Geodesic
Close cohomologous Morse forms with compact leaves
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 515-528
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We study the topology of foliations of close cohomologous Morse forms (smooth closed 1-forms with non-degenerate singularities) on a smooth closed oriented manifold. We show that if a closed form has a compact leave $\gamma $, then any close cohomologous form has a compact leave close to $\gamma $. Then we prove that the set of Morse forms with compactifiable foliations (foliations with no locally dense leaves) is open in a cohomology class, and the number of homologically independent compact leaves does not decrease under small perturbation of the form; moreover, for generic forms (Morse forms with each singular leaf containing a unique singularity; the set of generic forms is dense in the space of closed 1-forms) this number is locally constant.
We study the topology of foliations of close cohomologous Morse forms (smooth closed 1-forms with non-degenerate singularities) on a smooth closed oriented manifold. We show that if a closed form has a compact leave $\gamma $, then any close cohomologous form has a compact leave close to $\gamma $. Then we prove that the set of Morse forms with compactifiable foliations (foliations with no locally dense leaves) is open in a cohomology class, and the number of homologically independent compact leaves does not decrease under small perturbation of the form; moreover, for generic forms (Morse forms with each singular leaf containing a unique singularity; the set of generic forms is dense in the space of closed 1-forms) this number is locally constant.
DOI : 10.1007/s10587-013-0034-0
Classification : 57R30, 58E05, 58K65
Keywords: Morse form foliation; compact leaf; cohomology class 
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Gelbukh, Irina. Close cohomologous Morse forms with compact leaves. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 515-528. doi: 10.1007/s10587-013-0034-0

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