On the structure of a Morse form foliation
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.
Keywordsnumber of minimal components number of maximal components compact leaves foliation graph rank of a form
MSC 200057R30 58K65
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- I. Gelbukh: Ranks of collinear Morse forms. Submitted.Google Scholar
- F. Harary: Graph theory. Addison-Wesley Publ. Comp., Massachusetts, 1994.Google Scholar
- A. Katok: Invariant measures of flows on oriented surfaces. Sov. Math. Dokl. d14 (1973), 1104–1108.Google Scholar