An indicator of the noncompactness of a foliation on \(M_ g^ 2\).

*(English. Russian original)*Zbl 0809.57018
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.

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
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\textit{I. A. Mel'nikova}, Math. Notes 53, No. 3, 158--160 (1993; Zbl 0809.57018); translation from Mat. Zametki 53, No. 3, 158--160 (1993)

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##### References:

[1] | H. Damir, J. Funct. Anal.,75, No. 2, 349-361 (1987). · Zbl 0637.58021 · doi:10.1016/0022-1236(87)90100-5 |

[2] | S. P. Novikov, Usp. Mat. Nauk,37, No. 5, 3-49 (1982). |

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