scholarly journals Rigidity properties of Anosov optical hypersurfaces

2008 ◽  
Vol 28 (3) ◽  
pp. 707-737 ◽  
Author(s):  
NURLAN S. DAIRBEKOV ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Closed Manifold ◽  
Action Spectra ◽  
Characteristic Foliation ◽  
Zero Integral ◽  
Magnetic Flows ◽  
Related Theorem

AbstractWe consider an optical hypersurface Σ in the cotangent bundle τ:T*M→M of a closed manifold M endowed with a twisted symplectic structure. We show that if the characteristic foliation of Σ is Anosov, then a smooth 1-form θ on M is exact if and only if τ*θ has zero integral over every closed characteristic of Σ. This result is derived from a related theorem about magnetic flows which generalizes our previous work [N. S. Dairbekov and G. P. Paternain. Longitudinal KAM cocycles and action spectra of magnetic flows. Math. Res. Lett.12 (2005), 719–729]. Other rigidity issues are also discussed.

scholarly journals Hyperkähler metrics associated to compact Lie groups

1996 ◽  
Vol 120 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Andrew Dancer ◽  
Andrew Swann
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Left And Right ◽  
Hyperkähler Metrics

It is well known that the cotangent bundle of any manifold has a canonical symplectic structure. If we specialize to the case when the manifold is a compact Lie group G, then this structure is preserved by the actions of G on T*G induced by left and right translation on G. We refer to these as the left and right actions of G on T*G.

SYMPLECTIC REDUCTION FOR YANG–MILLS ON A CYLINDER

2004 ◽  
Vol 01 (04) ◽  
pp. 289-298 ◽  
Author(s):  
AMBAR N. SENGUPTA
Keyword(s):  
Gauge Group ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Hamiltonian Dynamics ◽  
Symplectic Reduction ◽  
Yang Mills

An account of the Lagrangian and Hamiltonian dynamics of the pure Yang–Mills system is presented. This framework is applied to the case of (1+1)-dimensional cylindrical spacetime. Hamiltonian dynamics on the space of connections over a circle is often identified with dynamics on the cotangent bundle of the gauge group by means of the holonomy. In support of this procedure we show that the symplectic structure for Hamiltonian dynamics for connections on a circle is identifiable with the natural symplectic structure on the cotangent bundle of the gauge group.

Symplectic structure and gauge invariance on the cotangent bundle

1994 ◽  
Vol 35 (1) ◽  
pp. 426-434 ◽  
Author(s):  
L. Hernández Encinas ◽  
J. Muñoz Masqué
Keyword(s):  
Gauge Invariance ◽  
Cotangent Bundle ◽  
Symplectic Structure

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 Quantum Mechanics as Hamilton–Killing Flows on a Statistical Manifold

2021 ◽  
Vol 3 (1) ◽  
pp. 12
Author(s):  
Ariel Caticha
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Information Geometry ◽  
Complex Numbers ◽  
Born Rule ◽  
Statistical Manifold ◽  
Finite Dimensional ◽  
Starting Point

The mathematical formalism of quantum mechanics is derived or “reconstructed” from more basic considerations of the probability theory and information geometry. The starting point is the recognition that probabilities are central to QM; the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold—a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini–Study metric, the linearity of the Schrödinger equation, the emergence of complex numbers, Hilbert spaces and the Born rule are derived rather than postulated.

scholarly journals On the Geometry of Relativistic Anyon

1997 ◽  
Vol 12 (24) ◽  
pp. 1783-1789 ◽  
Author(s):  
A. Nersessian
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Irreducible Representations ◽  
Hamiltonian Reduction ◽  
Poincare Group ◽  
Covariant Model ◽  
Lobachevsky Plane ◽  
Spin Generator

A twistor model is proposed for the free relativistic anyon. The Hamiltonian reduction of this model by the action of the spin generator leads to the minimal covariant model; whereas that by the action of spin and mass generators leads to the anyon model with free phase space which is a cotangent bundle of the Lobachevsky plane with twisted symplectic structure. Quantum mechanics of that model is described by irreducible representations of the (2+1)-dimensional Poincaré group.

scholarly journals Bi-Hamiltonian Structure of Spin Sutherland Models: The Holomorphic Case

2021 ◽  
Author(s):  
L. Fehér
Keyword(s):  
Lie Group ◽  
Poisson Structure ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Hamiltonian Structures ◽  
Poisson Reduction ◽  
Sutherland Model ◽  
Heisenberg Double ◽  
Real Forms

AbstractWe construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) , which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) Poisson–Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

scholarly journals Longitudinal KAM-cocycles and action spectra of magnetic flows

2005 ◽  
Vol 12 (5) ◽  
pp. 719-730 ◽  
Author(s):  
Nurlan S. Dairbekov ◽  
Gabriel P. Paternain
Keyword(s):  
Action Spectra ◽  
Magnetic Flows

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.

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