Mathematische Zeitschrift

, 263:725 | Cite as

Equicontinuous foliated spaces

Article

Abstract

Riemannian foliations are characterized as those foliations whose holonomy pseudogroup consists of local isometries of a Riemannian manifold. Their main structural features are well understood since the work of Molina. In this paper we analyze the more general concept of equicontinuous pseudogroup of homeomorphisms, which gives rise to the notion of equicontinuous foliated space. We show that equicontinuous foliated spaces have structural properties similar to those known for Riemannian foliations: the universal covers of their leaves are in the same quasi-isometry class, leaf closures are homogeneous spaces, and the holonomy pseudogroup is indeed given by local isometries.

Keywords

Riemannian foliation Foliated space Pseudogroup of isometrics Equicontinuous pseudogroup Quasi-isometry 

References

  1. 1.
    Alcalde Cuesta F.: Groupoï de d’homotopie d’un feuilletage riemannien et réalisation symplectique de certaines variétés de Poisson. Publ. Mat. 33, 395–410 (1989)MATHMathSciNetGoogle Scholar
  2. 2.
    Álvarez López, J.A., Candel, A.: Topological description of Riemannian foliations with dense leaves (preprint)Google Scholar
  3. 3.
    Álvarez López, J.A., Candel, A.: Generic geometry of leaves (Forthcomming preprint)Google Scholar
  4. 4.
    Block J., Weinberger S.: Aperiodic tilings, positive scalar curvature, and amenability of spaces. J. Am. Math. Soc. 5, 907–918 (1992)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Candel, A., Conlon, L.: Foliations I, Graduate Studies in Mathematics. American Mathematical Society, Providence (2000)Google Scholar
  6. 6.
    Goodman S.E., Plante J.F.: Holonomy and averaging in foliated sets. J. Diff. Geom. 14, 401–407 (1979)MATHMathSciNetGoogle Scholar
  7. 7.
    Gromov M.: Asymptotic invariants of infinite groups. In: Niblo, G.A., Roller, M.A. (eds) Geometric Group Theory vol. 2, Cambridge University Press, Cambridge (1993)Google Scholar
  8. 8.
    Haefliger, A.: Pseudogroups of local isometries. In: Cordero, L.A. (ed.) Differential Geometry (Santiago de Compostela 1984). Research Notes in Math., vol. 131, pp. 174–197. Pitman Advanced Pub. Program, Boston (1985)Google Scholar
  9. 9.
    Haefliger A. : Leaf closures in Riemannian foliations. In: Matsumoto, Y., Mizutani, T., Morita S., S. (eds) A Fête on Topology, pp. 3–32. Academic Press, New York (1988)Google Scholar
  10. 10.
    Haefliger, A.: Foliations and compactly generated pseudogroups (2001, preprint)Google Scholar
  11. 11.
    Hector, G., Hirsch, U.: Introduction to the geometry of foliations, Part A. In: Aspects of Mathematics, vol. E1. Friedr. Vieweg and Sohn, Braunschweig (1981)Google Scholar
  12. 12.
    Hector, G., Hirsch, U.: Introduction to the geometry of foliations, Part B. In: Aspects of Mathematics, vol. E3. Friedr. Vieweg and Sohn, Braunschweig (1983)Google Scholar
  13. 13.
    Hirsch M.: Differential Topology, Graduate Texts in Mathematics, vol. 33. Springer, New York (1976)Google Scholar
  14. 14.
    Hurder S.: Coarse geometry of foliations. In: Mizutani, T., Masuda, K., Matsumoto, S., Inaba, T., Tsuboi, T., Mitsumatsu, Y. (eds) Geometric Study of Foliations (Tokyo 1993), pp. 35–96. World Scientific Publishing Co. Pte. Ltd, Singapore (1994)Google Scholar
  15. 15.
    Hurder S., Katok A.: Ergodic theory and Weil measures for foliations. Ann. Math. 126, 221–275 (1987)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Kanai M.: Rough isometries, and combinatorial approximations of geometries of non-compact manifolds. J. Math. Soc. Jpn 37, 391–413 (1985)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kellum M.: Uniformly quasi-isometric foliations. Ergodic Theory Dyn. Syst. 13, 101–122 (1993)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Molino, P.: Riemannian Foliations (with appendices by Cairns, G., Carrière, Y., Ghys, E., Salem, E., Sergiescu, V.). Progress in Mathematics, vol. 73. Birkhäuser, Boston (1988)Google Scholar
  19. 19.
    Moore C.C., Schochet C.: Global Analysis on Foliated Spaces, MSRI Publications, vol. 9. Springer, New York (1988)Google Scholar
  20. 20.
    Munkres J.R.: Topology: a First Course. Prentice-Hall, Inc., Englewood Cliffs (1975)MATHGoogle Scholar
  21. 21.
    Nagata J.: Modern General Topology, 2nd revised edn. Noth-Holland Publishing Company, Amsterdam (1974)Google Scholar
  22. 22.
    Robinson A.: Non-standard Analysis, (Reprint of the 1974 Edition). Princeton University Press, Princeton (1996)Google Scholar
  23. 23.
    Plante J.F.: Foliations with measure preserving holonomy. Ann. Math. 102, 327–361 (1975)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Roe J.: Coarse cohomology and index theory on complete riemannian manifolds. Mem. Amer. Math. Soc. 104(497), x+90 (1993)MathSciNetGoogle Scholar
  25. 25.
    Sacksteder R.: Foliations and pseudogroups. Am. J. Math. 87, 79–102 (1965)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Smirnov Y.M.: On metrization of topological spaces. Am. Math. Soc. Transl. Ser. One 8, 62–77 (1953)Google Scholar
  27. 27.
    Steen L.A., Seebach J.A. Jr: Counterexamples in Topology, 2nd edn. Springer, New York (1978)MATHGoogle Scholar
  28. 28.
    Tarquini C.: Feuilletages conformes. Ann. Inst. Fourier 54, 453–480 (2004)MathSciNetGoogle Scholar
  29. 29.
    Veech W.: Topological dynamics. Bull. Am. Math. Soc. 83, 775–830 (1977)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Weil, A.: L’Integration dans les Groupes Topologiques et ses Applications. Actualités Scientifiques et Industrielles, no. 1145, 2nd edn. Publications de l’Institut de Mathematique de l’Universite de Strasbourg 4, Hermann, Paris (1951)Google Scholar
  31. 31.
    Willard S.: General Topology. Addison-Wesley Publishing Co., Reading (1970)MATHGoogle Scholar
  32. 32.
    Winkelnkemper H.E.: The graph of a foliation. Ann. Global Anal. Geom. 1, 51–75 (1983)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departamento de Xeometría e Topoloxía, Facultade de MatemáticasUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.Department of MathematicsCSUNNorthridgeUSA

Personalised recommendations