arXiv research

We define covariant Lie derivatives acting on vector-valued forms on Lie algebroids and study their properties. This allows us to obtain a concise formula for the Frölicher-Nijenhuis bracket on Lie algebroids.

Abstract: Review and definitions of generalised spin structures, their connections, and symmetry algebra.

This paper studies covariant derivatives for Lie groupoids with representation-valued forms.

We obtain a complete time expansion of the pull-back operator generated by a real analytic flow of real analytic automorphisms acting on analytic tensor sections of a manifold. Our expansion is given in terms of multiple Lie derivatives. Motivated by this expansion, we provide a rather simple and explicit estimate for …

Invariant covariant derivatives on homogeneous spaces are characterized.

Supplementary comments about generalized Lie algebroids are presented and a new point of view over the construction of the Lie algebroid generalized tangent bundle of a (dual) vector bundle is introduced. Using the general theory of exterior differential calculus for generalized Lie algebroids, a covariant derivative f…

The paper shows objective derivatives are covariant derivatives on Riemannian metrics.

In this paper we investigate the relations between semispray, nonlinear connection, dynamical covariant derivative and Jacobi endomorphism on Lie algebroids. Using these geometric structures, we study the symmetries of second order differential equations in the general framework of Lie algebroids.

Reduces field theories on principal bundles by a subgroup, deriving reduced equations.

Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.

We introduce a stochastic model for noisy vector fields on manifolds.

Horizontal endomorphisms, almost complex structures, vertical, horizontal and complete lifts on prolongation of a Lie algebroid are considered. Then using exact sequences, semisprays are constructed. Moreover, important geometrical objects such as classical distinguished connections, torsions and partial curvatures are…

We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. In the new version references ad…

Special orthogonal representations from octonions have geometric properties linked to binary cubics.

Differential conservation laws in Lagrangian field theory are usually related to symmetries of a Lagrangian density and are obtained if the Lie derivative of a Lagrangian density by a certain class of vector fields on a fiber bundle vanishes. However, only two field models meet this property in fact. In gauge theory of…

This thesis explores Weyl geometry and quantum anomalies in holography and gauge theories.

Study projective representations of infinite-dimensional Hilbert-Lie groups.

Class lecture notes at a beginning graduate level on the mathematical background needed to understand classical gauge theory. Covers group actions, fiber bundles, principal bundles, connections, gauge transformations, parallel transport, curvature, covariant derivatives, pseudo-riemannian manifolds, lagrangians, cliffo…

Motivated by positive energy representations, we classify those continuous central extensions of the compactly supported gauge Lie algebra that are covariant under a 1-parameter group of transformations of the base manifold.

In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant algebroids to generalize the notion of the Drinfeld double to Lie bialgebroids. As a first step towards understanding the complic…

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 develop a comprehensive geometric framework for defining spaces $\mathcal{G}(M,E)$ of nonlinear generalized sections of vector bundles $E \to M$ containing spaces of distributional sections $\mathcal{D}'(M, E)$. Our theory incorporates classical differential geometric operations (like tensor products, covariant deri…

A new mathematical approach to general covariance using stacks and Lie algebras.

The covariant phase space of a Lagrangian field theory is the solution space of the associated Euler-Lagrange equations. It is, in principle, a nice environment for covariant quantization of a Lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) "presymplectic structure" w …

We characterize the exact lumpability of smooth vector fields on smooth manifolds. We derive necessary and sufficient conditions for lumpability and express them from four different perspectives, thus simplifying and generalizing various results from the literature that exist for Euclidean spaces. We introduce a partia…

Covariance is shown as a commutator in random variable calculus.

The paper connects higher-dimensional mechanics to Lie n-algebroids.

Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesi…

Reduces necessary conditions for collision avoidance on curved spaces.

Rollings of reductive homogeneous spaces are studied using intrinsic curves.

As a difference with the positive-definite Riemannian case, in the Lorentzian case there exists proper second-order symmetric spacetimes, i.e., those with vanishing second covariant derivative of the Riemannian tensor ($R_{λμνρ;α;β}=0$) which are not locally symmetric ($R_{λμνρ;α}\neq 0$). In fact, they lie in the clas…

This paper is a continuation of our previous work, where eleven basic classes of almost paracontact metric manifolds with respect to the covariant derivative of the structure tensor field were obtained. First we decompose one of the eleven classes into two classes and the basic classes of the considered manifolds becom…

Study connections on Lie and Courant algebroids, defining basic curvature and Atiyah cocycle.

An action of a Lie algebra $\frak g$ on a manifold $M$ is just a Lie algebra homomorphism $ζ:\frak g\to \frak X(M)$. We define orbits for such an action. In general the space of orbits $M/\frak g$ is not a manifold and even has a bad topology. Nevertheless for a $\frak g$-manifold with equidimensional orbits we treat s…

Unified geometric framework for Brownian motion on various manifolds.

We construct algebraic and algebro-geometric models for the spaces of unparametrized paths. This is done by considering a path as a holonomy functional on indeterminate connections. For a manifold X, we construct a Lie algebroid P which serves as the tangent space to X (punctual paths) inside the space of all unparamet…

After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie derivative in pull-back formalism, we present ten equivalent conditions, each of which characterizes Berwald manifolds among Finsler manifolds. These range from Berwald's classical definition to the existence of a torsion-fre…

Given a multisymplectic manifold $(M,ω)$ and a Lie algebra $\frak{g}$ acting on it by infinitesimal symmetries, Fregier-Rogers-Zambon define a homotopy (co-)moment as an $L_{\infty}$-algebra-homomorphism from $\frak{g}$ to the observable algebra $L(M,ω)$ associated to $(M,ω)$, in analogy with and generalizing the notio…

Our paper is devoted to the study of the holonomy groups of Finsler surfaces using the methods of infinite dimensional Lie theory. The notion of infinitesimal holonomy algebra will be introduced, by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal c…

Develops a reduction theory for covariant field theories with gauge symmetries.

This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a global theory of algebras of generalised functions on manifolds based on the concept …

A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.

Classifies scalar second-order PDEs with low-dimensional symmetry groups.

A manifold with a ``Lie structure at infinity'' is a non-compact manifold $M_0$ whose geometry is described by a compactification to a manifold with corners M and a Lie algebra of vector fields on M, subject to constraints only on $M \smallsetminus M_0$. The Lie structure at infinity on $M_0$ determines a metric on $M_…

Constructs covariant derivatives for Ehresmann connections.

Homotopy equivalence between formalities with different covariant derivatives.

In the $3$-gauge theory, a $3$-connection is given by a $1$-form $A$ valued in the Lie algebra $\mathfrak g$, a $2$-form $B$ valued in the Lie algebra $\mathfrak h$ and a $3$-form $C$ valued in the Lie algebra $\mathfrak l$, where $(\mathfrak g,\mathfrak h, \mathfrak l)$ constitutes a differential $2$-crossed modu…

Derives a formula for the k-th covariant derivative of tensor fields.