LOCAL COHOMOLOGY AND THE VARIATIONAL BICOMPLEX

2008 ◽  
Vol 05 (04) ◽  
pp. 587-604 ◽  
Author(s):  
ROBERTO FERREIRO PÉREZ
Keyword(s):  
Principal Bundle ◽  
Local Cohomology ◽  
Differential Forms ◽  
Riemannian Metrics ◽  
Jet Bundle ◽  
Variational Bicomplex

The differential forms on the jet bundle J∞E of a bundle E → M over a compact n-manifold M of degree greater than n determine differential forms on the space Γ(E) of sections of E. The forms obtained in this way are called local forms on Γ(E), and its cohomology is called the local cohomology of Γ(E). More generally, if a group [Formula: see text] acts on E, we can define the local [Formula: see text]-invariant cohomology. The local cohomology is computed in terms of the cohomology of the jet bundle by means of the variational bicomplex theory. A similar result is obtained for the local [Formula: see text]-invariant cohomology. Using these results and the techniques for the computation of the cohomology of invariant variational bicomplexes in terms of relative Gelfand–Fuchs cohomology introduced in [4], we construct non trivial local cohomology classes in the important cases of Riemannian metrics with the action of diffeomorphisms, and connections on a principal bundle with the action of automorphisms.

Basic Forms

2020 ◽  
pp. 97-102
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Group ◽  
Principal Bundle ◽  
Differential Forms ◽  
Total Space ◽  
Lie Derivative ◽  
Connected Lie Group

This chapter describes basic forms. On a principal bundle π‎: P → M, the differential forms on P that are pullbacks of forms ω‎ on the base M are called basic forms. The chapter characterizes basic forms in terms of the Lie derivative and interior multiplication. It shows that basic forms on a principal bundle are invariant and horizontal. To understand basic forms better, the chapter considers a simple example. The plane ℝ2 may be viewed as the total space of a principal ℝ-bundle. A connected Lie group is generated by any neighborhood of the identity. This example shows the necessity of the connectedness hypothesis.

scholarly journals POISSON BRACKET IN CLASSICAL FIELD THEORY AS A DERIVED BRACKET

2008 ◽  
Vol 05 (07) ◽  
pp. 1051-1055
Author(s):  
S. A. POL'SHIN
Keyword(s):  
Field Theory ◽  
Poisson Bracket ◽  
Maxwell Equations ◽  
Differential Forms ◽  
Classical Field Theory ◽  
Classical Field ◽  
Hamiltonian Form ◽  
Jet Bundle ◽  
Finite Dimensional ◽  
Derived Bracket

We construct a Leibniz bracket on the space Ω• (Jk (π)) of all differential forms over the finite-dimensional jet bundle Jk (π). As an example, we write Maxwell equations with sources in the covariant finite-dimensional hamiltonian form.

Cartan's soldered spaces and conservation laws in physics

2015 ◽  
Vol 12 (09) ◽  
pp. 1550089 ◽  
Author(s):  
Joseph Kouneiher ◽  
Cécile Barbachoux
Keyword(s):  
Conservation Laws ◽  
Local Cohomology ◽  
Differential Forms ◽  
Einstein Equations ◽  
Vacuum Field ◽  
Order System ◽  
Mathematical Framework ◽  
Field Equations ◽  
Yang Mills ◽  
Integrability Conditions

In this paper, we will introduce a generalized soldering p-forms geometry, which can be the right framework to describe many new approaches and concepts in modern physics. Here we will treat some aspects of the theory of local cohomology in fields theory or more precisely the theory of soldering-form conservation laws in physics. We provide some illustrative applications, primarily taken from the Einstein equations of general theory of relativity and Yang–Mills theory. This theory can be considered to be a generalization of Noether's theory of conserved current to differential forms of any degree. An essential result of this, is that the conservation of the energy–momentum in general relativity, is linked to the fact that the vacuum field equations are equivalent to the integrability conditions of a first-order system of differential equations. We also apply the idea of the soldered space and the integrability conditions to the case of Yang–Mills theory. The mathematical framework, where these intuitive considerations would fit naturally, can be used to describe also the dynamics of changing manifolds.

scholarly journals Multisymplectic structures and the variational bicomplex

2009 ◽  
Vol 148 (1) ◽  
pp. 159-178 ◽  
Author(s):  
THOMAS J. BRIDGES ◽  
PETER E. HYDON ◽  
JEFFREY K. LAWSON
Keyword(s):  
Riemannian Manifold ◽  
Differential Forms ◽  
Relative Equilibria ◽  
Exterior Algebra ◽  
Variational Bicomplex ◽  
Main Observation ◽  
Hamiltonian Pdes ◽  
Definition Of ◽  
New View

AbstractMultisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.

scholarly journals Theory of Connections on Graded Principal Bundles

1998 ◽  
Vol 10 (01) ◽  
pp. 47-79 ◽  
Author(s):  
T. Stavracou
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Principal Bundle ◽  
Differential Forms ◽  
Lie Superalgebra ◽  
Principal Bundles ◽  
Graded Manifold ◽  
Definition Of ◽  
Manifold Theory ◽  
Free Actions

The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant–Berezin–Leites. We first review the basic elements of this theory establishing at the same time supplementary properties of graded Lie groups and their actions. Particular emphasis is given in introducing and studying free actions in the graded context. Next, we investigate the geometry of graded principal bundles; we prove that they have several properties analogous to those of ordinary principal bundles. In particular, we show that the sheaf of vertical derivations on a graded principal bundle coincides with the graded distribution induced by the action of the structure graded Lie group. This result leads to a natural definition of the graded connection in terms of graded distributions; its relation with Lie superalgebra-valued graded differential forms is also exhibited. Finally, we define the curvature for the graded connection and we prove that the curvature controls the involutivity of the horizontal graded distribution corresponding to the graded connection.

Glued linear connection on surface of the projective space

2020 ◽  
pp. 22-28
Author(s):  
K. V. Bashashina
Keyword(s):  
Projective Space ◽  
Fiber Bundle ◽  
Principal Bundle ◽  
Differential Forms ◽  
Linear Connection ◽  
Projective Connection ◽  
The Given

We consider a surface as a variety of centered planes in a multidi­mensional projective space. A fiber bundle of the linear coframes appears over this manifold. It is important to emphasize the fiber bundle is not the principal bundle. We called it a glued bundle of the linear coframes. A connection is set by the Laptev — Lumiste method in the fiber bundle. The ifferential equations of the connection object components have been found. This leads to a space of the glued linear connection. The expres­sions for a curvature object of the given connection are found in the pa­per. The theorem is proved that the curvature object is a tensor. A condi­tion is found under which the space of the glued linear connection turns into the space of Cartan projective connection. The study uses the Cartan — Laptev method, which is based on cal­culating external differential forms. Moreover, all considerations in the article have a local manner.

scholarly journals SUPERMETRICS ON SUPERMANIFOLDS

2008 ◽  
Vol 05 (02) ◽  
pp. 271-286 ◽  
Author(s):  
G. SARDANASHVILY
Keyword(s):  
Gauge Theory ◽  
Lie Group ◽  
Closed Subgroup ◽  
Principal Bundle ◽  
Riemannian Metrics ◽  
Global Section ◽  
General Linear ◽  
Base Manifold ◽  
General Linear Supergroup ◽  
Higgs Fields

By virtue of the well-known theorem, a structure Lie group K of a principal bundle P → X is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/K → X. In gauge theory, such sections are treated as Higgs fields, exemplified by pseudo-Riemannian metrics on a base manifold X. Under some conditions, this theorem is extended to principal superbundles in the category of G-supermanifolds. Given a G-supermanifold M and a graded frame superbundle over M with a structure general linear supergroup, a reduction of this structure supergroup to an orthogonal-symplectic supersubgroup is associated to a supermetric on a G-supermanifold M.

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

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