Weil bundles and jet spaces

Muñoz, J.; Rodríguez, Josemar; Muriel, F.J.

doi:10.1023/a:1022408527395

Cited by 18 publications

(29 citation statements)

References 9 publications

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“…In this paper we characterize the natural transformations that are affine bundles. It is done easily by adopting a different point of view on the tangent space of M A than in [6]. Our result is as follows: there is a canonical affine structure for natural transformations M A → M B induced by a surjective morphism A → B whose kernel has null square.…”

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confidence: 97%

“…C. Ehresmann formalized contact elements of S. Lie, introducing the spaces of jets of sections; simultaneously A. Weil showed in [8] that the theory of S. Lie could be formalized easily by replacing the spaces of contact elements by the more formal spaces of "points proches", known as Weil bundles. The general theory of jet spaces [6] recovers the classical spaces of contact elements J l m M of S. Lie applying the ideas and methodology of A. Weil. In the theory of Weil bundles, morphisms A → B of Weil algebras induce natural transformations [5] between Weil bundles.…”

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confidence: 99%

“…The width of A is the dimension of the vector space m A /m 2 A . Thus, R l m is a Weil algebra of height l and width m. If A is of height l and width m there exists a surjective morphism R l m → A (see [4], [6]). Definition 1.…”

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confidence: 99%

“…Theorem 1 ( [6]). The space M A is endowed with a unique structure of a smooth manifold such that the real components of smooth functions on M are smooth functions on M A .…”

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confidence: 99%

“…A Weil algebra morphism φ : A → B induces by composition a smooth mapφ : M A → M B [5,6] which is called a natural transformation,…”

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confidence: 99%

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Affine structures on jet and Weil bundles

Blázquez-Sanz

2009

Colloq. Math.

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Abstract. Weil algebra morphisms induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cases, this structure of affine bundle passes to jet spaces. We give a characterization of this fact in algebraic terms. This algebraic condition also determines an affine structure on the groups of automorphisms of related Weil algebras.Introduction. The theory of Weil bundles and jet spaces is developed in order to understand the geometry of PDE systems. C. Ehresmann formalized contact elements of S. Lie, introducing the spaces of jets of sections; simultaneously A. Weil showed in [8] that the theory of S. Lie could be formalized easily by replacing the spaces of contact elements by the more formal spaces of "points proches", known as Weil bundles. The general theory of jet spaces [6] recovers the classical spaces of contact elements J l m M of S. Lie applying the ideas and methodology of A. Weil.In the theory of Weil bundles, morphisms A → B of Weil algebras induce natural transformations [5] between Weil bundles. There are well known cases in which these natural transformations are affine bundles that often appear in differential geometry [5]. In [4] I. Kolář showed that this is the behaviour of M A l → M A l−1 . In this paper we characterize the natural transformations that are affine bundles. It is done easily by adopting a different point of view on the tangent space of M A than in [6]. Our result is as follows: there is a canonical affine structure for natural transformations M A → M B induced by a surjective morphism A → B whose kernel has null square. This is true for M A l → M A k with 2k + 1 ≥ l > k ≥ 0.In some cases the natural transformations induce maps between jet spaces. This holds in the cases studied in [4]. We characterize this situation, and moreover, we determine when an affine structure on the Weil bundle morphism passes to the jet space morphism. In addition, we prove that in this case there also exists an affine structure in the morphisms be-

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confidence: 99%

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confidence: 99%

“…Theorem 1 ( [6]). The space M A is endowed with a unique structure of a smooth manifold such that the real components of smooth functions on M are smooth functions on M A .…”

mentioning

confidence: 99%

“…A Weil algebra morphism φ : A → B induces by composition a smooth mapφ : M A → M B [5,6] which is called a natural transformation,…”

mentioning

confidence: 99%

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Affine structures on jet and Weil bundles

Blázquez-Sanz

2009

Colloq. Math.

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Integrability of Lie Equations and Pseudogroups

Muñoz

Muriel

Rodríguez

2000

Journal of Mathematical Analysis and Applications

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In this paper the theory of jets based on Weil's near points is applied to Lie equations and pseudogroups. Linear systems of partial differential equations are interpreted, in a canonical way, as distributions on the fibre bundles of invertible jets invariant under translations. We prove the two fundamental theorems for Lie equations and generalize the results of Rodrigues; a geometric correspondence between linear and nonlinear Lie equations is given, and the symbols of a linear Lie equation and its prolongations are canonically identified with the symbols of their attached nonlinear equations. From this fact we deduce that a linear Lie equation verifies the conditions of Goldsmichmidt's criterion on formal integrability if and only if its attached nonlinear Lie equation satisfies them locally. Finally, we define the Cartan 1-form on the fibre bundle of invertible jets and give a global form to the equivalence between the Lie and Cartan definitions of continuous groups. ᮊ

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A Remark on Goldschmidt's Theorem on Formal Integrability

Muñoz

Muriel

Rodríguez

2001

Journal of Mathematical Analysis and Applications

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In this paper we apply Weil bundle techniques to the study of formal integrability for systems of nonlinear partial differential equations. We clarify the role of the curvature and show that both the statement and proof of Goldschmidt's criterion on formal integrability are of algebraic nature. This fact allows us to obtain a version of this theorem which holds for smooth, algebraic, or analytic manifolds. Finally, we give a definition for the characteristic co-vectors of a system of partial differential equations and how their relationship with the Cauchy᎐Kowalevski normal form. ᮊ

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Weil bundles and jet spaces

Cited by 18 publications

References 9 publications

Affine structures on jet and Weil bundles

Affine structures on jet and Weil bundles

Integrability of Lie Equations and Pseudogroups

A Remark on Goldschmidt's Theorem on Formal Integrability

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