zbMATH — the first resource for mathematics

Properties of Morse forms that determine compact foliations on \(M_g^2\). (English. Russian original) Zbl 0898.57012
Math. Notes 60, No. 6, 714-716 (1996); translation from Mat. Zametki 60, No. 6, 942-945 (1996).
Summary: P. Arnoux and G. Levitt [Invent. Math. 84, 141-156 (1986; Zbl 0561.58024); ibid. 88, 635-667 (1987; Zbl 0594.57014)] showed that the topology of the foliation of a Morse form \(\omega\) on a compact manifold is closely related to the structure of the integration mapping \([\omega]: H_1(M)\to \mathbb{R}\). In this paper, the author considers the foliation of a Morse form on a two-dimensional manifold \(M_g^2\). He studies the relationship of the subgroup \(\text{Ker} [\omega] \subset H_1(M_g^2)\) with the topology of the foliation, considers the structure of the subgroup \(\text{Ker} [\omega]\) for a compact foliation and proves a criterion for the compactness of a foliation.
57R30 Foliations in differential topology; geometric theory
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
Full Text: DOI
[1] P. Arnoux and G. Levitt, Invent. Math.,84, 141–156 (1986). · Zbl 0577.58021 · doi:10.1007/BF01388736
[2] G. Levitt,Invent. Math.,88, 635–667 (1987). · Zbl 0594.57014 · doi:10.1007/BF01391835
[3] S. P. Novikov,Uspekhi Mat. Nauk [Russian Math. Surveys],37, No. 5, 3–49 (1982).
[4] I. A. Mel’nikova,Mat. Zametki [Math. Notes],53, No. 3, 158–160 (1983).
[5] I. A. Mel’nikova,Compact foliations of Morse forms, Kandidat thesis in the physico-mathematical sciences [in Russian], Moscow State Univ., Moscow (1996).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.