Integrability of Lie Equations and Pseudogroups

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

doi:10.1006/jmaa.2000.6805

Cited by 6 publications

(4 citation statements)

References 13 publications

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“…Since ω j , dx i , dy h,l are linearly independent in R * , by equaling their components in (17) to zero we have …”

Section: Remarks 63mentioning

confidence: 99%

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Symmetries of PDE systems and correspondences between jet spaces

Jiménez

2011

Journal of Mathematical Analysis and Applications

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Let M be a manifold. A PDE system R ⊆ J 1 m M can be prolonged to another one R * ⊆ T * M [10]). In analogy with the higher-order symmetries, symmetries of R * will be called higher-dimensional symmetries of R. For a broad class of PDE systems we prove that every (infinitesimal or finite) symmetry of R comes from another one of R * . We show that R * does not have internal (infinitesimal) symmetries (modulo trivial symmetries). This fact allows us, in the infinitesimal case, to compute the internal symmetries of R as external symmetries of R * . We also give an algorithmic method to obtain solutions of R invariant by a given internal symmetry.Let M be a smooth manifold with dim M = n and J k m M the space of k-jets of m-dimensional submanifolds of M. Let R ⊆ J k m M be a PDE system. Classically, a symmetry of R is a diffeomorphism of M whose prolongation to J k m M leaves R invariant or, infinitesimally, a vector field in M whose (m, k)-prolongation is tangent to R. These symmetries are called point symmetries because they come from transformations of the manifold M. However, more general symmetries can be considered: Lie himself dealt with symmetries not as point transformations, but as contact transformations; that is, diffeomorphisms of J k m M that preserve the contact system. In light of Bäcklund's theorem, we obtain new transformations only in the case m = n − 1. In any case, we continue regarding the system R from outside, considering geometrical transformations of the ambient J k m M that leave R invariant. These symmetries are called external symmetries. However, it seems more reasonable to consider transformations of R itself that preserve the contact system restricted to R. Such transformations are called internal symmetries of R. It is obvious that, by restriction, each external symmetry of R gives an internal one, and, in many cases, these are the only internal symmetries (see [6], [7, pp. 116-121]).Another important class of symmetries that generalize contact transformations is the class of higher-order symmetries. A PDE system can be considered not only as itself but together with its prolongations to all orders. The transformations that preserve the contact system in the infinite jet space and leave the infinite prolongation of the system invariant are called higher-order symmetries. Unlike the classical theory, there are no internal higher-order symmetries (see [7, p. 162]). The relationship between higher-order symmetries and internal symmetries is investigated in depth, in the infinitesimal case, in [6]. There the authors prove for a broad class of equations that every internal symmetry arises from a first-order generalized symmetry.This paper deals with a new class of symmetries and its connection with other kinds of symmetries. Let R ⊆ J 1 m M be a PDE system. Following [9,10] we can consider the 'prolongation' of R to J 1 r M (r > m) (prolongation with respect to

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“…Since ω j , dx i , dy h,l are linearly independent in R * , by equaling their components in (17) to zero we have …”

Section: Remarks 63mentioning

confidence: 99%

“…Here we shall only use jets of submanifolds; next we shall compile the main definitions and summarize, without proofs, some of the results that will be used later on. For further details we refer to [16], and to [1][2][3]9,10,[17][18][19] for the applications. Definition 1.1.…”

Section: Weil Near Points and Jets Of Submanifoldsmentioning

confidence: 99%

Symmetries of PDE systems and correspondences between jet spaces

Jiménez

2011

Journal of Mathematical Analysis and Applications

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“…Some of our techniques and results were used by R. Alonso in [1] to give a more natural definition of the Poincaré-Cartan form; in [11] we show how this point of view simplifies the study of Lie equations, and in further papers we will apply it to other topics such as differential invariants and formal integrability, for example.…”

Section: Introductionmentioning

confidence: 95%

Weil bundles and jet spaces

Muñoz

Rodríguez

Muriel

2000

Czechoslovak Mathematical Journal

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“…That theory relies essentially on a change of perspective: jets on a manifold M are ideals of the ring C ∞ (M) [15]. This point of view is close to the theory of Weil bundles [17] (refer to [4,5,[9][10][11][12][13][14] for its applications).…”

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confidence: 99%

A transformation of PDE systems related to the Drach theorem

Alonso-Blanco

2003

Journal of Mathematical Analysis and Applications

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We will define a new transformation of PDE systems as follows. Given a particular PDE system F, there is a new systemF whose solutions are the spaces of elements attached to the solutions of F. We will show that the systemF is a second order PDE system in a single unknown. As an application, we will derive as well a global version of the Drach theorem.In his research on PDE systems, Lie considered not only the ordinary points, but also the set of hyperplanes tangent to a given submanifold ('elements' in his terminology). This way, he gained a more ample point of view [6]. At this time, we will continue with this idea to obtain a new kind of transformation of PDE systems. This goes as follows: Given a PDE system F it is possible to construct a new systemF whose solutions are the spaces of elements attached to the solutions of F . Specifically,F will be determined by a suitable set of 1-jets of Lagrangian submanifolds (see below).On the other hand, we will show howF can be naturally understood as a second order PDE system. Let us first outline an explanation: It is well known, that a first order PDE system in a single unknown, which does not explicitly appear, is a submanifold of a symplectic manifold E [7,8]. Similarly, we propose a second order PDE system in a single unknown, which does not explicitly appear, as a submanifold of the space of 1-jets of Lagrangian submanifolds of E. This geometric structure seems to be very appropriate in order to study

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Integrability of Lie Equations and Pseudogroups

Cited by 6 publications

References 13 publications

Symmetries of PDE systems and correspondences between jet spaces

Symmetries of PDE systems and correspondences between jet spaces

Weil bundles and jet spaces

A transformation of PDE systems related to the Drach theorem

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