arXiv research

In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive Jacobi-like identities of tangent-vector-valued forms from the general Jacobi identity.

Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.

Survey reviews Hamilton-Jacobi theory in various geometric settings, focusing on Jacobi and Leibniz identities.

As the fourth paper of our series of papers concerned with axiomatic differential geometry, this paper is devoted to the general Jacobi identity supporting the Jacobi identity of vector fields. The general Jacobi identity can be regarded as one of the few fundamental results belonging properly to smootheology.

On the basis of the generalizations of the Jacobi identity found by the author some identities satisfied by the curvature and torsion of a covariant differentiation are derived. A kind of the generalized covariant differentiation is proposed and a method of finding some of satisfied by them identities is given in the c…

The most general Jacobi brackets in $\mathbb{R}^3$ are constructed after solving the equations imposed by the Jacobi identity. Two classes of Jacobi brackets were identified, according to the rank of the Jacobi structures. The associated Hamiltonian vector fields are also constructed.

This short note contains an explicit proof of the Jacobi identity for variational Schouten bracket in $Z_2$-graded commutative setup. For the reasoning to be rigorous, it refers to the product bundle geometry of iterated variations (see arXiv:1312.1262 [math-ph]); no ad hoc regularizations occur anywhere in this theory…

Researchers solved a number-theoretic hypothesis to determine the spin parity of k-differentials.

The paper proves a vanishing identity for twist knots using character varieties.

Characterizes Filippov n-algebroids using connections and formulas.

We explore a Pluecker-type relation which occurs naturally in the study of maximally supersymmetric solutions of certain supergravity theories. This relation generalises at the same time the classical Pluecker relation and the Jacobi identity for a metric Lie algebra and coincides with the Jacobi identity of a metric n…

Frolicher and Nijenhuis recognized well in the middle of the previous century that the Lie bracket and its Jacobi identity could and should exist beyond Lie algebras. Nevertheless the conceptual meaning of their discovery has been obscured by the messy techniques they exploited. The principal objective in this paper is…

New action functionals for sigma models on Jacobi bundles.

We construct a symplectic realization and a bi-hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 P…

The paper extends ternary algebra concepts using cube roots of unity.

We describe a method for solving the Maurer-Cartan structure equation associated with a Lie algebra that isolates the role of the Jacobi identity as an obstruction to integration. We show that the method naturally adapts to two other interesting situations: local symplectic realizations of Poisson structures, in which …

We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n,n) for any n which is at least 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither sa…

Notes on Whitney towers in 4-manifolds, focusing on local surface manipulations and invariants.

In this text we introduce the torsion of spinor connections. In terms of the torsion we give conditions on a spinor connection to produce Killing vector fields. We relate the Bianchi type identities for the torsion of spinor connections with Jacobi identities for vector fields on supermanifolds. Furthermore, we discuss…

The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX-relation). In addition, this relation was recently found to arise naturally in a theory of em…

The geometric approach [1312.1262] to iterated variations of local functionals -- e.g., of the (master-)action functional -- resulted in an extension of the deformation quantisation technique to the set-up of Poisson models of field theory [IHES/M/15/13]. It also allowed of a rigorous proof ([1312.1262],[1210.0726]) fo…

By resorting to Noether's Second Theorem, we relate the generalized Bianchi identities for Lagrangian field theories on gauge-natural bundles with the kernel of the associated gauge-natural Jacobi morphism. A suitable definition of the curvature of gauge-natural variational principles can be consequently formulated in …

New algebraic framework for Jacobi manifolds connects geometric mechanics and dimensional analysis.

Investigates harmonic self-maps' stability on cohomogeneity one manifolds.

Introduces a new bracket for multicontact geometry and applies it to field theories.

Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avant-garde branch of differential geometry, in which nilpotent infinitesimals are available in abundance. The principal objective in this paper is to show within our favorite framewor…

We generalize Hansen--Strobl's definition of $H$-twisted Courant algebroid such that the twist $H$ of the Jacobi identity is a 4-form in the kernel of the anchor map and is closed under a naturally occurring exterior covariant derivative. We give examples and define a cohomology.

There is a remarkable and canonical problem in 3D geometry and topology: To understand existing models of 3D fluid motion or to create new ones that may be useful. We discuss from an algebraic viewpoint the PDE called Euler's equation for incompressible frictionless fluid motion. In part I we define a "finite dimension…

The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of t…

We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in classical Lie theory is replaced by an identity similar to the Yang-Baxter equation. Every classical Lie algebroid has the structure of an …

New generalized Poisson structures are introduced by using suitable skew-symmetric contravariant tensors of even order. The corresponding `Jacobi identities' are provided by conditions on these tensors, which may be understood as cocycle conditions. As an example, we provide the linear generalized Poisson structures wh…

The closed string model in the background gravity field is considered as a bi-Hamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. The dual nonlocal Poisson brackets(PB), depending of the background fields and of their derivatives, are obtained. The in…

We give a complete description of differential operators generating a given bracket. In particular we consider the case of Jacobi-type identities for odd operators and brackets. This is related with homotopy algebras using the derived bracket construction. (Based on a talk at XXII Workshop on Geometric Methods in Physi…

We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element $Δ$. Given this, we introduce an infinite sequence of higher brackets on the image of the project…

We derive both {\em local} and {\em global} generalized {\em Bianchi identities} for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the {\em a priori} introduction of a connection. The proof is based on a {\em global} decom…

We define a new kind of algebroid which fulfills a Leibniz rule, a Jacobi identity twisted by a 3-form $H$ with values in the kernel of the anchor map, and the twist is closed under a naturally occurring exterior covariant derivative. We give examples and define three kinds of cohomology two via realization as Q-struct…

We show that one can skip the skew-symmetry assumption in the definition of Nambu-Poisson brackets. In other words, a n-ary bracket on the algebra of smooth functions which satisfies the Leibniz rule and a n-ary version of the Jacobi identity must be skew-symmetric. A similar result holds for a non-antisymmetric versio…

This paper constructs Brownian motion on complex flag manifolds and finds joint distribution of stochastic areas.

Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we …

Extends magnetic flow theory results to higher dimensions.

Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differential…

The study finds infinitely many p-harmonic maps between spheres for specific p and m.

As the third of our series of papers on differential geometry of microlinear Frolicher spaces, this paper is devoted to the Frolicher-Nijenhuis calculus of their named bracket. The main result is that the Frolicher-Nijenhuis bracket satisfies the graded Jacobi identity. It is also shown that the Lie derivation preserve…

Constructing $3$-Lie algebroids via connections

We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it …

We consider a vector space V over K=R or C, equipped with a skew symmetric bracket [.,.]: V x V --> V and a 2-form omega:V x V --> K. A simple change of the Jacobi identity to the form [A,[B,C]]+[C,[A,B]]+[B,[C,A]]=omega(B,C)A+omega(A,B)C+omega(C,A)B opens new possibilities, which shed new light on the Bianchi classifi…

We study a class of Poisson tensors on a fibered manifold which are compatible with the fiber bundle structure by the so-called almost coupling condition. In the case of a $5$-dimensional orientable fibered manifolds with $2$-dimensional bases, we describe a global behavior of almost coupling Poisson tensors and their …

We use the exterior product of double forms to reformulate celebrated classical results of linear algebra about matrices and bilinear forms namely the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi's formula for the determinant. This new formalism is then used to naturally g…