Local and global stability of compact leaves and foliations
The equivalence between local stability and completeness and Quasi-analyticity is proved for an arbitrary compact foliation. We prove that a compact foliation Locally stable if and only if it admits an Ehresmann connection and has a quasi-analytical holonomy pseudo-group. As an application we prove the local stability of complete compact foliations with a rigid transverse geometry, which included Cartan foliations. Without assumption about the existence of an Ehresmsnn conection we prove two theorems on stability of compact leaves of conformal foliations. The relation with results of other authors is established.
As an application of our previous results we prove theorems of local and global stability of leaves in sense of Ehresmann and Reeb for conformal foliations of codimention $q>2$. It has been shown that for transversally affine foliations the analogous statements on noncompact closed leaves are not valid. We also remind our rusults about local and global stability of compact leaves of foliations with quasi analytical holonomy pseudogroup admitting an Ehresmann connection and corresponding results of other authors.
We investigated conformal foliations $(M,F)$ of codimension $q\geq 3$ and proved a criterion for them to be Riemannian. In particular, the application of this criterion allowed us to proof the existence of an attractor that is a minimal set for each non-Riemannian conformal foliation. Moreover, if foliated manifold is compact then non-Riemannian conformal foliation $(M,F)$ is $(Conf(S^q),S^q)$-foliation with finitely many minimal sets. They are all attractors, and each leaf of the foliation belongs to the basin of at least one of them. The specificity of the proper conformal foliations is indicated. Special attention is given to complete conformal foliations.
In this paper a unified method for studying foliations with transversal parabolic geometry of rank one is presented.
Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations
We consider a Cartan foliation (M,F) of an arbitrary codimension q admitting an Ehresmann connection such that all leaves of (M,F) are embedded submanifolds of M. We prove that for any foliation (M,F) there exists an open, not necessarily connected, saturated, and everywhere dense subset M0 of M and a manifold L0 such that the induced foliation (M0, FM0) is formed by the fibers of a locally trivial fibration with the standard fiber L0 over (possibly, non-Hausdorff) smooth q-dimensional manifold. In the case of codimension 1, the induced foliation on each connected component of the manifold M0 is formed by the fibers of a locally trivial fibration over a circle or over a line.
The geometry of foliations generated by some differentials of Abelian type is considered. The case where all fibers are closed is studied.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.