Singular points of a Morsian form and foliations. (English. Russian original) Zbl 0914.58006

Mel’nikova, I. A.

Mosc. Univ. Math. Bull. 51, No. 4, 33-36 (1996); translation from Vestn. Mosk. Univ., Ser. I 1996, No. 4, 37-40 (1996).

Let \(M\) be a smooth compact connected orientable \(n\)-dimensional manifold on which the 1-form \(\omega\) with Morse peculiarities is defined. On the manifold \(M\) a foliation with peculiarities \(\mathcal F_\omega\) is specified. The irrationality power of the form \(\omega\) is determined by \[ \text{dirr } \omega = rk_Q\Biggl\{\int_{z_1} \omega,\dots, \int_{z_m} \omega \Biggr\} - 1, \] where \(z_1,\dots,z_m\) is the basis in \(H_1(M)\). It is proved that in the case of a compact foliation the irrationality power of the form and the number of homologically independent layers are determined by the correlation of the number of special points of index 0 and 1.

Reviewer: A.A.Martynyuk (Kyïv)

MSC:

58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
22E70	Applications of Lie groups to the sciences; explicit representations

Keywords:

Morse forms; foliation; singular points