Pfaffian systems with derived length one. The class of flag systems

Cañadas-Pinedo, María A.; Ruiz, Ceferino

doi:10.1090/s0002-9947-01-02638-1

Cited by 4 publications

(2 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The main ingredient in the definition of the singularity type will be the characteristic distributions C i defined by the following result, which is apparently due to Cartan [11], although he did not state it explicitly in his published works. Its proof can be found in [32] and [43] (see also [8], [31], and Appendix A), were slightly stronger versions are proved using the dual language of Pfaffian systems. i+1) and has constant corank one in D (i) .…”

Section: Characteristic Distributionsmentioning

confidence: 99%

“…That is, the singularity type of a contact or an Engel structure does not depend on the point at which the distribution is considered. This should be compared with the singularity type of a Goursat structure on a five-manifold, which can be either a 0 a 0 or a 0 a 1 at a given point p, depending on whether or not the Goursat structure can be converted into Goursat normal form in a small enough neighborhood of p. Indeed, for the Goursat structure spanned by the regular Kumpera-Ruiz normal form (7) the canonical submanifold S (0) 0 is empty, and thus the singularity type equals a 0 a 0 at each point of R 5 ; for the Goursat structure spanned by the singular Kumpera-Ruiz normal form (8) we have S (0) 0 = {x 5 = 0}, and thus the singularity type equals a 0 a 1 if x 5 = 0; and a 0 a 0 if x 5 = 0.…”

Section: Low Dimensional Examplesmentioning

confidence: 99%

See 1 more Smart Citation

On the Geometry of Goursat Structures

Pasillas-Lépine

Respondek

2001

ESAIM: COCV

View full text Add to dashboard Cite

Goursat structures appeared first in the work of von Weber and Cartan, who have shown that on an open and dense subset they can be converted into the so-called Goursat normal form. Later, Goursat structures have been studied by Kumpera and Ruiz. In the paper, we introduce a new local invariant for Goursat structures, called the singularity type, and prove that the growth vector and the abnormal curves of all elements of the derived flag are determined by this invariant. We provide a detailed analysis of all abnormal and rigid curves of Goursat structures. We show that neither abnormal curves, if n ≥ 6, nor abnormal curves of all elements of the derived flag, if n ≥ 9, determine the local equivalence class of a Goursat structure. The latter observation is deduced from a generalized version of Bäcklund's theorem. We also propose a new proof of a classical theorem of Kumpera and Ruiz. All results are illustrated by the n-trailer system, which, as we show, turns out to be a universal model for all local Goursat structures.Résumé. Une structure de Goursat sur une variété de dimension n est une distribution D de rang deux telle que dim. Les structures de Goursat sont d'abord apparues dans les travaux de von Weber et de Cartan, qui ont montré que sur un ouvert dense elles peuventêtre transformées en la forme normale de Goursat. Ensuite, les structures de Goursat ontété etudiées par Kumpera et Ruiz. Dans cet article, nous introduisons un nouvel invariant local pour les structures des Goursat, appelé le type de singulatité, et montrons que le vecteur de croissance et les courbes anormales de tous leséléments du drapeau dérivé sont déterminés par cet invariant. Nous donnons une analyse détaillée de toutes les courbes anormales et rigides. Nous montrons que ni les courbes anormales, lorsque n ≥ 6, ni les courbes anormales de tous leséléments du drapeau dérivé, lorsque n ≥ 9, determinent la classe d'équivalence locale d'une structure de Goursat. Cette dernière observation est déduite d'une version généralisée du théorème de Bäcklund. Nous proposons aussi une nouvelle preuve d'un théorème classique de Kumpera et Ruiz. Tous nos résultats sont illustrés par le camion avec n remorques, qui, comme nous le montrons, s'avèreêtre un modèle universel pour toutes les structures de Goursat.Mathematics Subject Classification. 58A30, 53C15, 93B29, 58A17.

show abstract

Section: Characteristic Distributionsmentioning

confidence: 99%

Section: Low Dimensional Examplesmentioning

confidence: 99%

On the Geometry of Goursat Structures

Pasillas-Lépine

Respondek

2001

ESAIM: COCV

View full text Add to dashboard Cite

show abstract

Pfaffian systems from twistor fibrations

Carrillo-Catalán

Yabe

2008

Differential Geometry and its Applications

View full text Add to dashboard Cite

show abstract

Automorphisms of flag systems

Kumpera

2019

Journal of Differential Equations

View full text Add to dashboard Cite

We first discuss the problems in the theory of ordinary differential equations that gave rise to the concept of a flag system and illustrate these with the Cartan criterion for Monge equations (1st order) as well as the Cartan statement concerning the local equivalence of Monge-Ampère type equations (2nd order). Next, we describe a prolongation functor operating on the infinitesimal symmetries (automorphisms) of the Darboux flag and extending these, isomorphically, to all the symmetries of any other flag. Hence, flag systems cannot be distinguished by their symmetry algebras and the local classification of these objects is approached by considering higher order isotropies of these algebras as well as the groupoids of k − th order formal equivalences since the differential equations defining the latter provide precious information for the application of flag systems to differential equations (e.g., Cartan's criterion for non-linear Monge equations). In examining the behaviour of the isotropy algebras, that can either diminish or remain the same, when passing from a derived system Sν to the previous system S ν−1 , we obtain a full set of numerical invariants for the elementary flag systems that moreover specify the local models. flag systems and models and infinitesimal automorphisms and isotropy and Darboux prolongations and k − th order equivalence groupoids.

show abstract

Pfaffian systems with derived length one. The class of flag systems

Cited by 4 publications

References 8 publications

On the Geometry of Goursat Structures

On the Geometry of Goursat Structures

Pfaffian systems from twistor fibrations

Automorphisms of flag systems

Contact Info

Product

Resources

About