilletages riemanniens sur des variétés compactes 1-connèxes,Géométrie et Physique, Paris 1986, Travaux en cours, Herman 1988Google Scholar
  10. [10]
    S. G. Hancock, Construction of invariant sets for Anosov diffeomorphisms,J. London Math. Soc. 18 (1978), 339–348MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    S. Kobayashi,Transformation Groups in Differential Geometry. Springer 1972Google Scholar
  12. [12]
    J.L. Koszul,Lectures on Groups of Transformations, Tata Institute of Fundamental Research, Bombay, 1960Google Scholar
  13. [13]
    R. Mañé, Invariant sets of Anosov diffeomorphisms,Invent. Math. 46 (1978), 147–152MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    P. Molino, Actions des groupes de Lie et presque-connexions, Springer Lectures Notes484 (1975), 153–161MathSciNetGoogle Scholar
  15. [15]
    P. Molino, Etude des feuilletages transversalement complets et applications,Ann. Sci. Ecole Norm. Sup. 10 (1977), 289–307MATHMathSciNetGoogle Scholar
  16. [16]
    P. Molino, Feuilletages riemanniens sur les variétés compactes: champs de Killing transverse,C. R. Acad. Sc. Paris 289 (1979), 421–423MATHMathSciNetGoogle Scholar
  17. [17]
    P. Molino, Feuilletages de Lie à feuilles denses,Séminaire de Géométrie Différentielle 1982–83, MontpellierGoogle Scholar
  18. [18]
    P. Molino,Riemannian Foliations, Progress in Math. vol73, Birkhäuser 1988Google Scholar
  19. [19]
    P. Molino, V. Sergiescu, Deux remarques sur les flots riemanniens,Manus. Math. 51 (1985), 145–161.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    R. Palais, On the existence of slices for action of non-compact Lie groups,Ann. of Math. 73 (1961), 295–323CrossRefMathSciNetGoogle Scholar
  21. [21]
    M. Pierrot, Orbites des champs feuilletés pour un feuilletage riemannien sur une variété compacte,C. R. Acad. Sc. Paris 301, 1 (1985), 443–445MATHMathSciNetGoogle Scholar
  22. [22]
    F. Przytycki, Construction of invariant sets for Anosov diffeomorphisms and hyperbolic attractors,Studia Math. 46 (1980), 199–213MathSciNetGoogle Scholar
  23. [23]
    E. Salem, Une généralization du théorème de Myers-Steenrod aux pseudogroupes d'isométries locales,Ann. Inst. Fourier 38, 2 (1988), 185–200MATHMathSciNetGoogle Scholar
  24. [24]
    S. Sternberg,Lectures on Differential Geometry, Chelsea Publ. C. (second edition), New York 1983MATHGoogle Scholar
  25. [25]
    M. Takeuchi, On foliations with the structure group of automorphisms of a geometric structure,J. Math. Soc. Japan 32, 1 (1980), 119–152.MATHMathSciNetCrossRefGoogle Scholar
  26. [26]
    R. Wolak, OnG-foliations,Ann. Pol. Math. 46 (1985), 371–377MathSciNetGoogle Scholar
  27. [27]
    R. Wolak, Foliations admitting systems of transverse differential equations,Comp. Math. 67 (1988), 89–101MATHMathSciNetGoogle Scholar
  28. [28]
    R. Wolak, Le graphe d'un feuilletage admettant un système transverse d'équations différentielles,Math. Z. 201 (1989), 177–182MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    R. Wolak, Maximal subalgebras in the algebra of foliated vector fields of a Riemannian foliation, to be publ. inComm. Math. Helv. Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Robert A. Wolak
    • 1
  1. 1.Departmento de Xeometria e Topoloxia Facultade de MatematicasUniversidade de Santiago de CompostelaSantiago de CompostelaSpain

Personalised recommendations