ON LAGRANGIAN EMBEDDINGS OF PARALLELIZABLE MANIFOLDS

2013 ◽  
Vol 24 (09) ◽  
pp. 1350073
Author(s):  
NAOHIKO KASUYA ◽  
TORU YOSHIYASU
Keyword(s):  
Euclidean Space ◽  
Symplectic Structure ◽  
Lagrangian Submanifold ◽  
Symplectic Structures

We prove that for any closed parallelizable n-manifold Mn, if the dimension n ≠ 7, or if n = 7 and the Kervaire semi-characteristic χ½(M7) is zero, then Mn can be embedded in the Euclidean space ℝ2n with a certain symplectic structure as a Lagrangian submanifold. By the results of Gromov and Fukaya, our result gives rise to symplectic structures of ℝ2n(n ≥ 3) which are not conformally equivalent to open domains in standard ones.

SYMPLECTIC STRUCTURES FROM CENTRAL EXTENSION OF W1+∞ ALGEBRA AND KP EQUATION

1995 ◽  
Vol 10 (04) ◽  
pp. 273-278
Author(s):  
J. GAWRYLCZYK ◽  
J. LUKIERSKI
Keyword(s):  
Central Extension ◽  
Symplectic Structure ◽  
Parameter Family ◽  
Kp Equation ◽  
Symplectic Structures ◽  
Kp Hierarchy ◽  
The One

We modify the first symplectic structure of KP hierarchy by considering its relation with W1+∞ algebra and introducing its central extension [Formula: see text]. We show that at least the first five Hamiltonians of modified KP hierarchy can be chosen to be conserved, in involution with respect to the symplectic bracket generated by [Formula: see text]. It appears that from the first four flows of modified KP hierarchy we shall obtain the same (2+1)-dimensional standard KP equation. We provide therefore the one-parameter family of Hamiltonians and symplectic structures describing the standard KP equation.

Generating families for images of Lagrangian submanifolds and open swallowtails

1986 ◽  
Vol 100 (1) ◽  
pp. 91-107 ◽  
Author(s):  
Stanisław Janeczko
Keyword(s):  
Generating Functions ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Normal Forms ◽  
Lagrangian Submanifolds ◽  
Lagrangian Submanifold ◽  
Classification Theorem ◽  
Local Models ◽  
Binary Forms

SummaryIn this paper we study the symplectic relations appearing as the generalized cotangent bundle liftings of smooth stable mappings. Using this class of symplectic relations the classification theorem for generic (pre) images of lagrangian submanifolds is proved. The normal forms for the respective classified puilbacks and pushforwards are provided and the connections between the singularity types of symplectic relation, mapped lagrangian submanifold and singular image, are established. The notion of special symplectic triplet is introduced and the generic local models of such triplets are studied. We show that the open swallowtails are canonically represented as pushforwards of the appropriate regular lagrangian submanifolds. Using the SL2(ℝ) invariant symplectic structure of the space of binary forms of n appropriate dimension we derive the generating families for the open swallowtails and the respective generating functions for its regular resolutions.

scholarly journals Poly-symplectic geometry and the AKSZ formalism

2021 ◽  
pp. 2150030
Author(s):  
Ivan Contreras ◽  
Nicolás Martínez Alba
Keyword(s):  
Phase Space ◽  
General Theory ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Sigma Model ◽  
Type Theorem ◽  
Poisson Structures ◽  
Action Functional ◽  
Symplectic Structures ◽  
Poisson Sigma Model

In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular, we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.

scholarly journals Canonical multi-symplectic structure on the total exterior algebra bundle

2006 ◽  
Vol 462 (2069) ◽  
pp. 1531-1551 ◽  
Author(s):  
Thomas J Bridges
Keyword(s):  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Hamiltonian Dynamics ◽  
Partial Differential Operator ◽  
Exterior Algebra ◽  
Legendre Transform ◽  
Natural Setting ◽  
Elliptic Pde ◽  
Symplectic Structures ◽  
Partial Differential

The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n -dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M . On the fibre, which has dimension 2 n , the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n -symplectic structures define a partial differential operator, J ∂ , which turns out to be a Dirac operator with multi-symplectic structure. The operator J ∂ generalizes the product operator J (d/d t ) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J ∂ is elliptic, and the generalization of Hamiltonian systems, J ∂ Z =∇ S ( Z ), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S ( Z ) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on k -forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.

scholarly journals Representations of the Fundamental Group of an L–Punctured Sphere Generated by Products of Lagrangian Involutions

2007 ◽  
Vol 59 (4) ◽  
pp. 845-879 ◽  
Author(s):  
Florent Schaffhauser
Keyword(s):  
Fixed Point ◽  
Moduli Space ◽  
Symplectic Structure ◽  
Lagrangian Submanifold ◽  
Unitary Representations ◽  
Fixed Point Set ◽  
Hamiltonian Description ◽  
Point Set ◽  
Reduced Space ◽  
By Products

AbstractIn this paper, we characterize unitary representations of π := π1(S2\{s1, … , sl}) whose generators u1, … , ul (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions uj = σjσj+1 with σl+1 = σ1. Our main result is that such representations are exactly the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space ℳᘓ := Homᘓ(π, U(n))/U(n). Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of ℳᘓ. To prove this, we use the quasi-Hamiltonian description of the symplectic structure of ℳᘓ and give conditions on an involution defined on a quasi-Hamiltonian U-space (M, ω, μ: M → U) for it to induce an anti-symplectic involution on the reduced space M//U := μ–1({1})/U.

scholarly journals Symplectic structures and symmetries of solutions of the complex Monge-Ampére equation

1998 ◽  
Vol 150 ◽  
pp. 63-83
Author(s):  
Stanley M. Einstein-Matthews
Keyword(s):  
Dirichlet Problem ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Symplectic Structures ◽  
Absolutely Continuous ◽  
Nonlinear Dirichlet Problem ◽  
Homogeneous Complex ◽  
Regularity Results ◽  
Monge Ampere

Abstract.The graphs that arise from the gradients of solutions u of the homogeneous complex Monge-Ampère equation are characterized in terms of the natural symplectic structure on the cotangent bundle. This characterization is invariant under symplectic biholomorphisms. Using the symplectic structures we construct symmetries (to be called Lempert transformations) for real valued functions u which are absolutely continuous on lines. We then use these symmetries to generate interesting solutions to the homogeneous complex Monge-Ampère equation and to transform the Poincaré-Lelong equation and the ∂-equation. An example of Lempert transform is given and the main theorem is applied to prove regularity results for exterior nonlinear Dirichlet problem for the homogeneous complex Monge-Ampère equation.

scholarly journals An invitation to singular symplectic geometry

2019 ◽  
Vol 16 (supp01) ◽  
pp. 1940008 ◽  
Author(s):  
Roisin Braddell ◽  
Amadeu Delshams ◽  
Eva Miranda ◽  
Cédric Oms ◽  
Arnau Planas
Keyword(s):  
Celestial Mechanics ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Symplectic Manifolds ◽  
Symplectic Structures ◽  
Contact Manifolds ◽  
Symplectic Forms

In this paper, we analyze in detail a collection of motivating examples to consider [Formula: see text]-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every [Formula: see text]-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to [Formula: see text]-symplectic manifolds: [Formula: see text]-contact manifolds.

scholarly journals Deformations of Pre-symplectic Structures and the Koszul L∞-algebra

2018 ◽  
Vol 2020 (14) ◽  
pp. 4191-4237 ◽  
Author(s):  
Florian Schätz ◽  
Marco Zambon
Keyword(s):  
Lie Algebra ◽  
Deformation Theory ◽  
Symplectic Structure ◽  
Poisson Manifold ◽  
Symplectic Structures ◽  
Characteristic Foliation ◽  
Infinitesimal Deformations ◽  
Fixed Rank

Abstract We study the deformation theory of pre-symplectic structures, that is, closed 2-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_{\infty }$-algebra, which we call the Koszul $L_{\infty }$-algebra. This $L_{\infty }$-algebra is a cousin of the Koszul dg Lie algebra associated to a Poisson manifold. In addition, we show that a quotient of the Koszul $L_{\infty }$-algebra is isomorphic to the $L_{\infty }$-algebra that controls the deformations of the underlying characteristic foliation. Finally, we show that the infinitesimal deformations of pre-symplectic structures and of foliations are both obstructed.

scholarly journals Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Maxence Mayrand
Keyword(s):  
Symplectic Manifold ◽  
Cotangent Bundle ◽  
Lagrangian Submanifolds ◽  
Poisson Manifold ◽  
Lagrangian Submanifold ◽  
Zero Section ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Symplectic Groupoids ◽  
Hyperkähler Metrics

Abstract The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.

scholarly journals CANONICAL SYMPLECTIC STRUCTURES AND DEFORMATIONS OF ALGEBRAIC SURFACES

2009 ◽  
Vol 11 (03) ◽  
pp. 481-493 ◽  
Author(s):  
FABRIZIO CATANESE
Keyword(s):  
Minimal Surface ◽  
Symplectic Structure ◽  
General Type ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Symplectic Structures ◽  
Surface Of General Type ◽  
Smooth Deformation

We show that a minimal surface of general type has a canonical symplectic structure (unique up to symplectomorphism) which is invariant for smooth deformation. We show that the symplectomorphism type is also invariant for deformations which allow certain normal singularities, provided one remains in the same smoothing component. We use this technique to show that the Manetti surfaces yield examples of surfaces of general type which are not deformation equivalent but are canonically symplectomorphic.

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