Involution

Seiler, Werner M.

doi:10.1007/978-3-642-01287-7

Cited by 74 publications

(40 citation statements)

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“…, x p ). By Cartan-Kuranishi's prolongation theorem, [2,35], we conclude that, generically, there exists n i ≥ n i such that the system of differential equations g (n ) · z (n ) = z (n ) is formally integrable (and eventually involutive for some n ≥ n i ). In the following, n i is assumed to be independent of the submanifold jet z (∞) ∈ S ∞ i and n i is called the order of partial freeness.…”

Section: Singular Submanifold Jetsmentioning

confidence: 91%

“…In Section 6, following Seiler's book [35], Cartan's test based on the algebraic theory of involution is introduced. This offers an alternative way of verifying, for example, that the structure equations (4.17) are involutive.…”

Section: Involutionmentioning

confidence: 99%

“…We also fix a convenient degree compatible term ordering on the monomials of S. For example, one could choose the degree lexicographic ordering, [35]. Given an arbitrary vector field…”

Section: Regular Submanifold Jetsmentioning

confidence: 99%

“…It guarantees that the equivalence map constructed by specifying its jets (or Taylor series coefficients) converges, and the Cartan characters give the "dimensional freedom" of the equivalence map in Theorem 4.11. The following exposition follows Seiler's book, [35].…”

Section: Involutivitymentioning

confidence: 99%

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Solving Local Equivalence Problems with the Equivariant Moving Frame Method

Valiquette¹

2013

SIGMA

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Abstract. Given a Lie pseudo-group action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate modifications and assumptions, the equivariant moving frame constructions extend to submanifold jets where the pseudo-group does not act freely at any order. Once this is done, we review the solution to the local equivalence problem of submanifolds within the equivariant moving frame framework. This offers an alternative approach to Cartan's equivalence method based on the theory of G-structures.

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Section: Singular Submanifold Jetsmentioning

confidence: 91%

Section: Involutionmentioning

confidence: 99%

“…We also fix a convenient degree compatible term ordering on the monomials of S. For example, one could choose the degree lexicographic ordering, [35]. Given an arbitrary vector field…”

Section: Regular Submanifold Jetsmentioning

confidence: 99%

Section: Involutivitymentioning

confidence: 99%

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Solving Local Equivalence Problems with the Equivariant Moving Frame Method

Valiquette¹

2013

SIGMA

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“…In the context of formal theory of PDEs [17] it is convenient to use the so-called jet notation for derivatives. More concretely, given a vector field u = (u 1 , .…”

Section: 3mentioning

confidence: 99%

2d incompressible Euler equations: New explicit solutions

Martín¹,

Tuomela²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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There are not too many known explicit solutions to the 2-dimensional incompressible Euler equations in Lagrangian coordinates. Special mention must be made of the well-known ones due Gerstner and Kirchhoff, which were already discovered in the 19th century. These two classical solutions share a common characteristic, namely, the dependence of the coordinates from the initial location is determined by a harmonic map, as recognized by Abrashkin and Yakubovich, who more recently -in the 1980s-obtained new explicit solutions with a similar feature.We present a more general method for constructing new explicit solutions in Lagrangian coordinates which contain as special cases all previously known ones. This new approach shows that in fact "harmonic labelings" are special cases of a much larger family.In the classical solutions, the matrix Lie groups were essential in describing the time evolution. We see that also the geodesics in these groups are important.

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