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We call a foliation (M, F) on a manifold M chaotic if it is topologically transitive and the union of closed leaves is dense in M. The foliated manifold M is not assumed to be compact. The chaotic foliations can be considered as multidimensional generalization of chaotic dynamical systems in the sense of Devaney. For foliations covered by fibrations we prove that a foliation is chaotic if and only if its global holonomy group is chaotic. We introduce the concept of the integrable Ehresmann connection for a foliation as a natural generalization of the integrable Ehresmann connection for smooth foliations. A description of the global structure of foliations with the integrable Ehresmann connection and a criterion for the chaotic behavior of such foliations are obtained. Applying the method of suspension, a new countable family of pairwise nonisomorphic chaotic foliations of codimension two on 3-dimensional closed and nonclosed manifolds is constructed. [ABSTRACT FROM AUTHOR]

We prove that the set of leaves of a singular foliation with the Nishimori relation is partially ordered if and only if all leaves are proper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a certain topology on the group DiffF (M) of all Cr-diffeomorphisms of the foliated manifold (M; F), where r ≥ 0. This topology depends on the foliation and is called the F-compact-open topology. It coincides with the compact-open topology when F is an n-dimensional foliation. If the codimension of the foliation is n, then the convergence in this topology coincides with the pointwise convergence, where n = dim M. We prove that some subgroups of the group DiffF (M) are topological groups with the F-compact-open topology. Throughout this paper, we use smoothness of the class C∞. [ABSTRACT FROM AUTHOR]

This survey is devoted to particular problems of extrinsic geometry of foliations, which, roughly speaking, describes how leaves (or single submanifolds) are located in the ambient pseudo- Riemannian space. We discuss the following topics with the mixed scalar curvature: integral formulas and splitting of foliations, prescribing the mixed curvature of foliations, and variations of functionals defined on foliations, which seem to be central in extrinsic geometry. [ABSTRACT FROM AUTHOR]

We construct infinite countable families of pairwise topologically nonconjugate free chaotic groups of homeomorphisms on closed topological manifolds of different dimension. We find new invariants of suspended foliations in some complete subcategory of the category of foliations and use these results to construct infinite countable families of pairwise nonisomorphic chaotic topological foliations of an arbitrary even codimension on closed manifolds. [ABSTRACT FROM AUTHOR]

The subject of this paper is the geometry of orbits of a family of smooth vector fields defined on a smooth manifold and singular foliations generated by the orbits. As is well known, the geometry of orbits of vector fields is one of the main subjects of investigation in geometry and control theory. Here we propose several author's results on this problem. Throughout this paper, smoothness means C∞-smoothness. [ABSTRACT FROM AUTHOR]

We study the graph G픐(F) of a foliation F with Ehresmann connection 픐. We prove that for any suspended foliation (M,F) the induced foliation (G픐(F),픽) is also suspended and describe the structure of the graph G픐(F), as well as leaves of the induced foliation. We show that the induced foliation is a suspended foliation with the same transversal manifold and the same structure group as the original foliation. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold started in a previous paper. Under the assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distribution defines an unbounded, regular, self-adjoint operator in some Hilbert module over the C*-algebra of the foliation. [ABSTRACT FROM AUTHOR]

A local description of the foliation of the group SL(n) into conjugacy classes, and also the foliation of 𝔰𝔩∗(n) into coadjoint orbits, requires introducing parameters on a conjugacy class (a coadjoint orbit). Under the assumption that the parameters are rational functions of natural coordinates (the matrix elements) on SL(n), the problem is reduced to solving a system of linear equations. This system arises from the requirement that the parameters be invariant with respect to translations along vector fields normal to the conjugacy class. Similarly, the problem of parametrization of coadjoint orbits in 𝔰𝔩∗(n) can be solved using the Cartan–Weyl basis for 𝔰𝔩(n). The adjoint action is the differential of the conjugation action. Consequently, the parameters on the conjugacy classes and the coadjoint orbits are related by a transformation determined by the mapping from the algebra (n) to the group SL(n). The groups SL(3), SL(4) are considered as examples. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

In this paper, the following assertion is proved: Let GW and GW ˜ be Grassmannian three-webs defined respectively in domains D and D ˜ of a Grassmannian manifold of straight lines of the projective space Pr+1; Φ : D → D ˜ be a local diffeomorphism that maps foliations of the web GW to foliations of the web GW ˜ . Then Φ maps bundles of lines to bundles of lines, i.e., induces a point transformation, which is a projective transformation. In the case where r = 1, the proof is much more complicated than in the multidimensional case. In the case where r = 1, the dual theorem is formulated as follows: Let LW be a rectilinear three-web on a plane, i.e., three families of lines in the general position, and let this web be not regular, i.e., not locally diffeomorphic to a three-web formed by three families of parallel straight lines. Then each local diffeomorphism that maps a three-web LW to another rectilinear three-web LW ˜ is a projective transformation. As a consequence, we obtain the positive solution of the Gronwall problem (Gronwall, 1912): If W is a linearizable irregular three-web and θ and θ ˜ are local diffeomorphisms that map the three-web W to some rectilinear three-webs, then θ ˜ = π ° θ, where π is a projective transformation. [ABSTRACT FROM AUTHOR]

In this paper, we examine geometry and topology of foliations generated by level surfaces of metric functions. [ABSTRACT FROM AUTHOR]

In this paper, we describe all invariant distributions of nondegenerate bi-Hamiltonian structures and investigate their integrability in the neighborhood of a generic point. [ABSTRACT FROM AUTHOR]

We discuss connections between Fomenko invariants of Liouville foliations and asymptotic solutions of the Navier-Stokes equations that describe vortex structures in an incompressible fluid. [ABSTRACT FROM AUTHOR]

We study the problem of classification of complete non-Riemannian conformal foliations of codimension q > 2 with respect to transverse equivalence. It is proved that two such foliations are transversally equivalent if and only if their global holonomy groups are conjugate in the group of conformal transformations of the q-dimensional sphere Conf ( $$ {\mathbb{S}}^{\mathrm{q}} $$). Moreover, any countable essential subgroup of the group Conf ( $$ {\mathbb{S}}^{\mathrm{q}} $$) is realized as the global holonomy group of some non-Riemannian conformal foliation of codimension q. Bibliography: 16 titles. [ABSTRACT FROM AUTHOR]

Generalizing powers of a single hyperbolic automorphism of the two-dimensional torus, we consider some class sequences of such automorphism. As a substitute for the pair of foliations in the classical hyperbolic theory, every sequence of this class has a stable family of foliations. We prove a kind of the Poisson limit theorem for such sequences extending a method used earlier by A. Sharova and the present authors to prove the Poisson limit theorem for powers of a single hyperbolic automorphism of the torus. Possible generalizations are briefly discussed. [ABSTRACT FROM AUTHOR]

The notion of the factor-web $$ \bar{W} $$ ( ρ, r, r) (1 ≤ ρ < r) is defined for the three-web W( r, r, r) formed on a 2 r-dimensional differentiable manifold by three r-dimensional smooth foliations. Embedding of the factor-web in the initial web W( r, r, r) is constructed. This construction is a well-known geometric analog of the canonical extension of a Lie group of transformations to its parameter group. [ABSTRACT FROM AUTHOR]

We show how the Cartan-Laptev method that generalizes Elie Cartan's method of external forms and moving frames is applied to the study of closed G-structures defined by multidimensional three-webs formed on a C-smooth manifold of dimension 2 r, r ≥ 1, s ≥ 3, by a triple of foliations of codimension r. We say that a tensor T belonging to a differential-geometric object of order s of a three-web W is closed if it can be expressed in terms of components of objects of lower order s. We find all closed tensors of a three-web and the geometric sense of one of relations connecting three-web tensors. We also point out some sufficient conditions for the web to have a closed G-structure. It follows from our results that the G-structure associated with a hexagonal three-web W is a closed G-structure of class 4. It is proved that basic tensors of a three-web W belonging to a differential-geometric object of order s of the web can be expressed in terms of an s-jet of the canonical expansion of its coordinate loop, and conversely. This implies that the canonical expansion of every coordinate loop of a three-web W with closed G-structure of class s is completely defined by an s-jet of this expansion. We also consider webs with one-digit identities of kth order in their coordinate loops and find the conditions for these webs to have the closed G-structure. [ABSTRACT FROM AUTHOR]

An optimal control problem with two phase constraints and a two-dimensional control is considered. We prove that its optimal trajectories have a countable number of points of contact with the phase constraint bound on a finite time interval. The optimal synthesis of the problem after applying the quotient mapping by the action of the scale group has a complicated topological structure and, in some neighborhood of singular extremals, it is the Reeb foliation. [ABSTRACT FROM AUTHOR]

The geometry of foliations generated by some differentials of Abelian type is considered. The case where all fibers are closed is studied. [ABSTRACT FROM AUTHOR]

In this paper, we prove that in the space of polynomial foliations of a fixed degree of the complex two-dimensional space, foliations with separatrix connection, i.e., foliations in which any two distinct points have a common separatrix, are dense. The main tool of the proof is the analysis of the monodromy group of the foliation in a neighborhood of the infinitely distant point of the ambient projective space. [ABSTRACT FROM AUTHOR]

In this paper, we prove that in the space of polynomial foliations of a fixed degree of the complex two-dimensional space, foliations with separatrix connection, i.e., foliations in which any two distinct points have a common separatrix, are dense. The main tool of the proof is the analysis of the monodromy group of the foliation in a neighborhood of the infinitely distant point of the ambient projective space. [ABSTRACT FROM AUTHOR]

The article focuses on the differential topological classification of curvilinear three-webs within families of curves and equivalence relation. It explores the accuracy of the web with the utilization of local deomorphisms with configurations comprised of foliations. It also discusses the differential-geometric web theory and its related works including the identification on the main types of webs and their classification.

The article discusses the process of understanding the geometry and algebra of the classical three-web theory with foliations. It provides a definition of three-web under the manifold of three generic foliations. It also explores the classical theory of the web, while considering its accuracy based on local diffeomorphisms and isotopic transformations.

In this paper, we study basic differential invariants of the pair (vector field, foliation). As a result, we establish a dynamic interpretation and a generalization of the Levi-Civita connection and Riemannian curvature treated as invariants of the geodesic flow on the tangent bundle. [ABSTRACT FROM AUTHOR]

We consider the phase space of the quadratic area-preserving Henon map on the plane. We construct the stable and unstable foliation to infinity and prove their differentiability in the real case. Main conjectures on the behavior of the foliation are discussed for the complex case. We use the presentation of a dynamical system in the form of a continued fraction. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]