The number of split points of a Morse form and the structure of its foliation. (English) Zbl 1324.57004

Gelbukh, Irina

Math. Slovaca 63, No. 2, 331-348 (2013).

In the paper under review the author studies closed manifolds equipped with a closed smooth one-form, having Morse-type singularities. This means that locally in a neighborhood of a singular point, the one-form \(\omega \) writes \(\omega =df\) where \(f\) is a non-degenerate quadratic form. Such a closed one-form defines a codimension one foliation outside its singular set, and some work has been done in the comprehension of the relation between the topology of the leaves of such a foliation and the combinatorics involving the arrangements of the singularities. Such singularities are originally divided into two classes: centers and saddles. In the paper under review the author introduces a class of so called split conic singularities, which are saddle-type singularities which are somehow accumulated by leaves coming from two distinct center-type singularities. The motivation is that when passing such a split point, a leaf splits into two in a sense made clear by the author. In the paper the author introduces and studies the number \(d(\omega)= \frac { a(\omega) - b(\omega)}{2} + 1\), where \(a(\omega)\) is the number of split points of \(\omega \) and \(b(\omega)\) is the number of centers of \(\omega \). The author proves \(d(\omega)\) is non-negative and shows some other inequalities associating \(d(\omega)\) to some important characteristics of the foliation. Relations with Novikov’s theory of closed 1-forms and their singularities are then clear. The sharpness of the results is discussed. The paper can be quite useful for those interested in the field and related areas.

Reviewer: Bruno Scardua (Rio de Janeiro)

MSC:

57R30

Foliations in differential topology; geometric theory

Keywords:

Morse form; singularities; foliation; foliation graph

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