Journal of Mathematical Sciences

, Volume 208, Issue 1, pp 115–130 | Cite as

Transverse Equivalence of Complete Conformal Foliations

  • N. I. ZhukovaEmail author

We study the problem of classification of complete non-Riemannian conformal foliations of codimension q > 2 with respect to transverse equivalence. It is proved that two such foliations are transversally equivalent if and only if their global holonomy groups are conjugate in the group of conformal transformations of the q-dimensional sphere Conf (\( {\mathbb{S}}^{\mathrm{q}} \)). Moreover, any countable essential subgroup of the group Conf (\( {\mathbb{S}}^{\mathrm{q}} \)) is realized as the global holonomy group of some non-Riemannian conformal foliation of codimension q. Bibliography: 16 titles.


Manifold Riemannian Manifold Conformal Transformation Global Attractor Canonical Projection 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. I. Zhukova, “Attractors and an analog of the Lichnérowicz conjecture for conformal foliations” [in Russian], Sib. Mat. Zh. 52, No. 3, 555–574 (2011); Englsih transl.: Sib. Math. J. 52, No. 3, 436–450 (2011).Google Scholar
  2. 2.
    C. Tarquini, “Feuilletages conformes.” Ann. Inst. Fourier 54, No. 2, 341–351 (2004).zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    C. Frances and C. Tarquini, “Autour du téorème de Ferrand-Obata.” Ann. Global Anal. Geom. 21, No. 1, 51–62 (2002).Google Scholar
  4. 4.
    R. W. Sharpe, Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program, Springer, Berlin etc. (1997).zbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    R. A. Blumenthal, “Cartan submersions and Cartan foliations.” Ill. Math. J. 31, 327–343 (1987).zbMATHMathSciNetGoogle Scholar
  7. 7.
    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
  8. 8.
    R. A. Blumenthal and J. J. Hebda, “Ehresmann connections for foliations.” Indiana Univ. Math. J. 33, No. 4, 597–611 (1984).zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    P. Molino, Riemannian Foliations, Birkhäuser, Boston (1988).zbMATHCrossRefGoogle Scholar
  10. 10.
    N. I. Zhukova, “Complete foliations with transverse rigid geometries and their basic automorphisms,” Vetsn. Ross. Univ. Druzh. Nar., Ser. Prikl. Mat. Inf. No. 2, 14–35 (2009).Google Scholar
  11. 11.
    L. Greenberg, “Discrete subgroups of the Lorentz group.” Math. Scand. 10, 85–107 (1962).zbMATHMathSciNetGoogle Scholar
  12. 12.
    E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co., New York etc. (1966).zbMATHGoogle Scholar
  13. 13.
    R. Palais, “A global formulation of the Lie theory of transformation groups.” Mem. Am. Math. Soc. 22, (1957).Google Scholar
  14. 14.
    A. Candel and L. Conlon, Foliations I, Am. Math. Soc., Providence, RI (2000).Google Scholar
  15. 15.
    J. Ferrand, “The action of conformal transformations on a Riemannian manifold.” Math. Ann. 304, No. 2, 277–291 (1996).zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    H. Bourdon, “Sur la dimension de Hausdorff de l’ensemble limite d’une famille de sousgroupes convexes co-compacts.” C. R. Acad. Sci. Paris. Ser. 1. Math. 325, 1097–1100 (1997).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

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

Personalised recommendations