相关论文: On Local Integrability Conditions Of Jet Groupoids
For each positive integer $k$, the bundle of $k$-jets of functions from a smooth manifold, $X$, to a Lie group, $G$, is denoted by $J^k(X,G)$ and it is canonically endowed with a Lie groupoid structure over $X$. In this work, we utilize a…
A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of…
This work is a spin-off of an on-going programme which aims at revisiting the original studies of Lie and Cartan on pseudogroups and geometric structures from a modern perspective. We encode geometric structures induced by transitive Lie…
We show that the fundamental groupoid~\(\Pi_1(X)\) of a locally path connected semilocally simply connected space~\(X\) can be equipped with a \emph{natural} topology so that it becomes a topological groupoid; we also justify the necessity…
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other…
In a stable theory, a stationary type $q \in S(A)$ internal to a family of partial types $\mathcal{P}$ over $A$ gives rise to a type-definable group, called its binding group. This group is isomorphic to the group…
The definition of Q-conditional symmetry for one PDE is correctly generalized to a special case of systems of PDEs and involutive families of operators. The notion of equivalence of Q-conditional symmetries under a group of local…
The aim of this paper is to explain, mostly through examples, what groupoids are and how they describe symmetry. We will begin with elementary examples, with discrete symmetry, and end with examples in the differentiable setting which…
An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with…
In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional…
Let G be a Lie groupoid with Lie algebroid g. It is known that, unlike in the case of Lie groups, not every subalgebroid of g can be integrated by a subgroupoid of G. In this paper we study conditions on the invariant foliation defined by a…
The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector…
We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in…
Principal bundles have at least three different definitions, depending on the category of geometric objects studied. In Differential Geometry, they are defined as locally trivial projection map of smooth manifolds with an atlas whose…
The jet bundle $J^kG$ of $k$-jets of curves in a Lie group $G$ has a natural Lie group structure. We present an explicit formula for the group multiplication in the right trivialization and for the group 2-cocycle describing the abelian Lie…
We define a quandle variety as an irreducible algebraic variety $Q$ endowed with an algebraically defined quandle operation $\rhd$. It can also be seen as an analogue of a generalized affine symmetric space or a regular $s$-manifold in…
In a natural way, the local diffeomorphisms of a manifold onto itself act on the reference frame bundles of any order and on the bundles associated with them. Due to the transitivity, the invariants by diffeomorphisms of an associated…
Jets of modules over a commutative ring are well known to make up the representative objects of linear differential operators on these modules. In noncommutative geometry, jets of modules provide the representative objects only of a certain…
For a closed connected manifold N, we establish the existence of geometric structures on various subgroups of the contactomorphism group of the standard contact jet space J^1N, as well as on the group of contactomorphisms of the standard…
Axioms of Lie algebroid are discussed in order to review some known aspects for non-experts. In particular, it is shown that a Lie QD-algebroid (i.e. a Lie algebra bracket on the Functions(M)-module F of sections of a vector bundle E over a…