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

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.


57R30 Foliations in differential topology; geometric theory
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