Advertisement

Journal of Mathematical Sciences

, Volume 219, Issue 1, pp 112–124 | Cite as

Typical Properties of Leaves of Cartan Foliations with Ehresmann Connection

  • N. I. ZhukovaEmail author
Article
  • 17 Downloads

We consider a Cartan foliation (M,F) of an arbitrary codimension q admitting an Ehresmann connection such that all leaves of (M,F) are embedded submanifolds of M. We prove that for any foliation (M,F) there exists an open, not necessarily connected, saturated, and everywhere dense subset M0 of M and a manifold L0 such that the induced foliation (M0, FM0) is formed by the fibers of a locally trivial fibration with the standard fiber L0 over (possibly, non-Hausdorff) smooth q-dimensional manifold. In the case of codimension 1, the induced foliation on each connected component of the manifold M0 is formed by the fibers of a locally trivial fibration over a circle or over a line.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. Millett, “Generic properties of proper foliations,” Santa Barbara Univ. Math. J. 138, 131–138 (1984).MathSciNetzbMATHGoogle Scholar
  2. 2.
    N. I. Zhukova, “Minimal sets of Cartan foliations” [in Russian], Tr. Mat. Inst. Steklova 256, 115-147 (2007); English transl.: Proc. Steklov Inst. Math. 256, 105-135 (2007).Google Scholar
  3. 3.
    R. A. Blumenthal and J. J. Hebda, “Ehresmann connections for foliations,” Indiana Univ. Math. J. 33, No 4, 597–611 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    I. Tamura, Topology of Foliations: An Introduction, Am. Math. Soc., Providence, RI (1992).Google Scholar
  5. 5.
    N. I. Zhukova, “Complete foliations with transverse rigid geometries and their basic automorphisms” [in Russian], Vestn. Ross. Univ. Druzh. Nar., Ser. Mat. Inform. Fiz. No 2, 14–35 (2009) [electronic version only]Google Scholar
  6. 6.
    G. Glimm, “Locally compact transformation group,” Trans. Am. Math. Soc. 101, 124–138 (1961).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    N. Zhukova, “On the stability of leaves of Riemannian foliations,” Ann. Global Anal. Geom. 5, No 3, 261–271 (1987).Google Scholar
  8. 8.
    N. I. Zhukova, “Local and global stability of compact leaves of foliations,” Zh. Mat. Fiz. Anal. Geom. 9, No 3. 400–420 (2013).Google Scholar
  9. 9.
    N. I. Zhukova, “Local and global stability of leaves of conformal foliations,” In: Foliations 2012, pp. 215–233, World Scientific, Singapore (2013).Google Scholar
  10. 10.
    R. A. Blumenthal, “Cartan submersions and Cartan foliations,” Ill. Math. J. 31, No 2, 327–343 (1987).MathSciNetzbMATHGoogle Scholar
  11. 11.
    N. I. Zhukova, “Global attractors of complete conformal foliations” [in Russian] Mat. Sb. 203, No. 3, 79–106 (2012); English transl.: Sb. Math. 203, No. 3, 380–405 (2012).Google Scholar
  12. 12.
    Sh. Kobayashi and K. Nomizu, Foundation of Differential Geometry. I, John Wiley & Sons, New York etc. (1963).Google Scholar
  13. 13.
    Sh. Kobayashi, Transformation Groups in Differential Geometry, Springer, Berlin (1995).zbMATHGoogle Scholar
  14. 14.
    R. Hermann, “On differential geometry of foliations,” Ann. Math. 72, No 3, 445–457 (1960).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    D. Epstein, K. Millett, and D. Tischler, “Leaves without holonomy,” J. London Math. Soc. 16, 548–552 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    S. Kashiwabara, “The decomposition of differential manifolds and its applications,” Tohoku Math. J. 11, No 1, 43–53 (1959).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    B. L. Reinhart, “Foliated manifolds with bundle-like metrics,” Ann. Math. 69, No. 1, 119–132 (1959).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsNizhny NovgorodRussia

Personalised recommendations