From PDEs to Pfaffian fibrations

Cattafi, Francesco; Crainic, Marius; Salazar, Maria Amelia

doi:10.4171/lem/66-1/2-10

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“…The theory of Pfaffian fibrations and Pfaffian groupoids was developed in [55]; see also [17]. Pfaffian fibrations encode the abstract properties of geometric PDEs, i.e.…”

Section: Diffeology Induced By a Support D σ -Topology And C ∞Convergencementioning

confidence: 99%

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Topics on Lie pseudogroups

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First of all, let us give the precise definition of pseudogroup, which is at the core of our work.Definition. Let X be a smooth manifold and Diff loc (X) be the set of locally defined diffeomorphisms of X. A pseudogroup Γ over X is a subset Γ ⊂ Diff loc (X) such that • Γ contains the identity and is closed under inversion and composition of transformations;• Γ is closed under restriction and gluing of transformations.By "closed under gluing of transformations" we mean that if φ : U ⊂ X → V ⊂ X is a local transformation of X and there is a cover {U i } i∈I of U such that φ| Ui ∈ Γ, then φ ∈ Γ.The class of examples we are mostly interested in is that of sets of symmetries. The tools: Lie groupoids and multiplicative forms In [44], Lie and Engel focused mainly on the subclass of finite type Lie pseudogroups, see e.g. [69]. Their work can be seen as the origin of the modern notion of Lie group. Later on,Élie Cartan published two papers [13, 14] presenting a structure theory for Lie pseudogroups in full generality; his work contains great intuitions and underlies most of the later developments. See [69] for a modern formulation of Cartan's work.The framework that we have chosen to deal with pseudogroups has been developed fairly recently [15,55,69]. The fundamental objects are Lie groupoids. When Γ is a pseudogroup over X and k ∈ N, the jet space J k Γ is a groupoid over Example. Let be k ∈ N and let M be a 2k-dimensional manifold. Let us assume that M admits a Γ sp -atlas (see the example above) denoted by {(U i , ϕ i )} i∈I . That is, {U i } i∈I is a cover of M , the maps ϕ i : U i → R 2k are charts on M and the changes of coordinates φ ij , for i, j ∈ I, are locally defined symplectomorphisms of (R 2k , ω std ).The Γ sp -atlas {(U i , ϕ i )} i∈I endows M with the structure of symplectic manifold. In fact, for all i ∈ I, the pullback ϕ * i (ω std ) is a symplectic form on the open set U i ⊂ M . Since the change of coordinates φ ij is a symplectomorphism, ϕ * i (ω std ) and ϕ * j (ω std ) agree on U i ∩ U j and can be "glued" together. In other words, the collection {ϕ * i (ω std )} is given by the restrictions ω| Ui of a symplectic form ω on M .There are several other remarkable examples of geometric structures that can be described as Γ-structures -e.g. flat Riemannian metrics, complex structures, foliations, contact structures. The recipe is always the same: one considers the Chapter 1Chapter 1 is joint work in progress with Francesco Cattafi. Our starting point is the description of geometric structures on differentiable manifolds in terms of Lie pseudogroups. Throughout the chapter, we focus on the case when the Lie pseudogroup is transitive -see Definition 1.1.3. In the literature, structures associated to transitive pseudogroups have been encoded into principal group bundles equipped with some additional object -a distribution/one form interacting nicely with the group action, e.g. [58,64]. On the other hand, in [15], Pfaffian groupoids and principal Pfaffian bundles (Definitions 1.2.1 and 1.2.22) were shown t...

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“…The theory of Pfaffian fibrations and Pfaffian groupoids was developed in [55]; see also [17]. Pfaffian fibrations encode the abstract properties of geometric PDEs, i.e.…”

Section: Diffeology Induced By a Support D σ -Topology And C ∞Convergencementioning

confidence: 99%

“…Below, we briefly recall the main elements of the theory. We refer to [15,17,55,69] for more details.…”

Section: Diffeology Induced By a Support D σ -Topology And C ∞Convergencementioning

confidence: 99%

Topics on Lie pseudogroups

Accornero¹

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Cartan geometries and multiplicative forms

Cattafi

2021

Differential Geometry and its Applications

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From PDEs to Pfaffian fibrations

Abstract: We explain how to encode the essential data of a PDE on jet bundle into a more intrinsic object called Pfaffian fibration. We provide motivations to study this new notion and show how prolongations, integrability and linearisations of PDEs generalise to this setting.

Cited by 2 publications

References 13 publications

Topics on Lie pseudogroups

Topics on Lie pseudogroups

Cartan geometries and multiplicative forms

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