Graded Riemann surfaces and Krichever-Novikov algebras

Following the work of Krichever and Novikov, Bonora, Martellini, Rinaldi and Russo defined a superalgebra associated to each compact Riemann surface with spin structure. Noting that this data determines a graded Riemann surface, we find a natural interpretation of the BMRR-algebra in terms of the geometry of graded Riemann surfaces. We also discuss the central extensions of these algebras (correcting the form of the central extension given by Bonoraet al.). It is hoped that this work will be the first step towards defining Krichever-Novikov algebras for (the more general) super-Riemann surfaces; in particular we emphasise the importance ofgraded conformal vectorfields.     

Symmetries of parabolic contact structures

We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion-free and some types of them have to be locally flat. We prove that each symmetry given at a point with non-zero harmonic curvature is involutive. Finally we give restrictions on the number of different symmetries which can exist at such a point.     

On surfaces whose twistor lifts are harmonic sections

We study surfaces whose twistor lifts are harmonic sections, and characterize these surfaces in terms of their second fundamental forms. As a corollary, under certain assumptions for the curvature tensor, we prove that the twistor lift is a harmonic section if and only if the mean curvature vector field is a holomorphic section of the normal bundle. For surfaces in four-dimensional Euclidean space, a lower bound for the vertical energy of the twistor lifts is given. Moreover, under a certain assumption involving the mean curvature vector field, we characterize a surface in four-dimensional Euclidean space in such a way that the twistor lift is a harmonic section, and its vertical energy density is constant.     

Inherited structures in deformations of Poisson pencils

In this paper, we study some properties of bi-Hamiltonian deformations of Poisson pencils of hydrodynamic type. More specifically, we are interested in determining those structures of the fully deformed pencils that are inherited through the interaction between structural properties of the dispersionless pencils (in particular exactness or homogeneity) and suitable finiteness conditions on the central invariants (like polynomiality). This approach enables us to gain some information about each term of the deformation to all orders in  ?$?$.     

Morita Invariant Properties of Twisted Poisson Manifolds

In this letter, we study some invariants of integrable twisted Poisson manifolds under Morita equivalence. We show that two twisted symplectic manifolds are Morita equivalent if and only if their fundamental groups are isomorphic. Moreover, the first cohomology group of twisted Poisson manifolds is invariant under Morita equivalence. Finally, we prove that if two twisted Poisson manifolds are Morita equivalent, then their representation categories are equivalent.      

Complete explicit classification of parallel Lorentz surfaces in arbitrary pseudo-Euclidean spaces

A Lorentz surface of an indefinite space form is called parallel if its second fundamental form is parallel. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in physics since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel spacelike surfaces in an arbitrary indefinite space form are classified in Chen (2010) [20]. Moreover, parallel Lorentz surfaces in 4D indefinite space forms are completely classified in a series of recent articles Chen (submitted for publication) [16], Chen (submitted for publication) [17], Chen (in press) [18], Chen (2010) [19], Chen and Van der Veken (2009) [15] (see also Graves (1979) [12], Graves (1979) [13] and Magid (1984) [14] for some partial results). In this paper, we achieve the complete classification of parallel Lorentz surfaces in a pseudo-Euclidean space with arbitrary codimension and arbitrary index.     

A tri-Hamiltonian formulation of the full Kostant-Toda lattice

We define Poisson structures which lead to a tri-Hamiltonian formulation for the full Kostant-Toda lattice. In addition, a hierarchy of vector fields called master symmetries are constructed and they are used to generate the nonlinear Poisson brackets and other invariants. Various deformation relations are investigated. The results are analogous to results for the finite nonperiodic Toda lattice.     

Perturbation to symmetries and adiabatic invariants of a type of nonholonomic singular system

Based on the theory of symmetries and conserved quantities, the perturbation to the symmetries and adiabatic invariants of a type of nonholonomic singular system are discussed. Firstly, the concept of higher order adiabatic invariants of the system is proposed. Secondly, the conditions for existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Thirdly, we study the inverse problems of the perturbation to symmetries of the system. An example is presented to illustrate these results.     

One-Loop β Functions of a Translation-Invariant Renormalizable Noncommutative Scalar Model

In this paper we explain how to define “lower dimensional” volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes do not involve noncommutative geometry or spin structures at all.      

Quasi-classical limit of Toda hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP hierarchy and itsW-infinity symmetries are extended to the Toda hierarchy. The Planck constant now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions, and tau function, are redefined.W _{1 +} symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted tow _{1 +} symmetries of the dispersionless hierarchy through their action on the tau function.     

On the twistor-spinors

Two interesting conformal invariants which are constant on the manifold are given for twistor-spinors on a spin manifold following the notion of a twistor-spinor associated to a twisted spin bundle. For a twisted spin bundle corresponding to a flat Hermitian vector bundle, the associated twistor-spinors admit the same conformal invariants.An analysis is made of the twistor-spinors given by , where f is a complex-valued function. There is only one case where is not a Killing spinor. An example is given of a compact spin manifold for which the situation is realized.     

Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for Lagrange systems

Based on the invariance of differential equations under infinitesimal transformations of group, Lie symmetries, exact invariants, perturbation to the symmetries and adiabatic invariants in form of non-Noether for a Lagrange system are presented. Firstly, the exact invariants of generalized Hojman type led directly by Lie symmetries for a Lagrange system without perturbations are given. Then, on the basis of the concepts of Lie symmetries and higher order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for the system with the action of small disturbance is investigated, the adiabatic invariants of generalized Hojman type for the system are directly obtained, the conditions for existence of the adiabatic invariants and their forms are proved. Finally an example is presented to illustrate these results.     

Equivariant quantization of orbifolds

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces, etc. In this work, we prove the existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in Poncin et al. (2009) [19]. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization.     

Lie Symmetries, Perturbation to Symmetries and Adiabatic Invariants of a Generalized Birkhoff System

We study the perturbation to symmetries and adiabatic invariants of a generalized Birkhoff system. Based on the invariance of differential equations under infinitesimal transformations, Lie symmetries, laws of conservations, perturbation to the symmetries and adiabatic invariants of the generalized Birkhoff system are presented. First, the concepts of Lie symmetries and higher order adiabatic invariants of the generalized Birkhoff system are proposed. Then, the conditions for the existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate the method and results.     

Symmetries of the super KP hierarchy

Symmetries of the super Kadomtsev-Petviashvili hierarchy are studied. A key role is played by a D-module structure, which connects the nonlinear system with the geometry of an infinite-dimensional super Grassmannian manifold. Infinitesimal action of a Lie superalgebra on the super Grassmannian manifold, via this connection, gives rise to symmetries of the nonlinear system.Supported in part by the Grant in Aid for Scientific Research, the Ministry of Education.     

Invariant currents and scattering off locally symmetric potential landscapes

We study the effect of discrete symmetry breaking in inhomogeneous scattering media within the framework of generic wave propagation. Our focus is on one-dimensional scattering potentials exhibiting local symmetries. We find a class of spatially invariant nonlocal currents, emerging when the corresponding generalized potential exhibits symmetries in arbitrary spatial domains. These invariants characterize the wave propagation and provide a spatial mapping of the wave function between any symmetry related domains. This generalizes the Bloch and parity theorems for broken reflection and translational symmetries, respectively. Their nonvanishing values indicate the symmetry breaking, whereas a zero value denotes the restoration of the global symmetry where the well-known forms of the two theorems are recovered. These invariants allow for a systematic treatment of systems with any local symmetry combination, providing a tool for the investigation of the scattering properties of aperiodic but locally symmetric systems. To this aim we express the transfer matrix of a locally symmetric potential unit via the corresponding invariants and derive quantities characterizing the complete scattering device which serve as key elements for the investigation of transmission spectra and particularly of perfect transmission resonances.     

相对论性Birkhoff系统的对称性摄动及其逆问题

研究在小干扰力作用下相对论性Birkhoff系统的对称性摄动问题.建立了相对论性Birkhoff系统的基本原理、运动方程和小扰动方程.讨论该系统的Lie对称性变换和守恒量.研究在无限小变换下该系统的对称性摄动，构造了s阶绝热不变量.给出了绝热不变量存在的条件和形式.研究该系统的对称性摄动逆问题，当系统存在s阶绝热不变量时，得到了该系统的无限小变换的对称性摄动.研究相对论性Birkhoff系统和经典Birkhoff系统对称性摄动之间的关系. 关键词： Lie对称性 摄动 绝热不变量 相对论     

微扰Kepler系统轨道微分方程的近似Lie对称性与近似不变量

把极角θ视为独立变量,得到Kepler系统的轨道微分方程.首先讨论Kepler系统轨道微分方程的Lie对称性和不变量,微扰Kepler系统轨道微分方程的精确Lie对称性和精确不变量,其次讨论微扰Kepler系统轨道微分方程的近似Lie对称性和近似不变量,并得到了微扰Kepler系统的9个一阶近似Lie对称性和6个一阶近似不变量,其中1个实为精确不变量,而其余5个分别等于微扰系数ε乘以Kepler系统相应的5个不变量。     

相空间中离散力学系统对称性的摄动与Hojman型绝热不变量

研究相空间中离散力学系统对称性的摄动与绝热不变量.列出相空间中未受扰离散力学系统的特殊Lie对称性导致的Hojman型精确不变量.基于相空间中力学系统的高阶绝热不变量的定义，研究在小扰动作用下系统Lie对称性的摄动，得到了相空间中离散力学系统的一类新的绝热不变量——Hojman型绝热不变量.举例说明结果的应用. 关键词： 相空间 Lie对称性 摄动 绝热不变量     

事件空间中完整系统的Lie对称性与绝热不变量

研究事件空间中完整力学系统Lie对称性的摄动与绝热不变量.基于力学系统的高阶绝热不变量的概念，研究在小扰动作用下系统Lie对称性的摄动，得到了事件空间中完整力学系统的一类Hojman形式的高阶绝热不变量，给出了绝热不变量存在的条件及形式.并举例说明结果的应用. 关键词： 事件空间 对称性 摄动 绝热不变量