On the Structure of a Morse Form Foliation

Author: Irina Gelbukh

Czech. Math. J., vol. 59, no. 1, pp. 207-220, 2009.

Abstract: The foliation of a Morse form $\omega$ on a closed manifold $M$ is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated. The number of minimal and maximal components is estimated in terms of characteristics of $M$ and $\omega$. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of $rk \omega$ and $Sing \omega$. The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that if $\omega$ has more centers than conic singularities then $b_1(M) = 0$ and thus the foliation has no minimal components and homologically non-trivial compact leaves, its foliation graph being a tree.

Keywords: Number of minimal components, number of maximal components, compact leaves, foliation graph, rank of a form

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