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On collinear closed one-forms. (English) Zbl 1226.57040
For a closed, oriented manifold \(M\), a smooth 1-form \(\alpha\) determines a foliation \(\mathcal F_\alpha\) on \(M\setminus\text{Sing}\;\alpha\) which extends to a singular foliation \(\overline{\mathcal F_\alpha}\), where Sing \(\alpha=\{x\in M\;/\;\alpha_x=0\}\). Conditions for \(\mathcal F_\alpha=\mathcal F_\beta\) or \(\overline{\mathcal F_\alpha}= \overline{\mathcal F_\beta}\), for \(\beta\) another 1-form, are considered. For example if Sing \(\alpha\) and Sing \(\beta\) are nowhere dense then \(\overline{\mathcal F_\alpha}= \overline{\mathcal F_\beta}\) if and only if \(\alpha\wedge\beta=0\). It is also shown that \(\mathcal F_\alpha\) has a compact leaf if and only if there is a smooth function \(f\) such that \(df\wedge\alpha=0\) and Supp \(\alpha\cap \text{Supp }\;df\not=\varnothing\).

MSC:
57R30 Foliations in differential topology; geometric theory
58A10 Differential forms in global analysis
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