Pseudo-Riemannian foliations and their graphs
We prove that a foliation (M;F) of codimension q on a ndimensional pseudo-Riemannian manifold is pseudo-Riemannian if and only if any geodesic that is orthogonal at one point to a leaf is orthogonal to every leaf it intersects.
We show that on the graph G = G(F) of a pseudo-Riemannian foliation there exists a unique pseudo-Riemannian metric such that canonical projections are pseudo-Riemannian submersions and the fibers of different projections
are orthogonal at common points. Relatively this metric the induced foliation (G;F) on the graph is pseudo-Riemannian and the structure of the leaves of (G;F) is described. Special attention is given to the structure of graphs of transversally (geodesically) complete pseudo-Riemannian foliations which are totally geodesic pseudo-Riemannian ones.
We investigate the structure of the holonomy groupoids of pseudo-Riemannian foliations
of arbitrary codimension on n-dimensional pseudo-Riemannian manifolds.
The structure of leaves of induced foliations on the holonomy groupoids is described.
Clarified specifics of the holonomy groupoids of transversally complete pseudo-Riemannian foliations.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.
We investigate totally geodesic foliations (M, F) of arbitrary codimemsion q on n-dimensional pseudo-Riemannian manifolds for which the induced metrics on leaves don't degenerate. We assume that the q-dimensional orthogonal distribution D to (M, F) is an Ehresmann connection for this foliation. Since the usual graph G(F) is not Hausdorff manifold in general, we investigate the graph G(F, D) of the foliation with an Ehresmann connection D introduced early by the author. This graph is always Hausdorff manifold. We prove that on the graph G(F, D) a pseudo-Riemannian metric is defined, with respect to which the induced foliation and the simple foliations formed by the fibers of the canonical projections are totally geodesic. It is proved that the leaves of the induced foliation on the graph are nondegenerately reducible pseudo-Riemannian manifolds and their structure is described. The application to parallel foliations on nondegenerately reducible pseudo-Riemannian manifolds is considered. It is shown that every foliation defined by the suspension of a homomorphism of the fundamental group of a pseudo-Riemannian manifold belongs to the investigated class of foliations.
We study totally geodesic foliations (M, F) of arbitrary codimension q on n-dimensional pseudo-Riemannian manifolds, for which the induced metrics on leaves is nondegenerate. We assume that the q-dimensional orthogonal distribution M to (M, F) is an Ehresmann connection for this foliation. Since the usual graph G(F) is not Hausdorff manifold in general, we study the graph GM(F) of the foliation with an Ehresmann connection M introduced early by the author. This graph is always a Hausdorff manifold. We prove that on the graph GM(F), a pseudo-Riemannian metric is defined, with respect to which the induced foliation and the simple foliations formed by the leaves of the canonical projections are totally geodesic. We show that the leaves of the induced foliation on the graph are non-degenerately reducible pseudo-Riemannian manifolds and their structure is described. The application to parallel foliations on nondegenerately reducible pseudo- Riemannian manifolds is considered. We also show that each foliation defined by the suspension of a homomorphism of the fundamental group of a pseudo-Riemannian manifold belongs to the considered class of foliations.
The subject of this article is a review of the results on foliations with transversal linear connection obtained by the author together with N.I. Zhukova, and their comparison with the results of other authors. The work consists of three parts. The first part focuses on to automorphism groups of foliations with a transversal linear connection in the category of foliations. In the second part, the question of the equivalence of the concept of completeness for the class of foliations under investigation is studied. The third part we present theorems on pseudo-Riemannian foliations that form an important class of foliations with a transversal linear connection.In particular, we present results on graphs of pseudo-Riemannian foliations that contain all information about foliations.
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.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.