Czechoslovak Mathematical Journal

, Volume 59, Issue 1, pp 207–220 | Cite as

On the structure of a Morse form foliation

Article

Abstract

The foliation of a Morse form ω on a closed manifold M is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated. The number of minimal and maximal components is estimated in terms of characteristics of M and ω. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of rk ω and Sing ω. The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that if ω has more centers than conic singularities then b1(M) = 0 and thus the foliation has no minimal components and homologically non-trivial compact leaves, its folitation graph being a tree.

Keywords

number of minimal components number of maximal components compact leaves foliation graph rank of a form 

MSC 2000

57R30 58K65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Arnoux and G. Levitt: Sur l’unique ergodicité des 1-formes fermées singulières. Invent. Math. 84 (1986), 141–156.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    M. Farber, G. Katz and J. Levine: Morse theory of harmonic forms. Topology 37 (1998), 469–483.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    I. Gelbukh: Presence of minimal components in a Morse form foliation. Diff. Geom. Appl. 22 (2005), 189–198.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    I. Gelbukh: Ranks of collinear Morse forms. Submitted.Google Scholar
  5. [5]
    F. Harary: Graph theory. Addison-Wesley Publ. Comp., Massachusetts, 1994.Google Scholar
  6. [6]
    K. Honda: A note on Morse theory of harmonic 1-forms. Topology 38 (1999), 223–233.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    H. Imanishi: On codimension one foliations defined by closed one forms with singularities. J. Math. Kyoto Univ. 19 (1979), 285–291.MATHMathSciNetGoogle Scholar
  8. [8]
    A. Katok: Invariant measures of flows on oriented surfaces. Sov. Math. Dokl. d14 (1973), 1104–1108.Google Scholar
  9. [9]
    G. Levitt: 1-formes fermées singulières et groupe fondamental. Invent. Math. 88 (1987), 635–667.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    G. Levitt: Groupe fondamental de l’espace des feuilles dans les feuilletages sans holonomie. J. Diff. Geom. 31 (1990), 711–761.MATHMathSciNetGoogle Scholar
  11. [11]
    I. Mel’nikova: A test for non-compactness of the foliation of a Morse form. Russ. Math. Surveys 50 (1995), 444–445.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    I. Mel’nikova: Maximal isotropic subspaces of skew-symmetric bilinear map. Vestnik MGU 4 (1999), 3–5.MathSciNetGoogle Scholar
  13. [13]
    S. Novikov: The Hamiltonian formalism and a multivalued analog of Morse theory. Russian Math. Surveys 37 (1982), 1–56.MATHCrossRefGoogle Scholar
  14. [14]
    A. Pazhitnov: The incidence coefficients in the Novikov complex are generically rational functions. Sankt-Petersbourg Math. J. 9 (1998), 969–1006.MathSciNetGoogle Scholar
  15. [15]
    D. Tischler: On fibering certain foliated manifolds over S 1. Topology 9 (1970), 153–154.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2009

Authors and Affiliations

  1. 1.CIC-IPNDFMexico

Personalised recommendations