Jacobi Structures in R3\mathbb{R}^3

The most general Jacobi brackets in $\mathbb{R}^3$ are constructed after solving the equations imposed by the Jacobi identity. Two classes of Jacobi brackets were identified, according to the rank of the Jacobi structures. The associated Hamiltonian vector fields are also constructed

On invariants of almost symplectic connections

We study the irreducible decomposition under Sp(2n, R) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insight for finding a preferred linear almost symplectic connection on M . We rediscover Ph. Tondeur's Theorem on almost symplectic connections. Properties of torsion of the vectorial kind are deduced

Harmonic G-structures

For closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold M, where G-structures are considered as sections of the quotient bundle O(M)/G. Then, we deduce the corresponding first and second variation formulae and the characterising conditions for critical points by means of tools closely related with the study of G-structures. In this direction, we show the role in the energy functional played by the intrinsic torsion of the G-structure. Moreover, we analyse the particular case G=U(n) for even-dimensional manifolds. This leads to the study of harmonic almost Hermitian manifolds and harmonic maps from M into O(M)/U(n).Comment: 27 pages, minor correction

Generalized Lenard Chains, Separation of Variables and Superintegrability

We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multi-separable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional \omega N manifold guarantees the separation of variables. As an application, we construct such chains for the H\'enon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.Comment: 14 pages Revte

A class of Poisson-Nijenhuis structures on a tangent bundle

Equipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis structure from a given type (1,1) tensor field J on Q. It is argued that the complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ, but plays a crucial role in the construction of a different tensor R, which appears to be the pullback under the Legendre transform of the lift of J to co-tangent manifold of Q. We show how this tangent bundle view brings new insights and is capable also of producing all important results which are known from previous studies on the cotangent bundle, in the case that Q is equipped with a Riemannian metric. The present approach further paves the way for future generalizations.Comment: 22 page

Geometric Quantization on the Super-Disc

In this article we discuss the geometric quantization on a certain type of infinite dimensional super-disc. Such systems are quite natural when we analyze coupled bosons and fermions. The large-N limit of a system like that corresponds to a certain super-homogeneous space. First, we define an example of a super-homogeneous manifold: a super-disc. We show that it has a natural symplectic form, it can be used to introduce classical dynamics once a Hamiltonian is chosen. Existence of moment maps provide a Poisson realization of the underlying symmetry super-group. These are the natural operators to quantize via methods of geometric quantization, and we show that this can be done.Comment: 17 pages, Latex file. Subject: Mathematical physics, geometric quantizatio

Development of the technology for conservation of spoil heaps in order to reduce its negative impact on the environment and preserve resource potential

Coal mining is accompanied by bulk waste formation in form of spoil heaps from low quality ore. As a result of aggressive natural factors impact specific mineral and chemical composition of rocks, which forms spoil heaps, leads to the environmental man-caused load in form of dust and gas emission, water pollution and adjacent area littering. Thus, lands, where spoil heaps are disposed, are excluded from economic use, and their resource potential gets lost because of wind and water erosion, negative physical, chemical and biochemical processes, and transformation of rocks, which forms spoil heaps, that causes the reduction of its value as secondary resources. It defines the urgency of developing methods aimed at the lowering of ecological load, caused by spoil heaps, to acceptable level and at the preservation of its resource potential as artificial deposits. One of the possible solutions is to fill underground worked-out mine area with stowage materials using low quality ore. But this technology pushes up the cost of coal production and is not applicable to new coal mining objects, which has lack of free worked-out area as well as to waste dumps, accumulated during long years of coal production. One of the most common ways of reducing the negative impact on the environment of already formed spoil heaps is its recultivation. However, recultivation of coal mining waste, which is followed by application of antipyrogens, ameliorants and mineral fertilizers to waste dumps, considerably limits their further utilization as artificail deposits. Antipyrogens, ameliorants and mineral fertilizers, when entering into physical and chemical reactions with coal mining waste, make this waste unsuitable for application as a secondary resource, particularly during the production of many construction materials. It defines the prospectivity of spoil heaps conservation, which allows not only to reduce the negative impact of spoil heaps on the environment but also to preserve their resource potential. This study offers a method, a technology and technical solutions for spoil heaps conservation that ensure their geoecological safety as sources of environmental pollution and allow to use its resource potential for production of target products

Volume preserving multidimensional integrable systems and Nambu--Poisson geometry

In this paper we study generalized classes of volume preserving multidimensional integrable systems via Nambu--Poisson mechanics. These integrable systems belong to the same class of dispersionless KP type equation. Hence they bear a close resemblance to the self dual Einstein equation. All these dispersionless KP and dToda type equations can be studied via twistor geometry, by using the method of Gindikin's pencil of two forms. Following this approach we study the twistor construction of our volume preserving systems

Dual branes in topological sigma models over Lie groups. BF-theory and non-factorizable Lie bialgebras

We complete the study of the Poisson-Sigma model over Poisson-Lie groups. Firstly, we solve the models with targets $G$ and $G^*$ (the dual group of the Poisson-Lie group $G$) corresponding to a triangular $r$-matrix and show that the model over $G^*$ is always equivalent to BF-theory. Then, given an arbitrary $r$-matrix, we address the problem of finding D-branes preserving the duality between the models. We identify a broad class of dual branes which are subgroups of $G$ and $G^*$, but not necessarily Poisson-Lie subgroups. In particular, they are not coisotropic submanifolds in the general case and what is more, we show that by means of duality transformations one can go from coisotropic to non-coisotropic branes. This fact makes clear that non-coisotropic branes are natural boundary conditions for the Poisson-Sigma model.Comment: 24 pages; JHEP style; Final versio