The Contact Metric Connection on the Heisenberg Group

We prove that there is only one contact metric connection with skew-torsion on the Heisenberg group endowed with a left-invariant Sasakian structure. We obtain the expression of this connection via the contact form and the metric tensor, and show that the torsion tensor and the curvature tensor are constant and the sectional curvature varies between ?1 and 0.     

Canonical connection on a class of Reimannian almost product manifolds

The canonical connection on a Reimannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Reimannian almost product manifolds with non-integrable almost product structure.     

Invariant tensorfields and the canonical connection of a 3-web

In the following we shall give a new approach to web geometry. Instead of using the reduced adapted coframe bundles over the web manifolds we define invariant tensorfields corresponding to the 3-web structure and express the covariant derivation of the Chern connection with the help of these tensorfields, working only on the tangent bundle of the web manifold. The invariant tensorfields of a 3-web define a more general, so-called {H, J}-structure on the manifold which can be considered as an infinitesimal, non-integrable version of a web structure. We introduce a canonical connection of a {H, J}-structure which reduces to the Chern connection in the case of a 3-web. Using the tensorial expression of the covariant derivation of the Chern connection we give direct proof for the torsion and curvature identities. Finally, we apply our formulae to an algebraic characterization of the parallel translation with respect to the Chern connection.     

关于发展方程的守恒率的一个标记

本文使用微分代数的技巧，研究了发展方程的守恒率与对应的行波所满足方程的首次积分之间的关系．通过文本给出的结果，我们研究了Burgers方程和Burgers—KdV方程的可积性，证明了这两类方程都只有一个守恒率．利用本文给出的方法，可以通过常微分方程的研究方法来研究某些非线性发展方程．     

Ball on a viscoelastic plane

We consider dynamical problems arising in connection with the interaction of an absolutely rigid ball and a viscoelastic support plane. The support is a relatively stiff viscoelastic Kelvin-Voigt medium that coincides with the horizontal plane in the undeformed state. We also assume that under the deformation the support induces dry friction forces that are locally governed by the Coulomb law. We study the impact appearing when a ball falls on the plane. Another problem of our interest is the motion of a ball “along the plane.” A detailed analysis of various stages of the motion is presented. We also compare this model with classical models of interaction of solid bodies.     

Connections on contact manifolds and contact twistor space

In this paper we generalize the definition of symplectic connection to the contact case. It turns out that any odd-dimensional manifold equipped with a contact form admits contact connections and that any Sasakian structure induces a canonical contact connection. Furthermore (as in the symplectic case), any contact connection induces an almost CR structure on the contact twistor space which is integrable if and only if the curvature of the connection is of Ricci-type.     

On a class of Riemannian metrics arising from Finsler structures

On the slit tangent bundle of Finsler manifolds, we introduce a class of metrics and study the relation between Levi-Civita connection, Vaisman connection, vertical foliation, and Reinhart spaces. We show that the Levi-Civita and the Vaisman connections induce the same connections in the structural bundle if and only if the base manifold is Landsbergian. Moreover every foliated Reinhart manifold reduces to a Riemannian manifold.     

Automorphisms of Ehresmann connections

We prove that a diffeomorphism of a manifold with an Ehresmann connection is an automorphism of the Ehresmann connection, if and only if, it is a totally geodesic map (i.e., sends the geodesies, considered as parametrized curves, to geodesies) and preserves the strong torsion of the Ehresmann connection. This result generalizes and to some extent strengthens the classical theorem on the automorphisms of a D-manifold (manifold with covariant derivative). The second author is supported by the Hungarian Nat. Sci. Found. (OTKA), Grant No. NK68040.     

A remark on the canonical decomposition of a connection at an irregular singular point

We give a remark on the proof of the theorem of existence and uniqueness of the slope decomposition and the Turritin-Levelt decomposition of a connection in several variables. Using a lemma from [2], we show that the slope decomposition and the Turritin-Levelt decomposition of the relative connection in one variable is stabilized by the absolute connection.     

On Geodesic Curves on Quotient Manifold of Nondegenerate Affinor Fields

We consider the quotient manifold of the manifold of nondegenerate affinor fields on a compact manifold with respect to the action of the group of nowhere vanishing functions. This manifold is endowed with a structure of infinite-dimensional Lie group. On this Lie group, we construct an object of linear connection with respect to which all left-invariant vector fields are covariantly constant (the Cartan connection). We also find the geodesics of the Cartan connection.     

Eigenvectors for a random walk on a left-regular band

We present a simple construction of the eigenvectors for the transition matrices of random walks on a class of semigroups called left-regular bands. These walks were introduced and analyzed by Brown, and they include the hyperplane chamber walks of Bidigare, Hanlon and Rockmore. This construction leads to new concise proofs of several of the known results about these walks. We also explain how tools from poset topology can be used to extract an eigenbasis for the transition matrices of the hyperplane chamber walks, and indicate the connection with a method recently described by Denham.     

Ehresmann connection for the canonical foliation on a manifold over a local algebra

For a canonical foliation on a manifoldM ^A over a local algebra, theA-affine horizontal distribution complementary to the leaves, similar to the horizontal distribution of a higher order connection on the fiber bundle ofA-jets, is defined. In the case of a complete manifoldM ^A, theA-affine horizontal distribution is proved to be an Ehresmann connection in the sense of Blumental-Hebda. It is shown that theA-affine horizontal distribution onM ^A exists if and only if the Atiyah class of a certain foliated principal bundle vanishes.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 303–310, February, 1996.     

Cartan connection and its geodesics on a manifold of nondegenerate affinor fields

We consider the quotient manifold of the manifold of nondegenerate affinor fields on a compact manifold with respect to the action of the group of nowhere vanishing functions. On this manifold, we construct a Cartan connection and find its torsion tensor. We also find the geodesics of the Cartan connection.     

A note on the uniqueness of the canonical connection of a naturally reductive space

We extend the result in Olmos and Reggiani (J. Reine Angew. Math. 664:29–53, 2012) to the non-compact case. Namely, we prove that the canonical connection on a simply connected and irreducible naturally reductive space is unique, provided the space is not a sphere, a compact Lie group with a bi-invariant metric or its symmetric dual. In particular, the canonical connection is unique for the hyperbolic space when the dimension is different from three. We also prove that the canonical connection on the sphere is unique for the symmetric presentation. Finally, we compute the full isometry group (connected component) of a compact and locally irreducible naturally reductive space.     

Connections on modules over singularities of finite CM representation type

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection on M?We consider the maximal Cohen-Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d≤2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d≥3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections.Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a -Gorenstein singularity of dimension d≥2. We show that this condition is sufficient for the canonical module ω_A to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ω_A admits an integrable connection if and only if A is Gorenstein.     

On the specificity of a possibility distribution

The specificity of a possibility distribution measures the degree to which the distribution allows one and only one element as its manifestation. As such it is a measure of amount of uncertainty or information. We investigate a number of issues related to specificity measures. We discuss the connection between the specificity of a possibility distribution and the entropy of a probability distribution. We describe unifying view for constructing specificity measures. We look at the relationship of the specificity of a distribution and its negation. We consider the case where the base set is continuous.     

Identification of a Connection from Cauchy Data on a Riemann Surface with Boundary

We consider a connection ?^X{\nabla^X} on a complex line bundle over a Riemann surface with boundary M ₀, with connection 1-form X. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) L : = ?^X*?^X + q{L := \nabla^X{^*\nabla^X} + q} , with q a complex-valued potential, uniquely determines the connection up to gauge isomorphism, and the potential q.     

带有异常点的平方度量设施选址问题

传统的设施选址问题一般假设所有顾客都被服务,考虑到异常点的存在不仅会增加总费用(设施的开设费用与连接费用之和),也会影响到对其他顾客的服务质量.研究异常点在最终方案中允许不被服务的情况,称之为带有异常点的平方度量设施选址问题.该问题是无容量设施选址问题的推广.问题可描述如下:给定设施集合、顾客集,以及设施开设费用和顾客...     

Normal linearization and transition map near a saddle connection with symmetric resonances

We consider a heteroclinic connection in a planar system, between two symmetric hyperbolic saddles of which the eigenvalues are resonant. Starting with a $C^{\infty }$ normal form, defined globally near the connection, we normally linearize the vector field by using finitely smooth tags of logarithmic form. We furthermore define an asymptotic entry–exit relation, and we discuss on two examples how to deal with counting limit cycles near a limit periodic set involving such a connection.     

On the solvability of a boundary value problem for a fourth-order equation on a graph

We consider a boundary value problem for a fourth-order equation on a graph modeling elastic deformations of a plane rod system with conditions of rigid connection at the vertices. Conditions for the unique solvability are stated. We also present sufficient conditions for the problem to be degenerate.