Problems of Lifts in Symplectic Geometry

Let(M,ω)be a symplectic manifold.In this paper,the authors consider the notions of musical(bemolle and diesis)isomorphisms ω~b:T M→T~*M and ω~?:T~*M→TM between tangent and cotangent bundles.The authors prove that the complete lifts of symplectic vector field to tangent and cotangent bundles is ω~b-related.As consequence of analyze of connections between the complete lift ~cω_(T M )of symplectic 2-form ω to tangent bundle and the natural symplectic 2-form dp on cotangent bundle,the authors proved that dp is a pullback o f~cω_(TM)by ω~?.Also,the authors investigate the complete lift ~cφ_T~*_M )of almost complex structure φ to cotangent bundle and prove that it is a transform by ω~?of complete lift~cφ_(T M )to tangent bundle if the triple(M,ω,φ)is an almost holomorphic A-manifold.The transform of complete lifts of vector-valued 2-form is also studied.     

Minimal surfaces and particles in 3-manifolds

We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the space of such cone-manifolds is parametrized by the cotangent bundle of Teichmüller space, and that moreover such cone-manifolds have a canonical foliation by space-like surfaces. We extend these results to de Sitter and Minkowski cone-manifolds, as well as to some related “quasifuchsian” hyperbolic manifolds with conical singularities along infinite lines, in this later case under the condition that they contain a minimal surface with principal curvatures less than 1. In the hyperbolic case the space of such cone-manifolds turns out to be parametrized by an open subset in the cotangent bundle of Teichmüller space. For all settings, the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichmüller space is the same as the 3-dimensional gravity one. The proofs use minimal (or maximal, or CMC) surfaces, along with some results of Mess on AdS manifolds, which are recovered here in a different way, using differential-geometric methods and a result of Labourie on some mappings between hyperbolic surfaces, that allows an extension to cone-manifolds.      

Exact Lagrangian submanifolds in simply-connected cotangent bundles

We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel–Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler [16], using a different approach.     

Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index

We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, then the asymptotic Maslov indices of periodic orbits are dense in the half line [0,+∞). Furthermore, if the Hamiltonian is the Fenchel dual of an electromagnetic Lagrangian, then every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of Mather’s theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector. Dedicated to Vladimir Igorevich Arnold     

Musical Isomorphisms and Problems of Lifts

Using the complete lift on tangent bundles, the authors construct the complete lift on cotangent bundles of tensor fields with the aid of a musical isomorphism. In this new framework, the authors have a new intrepretation of the complete lift of tensor fields on cotangent bundles.     

Quadratic algebras related to elliptic curves

We construct quadratic finite-dimensional Poisson algebras corresponding to a rank-N degree-one vector bundle over an elliptic curve with n marked points and also construct the quantum version of the algebras. The algebras are parameterized by the moduli of curves. For N = 2 and n = 1, they coincide with Sklyanin algebras. We prove that the Poisson structure is compatible with the Lie-Poisson structure defined on the direct sum of n copies of sl(N). The origin of the algebras is related to the Poisson reduction of canonical brackets on an affine space over the bundle cotangent to automorphism groups of vector bundles. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 163–183, August, 2008.     

Partitioned Runge–Kutta Methods in Lie-Group Setting

We introduce partitioned Runge–Kutta (PRK) methods as geometric integrators in the Runge–Kutta–Munthe-Kaas (RKMK) method hierarchy. This is done by first noticing that tangent and cotangent bundles are the natural domains for the differential equations to be solved. Next, we equip the (co)tangent bundle of a Lie group with a group structure and treat it as a Lie group. The structure of the differential equations on the (co)tangent-bundle Lie group is such that partitioned versions of the RKMK methods are naturally introduced. Numerical examples are included to illustrate the new methods.     

Singular cotangent bundle reduction & spin Calogero–Moser systems

We develop a bundle picture for singular symplectic quotients of cotangent bundles acted upon by cotangent lifted actions for the case that the configuration manifold is of single orbit type. Furthermore, we give a formula for the reduced symplectic form in this setting. As an application of this bundle picture we consider Calogero–Moser systems with spin associated to polar representations of compact Lie groups.     

Hamiltonian reduction of free particle motion on the group SL(2, ℝ)

We analyze the structure of the reduced phase space that arises in the Hamiltonian reduction of the phase space of free particle motion over the groupSL(2, ℝ). The reduction considered is based on introducing constraints that are analogous to those used in the reduction of the Wess-Zumino-Novikov-Witten model to Toda systems. It is shown that the reduced phase space is diffeomorphic either to a union of two two-dimensional planes or to a cylinder S¹×ℝ. We construct canonical coordinates for both cases and show that in the first case, the reduced phase space is symplectomorphic to the union of two cotangent bundles T*(ℝ) endowed with a canonical symplectic structure, while in the second case, it is symplectomorphic to the cotangent bundle T* (S¹), which is also endowed with a canonical symplectic structure. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 149–161, January, 1997.     

Homoclinic orbits and chaos in discretized perturbed NLS systems: Part II. Symbolic dynamics

Summary In Part I ([9], this journal), Li and McLaughlin proved the existence of homoclinic orbits in certain discrete NLS systems. In this paper, we will construct Smale horseshoes based on the existence of homoclinic orbits in these systems. First, we will construct Smale horseshoes for a general high dimensional dynamical system. As a result, a certain compact, invariant Cantor set Λ is constructed. The Poincaré map on Λ induced by the flow is shown to be topologically conjugate to the shift automorphism on two symbols, 0 and 1. This gives rise to deterministicchaos. We apply the general theory to the discrete NLS systems as concrete examples. Of particular interest is the fact that the discrete NLS systems possess a symmetric pair of homoclinic orbits. The Smale horseshoes and chaos created by the pair of homoclinic orbits are also studied using the general theory. As a consequence we can interpret certain numerical experiments on the discrete NLS systems as “chaotic center-wing jumping.”     

Homotopy classification of equivariant regular sections

In his paper [2], Bierstone proves the equivariant Gromov theorem which is an integrability theorem for “open regularity condition” of equivariant sections of a smooth G-fibre bundle under the assumption that all orbit bundles of base manifold are non-closed. Here, we prove the result without his assumption under a nice “open regularity condition” which we call “G-extensible”. One of the examples of “G-extensible condition” is given by notions of Thom-Boardman singularities.     

Universal connections in Fréchet principal bundles

A new methodology leading to the construction of a universal connection for Fréchet principal bundles is proposed in this paper. The classical theory, applied successfully so far for finite dimensional and Banach modelled bundles, collapses within the framework of Fréchet manifolds. However, based on the replacement of the space of continuous linear mappings by an appropriate topological vector space, we endow the bundle J ¹ P of 1-jets of the sections of a Fréchet principal bundle P with a connection form by means of which we may “reproduce” every connection of P.      

A generalization of Higgs bundles to higher dimensional varieties

Let X be a smooth n-dimensional projective variety defined over and let L be a line bundle on X. In this paper we shall construct a moduli space parametrizing -cohomology L-twisted Higgs pairs, i.e., pairs where E is a vector bundle on X and . If we take , the canonical line bundle on X, the variety is canonically identified with the cotangent bundle of the smooth locus of the moduli space of stable vector bundles on X and, as such, it has a canonical symplectic structure. We prove that, in the general case, in correspondence to the choice of a non-zero section , one can define, in a natural way, a Poisson structure on . We also analyze the relations between this Poisson structure on and the canonical symplectic structure of the cotangent bundle to the smooth locus of the moduli space of parabolic bundles over X, with parabolic structure over the divisor D defined by the section s. These results generalize to the higher dimensional case similar results proved in [Bo1] in the case of curves. Received November 4, 1997; in final form May 28, 1998     

Classes of General Natural Almost Anti-Hermitian Structures on the Cotangent Bundles

We give the characterizations for the general natural anti-Hermitian structures on the cotangent bundles, which are in the eight classes obtained by Ganchev and Borisov. Considering an anti-Hermitian structure of natural diagonal type on the cotangent bundle, we construct examples for every class of anti-Hermitian structures.     

Langlands duality for representations of quantum groups

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an “interpolating quantum group” depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.     

On Riesz groups

In this paper we investigate projective 4-dimensional manifolds X whose tangent bundles T_X are numerically effective and give an almost complete classification. An important technical tool is the “Mori theory” of projective manifolds X whose canonical bundles K_X are not numerically effective.     

Morse theory for the space of Higgs <Emphasis Type="Italic">G</Emphasis>–bundles

Fix a C ^∞ principal G–bundle E⁰_G{E^0_G} on a compact connected Riemann surface X, where G is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang–Mills–Higgs functional on the cotangent bundle of the space of all smooth connections on E⁰_G{E^0_G}. We prove that this flow preserves the subset of Higgs G–bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs G–bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.     

Dynamics of acyclic interval maps

We investigate conditions under which a map f in a possibly non-compact interval is acyclic— the only periodic orbits are fixed points. Several earlier results are generalized to maps with multiple fixed points. The chief tools are convergence results due to Coppel and Sharkovski, and the Schwarzian derivative. Illustrative examples are given and open problems are suggested.

He who can digest a second or third fluxion . . . need not, methinks, be squeamish about any point in divinity. ―Bishop George Berkeley, “The Analyst,” 1734

     

Coherent orientation of mixed moduli spaces in Morse-Floer theory

We give the coherent orientation for the spaces of intersections of gradient trajectories and holomorphic disks in cotangent bundle. This construction provides the Piunikhin-Salamon-Schwarz isomorphism between Morse homology and Floer homology for Lagrangian intersections in cotangent bundles, with integer coefficients. This work is partially supported by Ministry of Science and Environmental Protection of Republic of Serbia Project #144020.