An indicator of the noncompactness of a foliation on \(M_ g^ 2\). (English. Russian original) Zbl 0809.57018

Mel’nikova, I. A.

Math. Notes 53, No. 3, 356-358 (1993); translation from Mat. Zametki 53, No. 3, 158-160 (1993).

Let \(\omega\) be a closed form on a manifold \(M\) and possessing nondegenerate isolated singularities. A point \(p \in M\) is called a regular singularity of \(\omega\), if in some neighbourhood \(O(p) \omega = df\), where \(f\) is a Morse function having a singularity at \(p\). The form \(\omega\) determines a foliation \(F_ \omega\) on the set \(M-\text{Sing} \omega\).
Let \(M = M^ 2_ g\), the orientable closed surface of genus 2. The homology classes \([\gamma]\) of the nonsingular compact leaves of \(F_ \omega\) generate a subgroup of \(H_ 1 (M^ 2_ g)\) denoted by \(H_ \omega\). If \([z_ 1], \dots, [z_{2g}]\) is a basis of \(H_ 1 (M^ 2_ g)\) we define \(\text{dirr} \omega = rk_ \mathbb{Q} \{\int_{z_ 1} \omega, \dots, \int_{z_{2g}} \omega\} - 1\).
By \(M_ \omega\) is denoted the set obtained by discarding all maximal neighbourhoods consisting of diffeomorphic compact leaves and all leaves which can be compactified by adding singular points.
Theorem 1. \(M_ \omega = \emptyset \Leftrightarrow \text{rk} H_ \omega = g\).
Theorem 2. If \(\omega\) is a closed form with Morse singularities given on \(M^ 2_ g\) \((g \neq 0)\) such that \(\text{dirr} \omega \geq g\), then the foliation \(F_ \omega\) has a noncompact fiber.

Reviewer: V.G.Angelov (Sofia)

MSC:

57R30	Foliations in differential topology; geometric theory
57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)

Keywords:

closed form; nondegenerate isolated singularities; foliation; closed form with Morse singularities

References:

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