On the Geometry of Goursat Structures

William Pasillas-Lépine; Witold Respondek

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 6, page 119-181
  • ISSN: 1292-8119

Abstract

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A Goursat structure on a manifold of dimension n is a rank two distribution Ɗ such that dim Ɗ(i) = i + 2, for 0 ≤ i ≤ n-2, where Ɗ(i) denote the elements of the derived flag of Ɗ, defined by Ɗ(0) = Ɗ and Ɗ(i+1) = Ɗ(i) + [Ɗ(i),Ɗ(i)] . 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.

How to cite

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Pasillas-Lépine, William, and Respondek, Witold. "On the Geometry of Goursat Structures." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 119-181. <http://eudml.org/doc/116572>.

@article{Pasillas2010,
abstract = { A Goursat structure on a manifold of dimension n is a rank two distribution Ɗ such that dim Ɗ(i) = i + 2, for 0 ≤ i ≤ n-2, where Ɗ(i) denote the elements of the derived flag of Ɗ, defined by Ɗ(0) = Ɗ and Ɗ(i+1) = Ɗ(i) + [Ɗ(i),Ɗ(i)] . 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. },
author = {Pasillas-Lépine, William, Respondek, Witold},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Goursat structures; Kumpera-Ruiz normal forms; abnormal curves; nonholonomic control systems; trailer systems.; trailer systems},
language = {eng},
month = {3},
pages = {119-181},
publisher = {EDP Sciences},
title = {On the Geometry of Goursat Structures},
url = {http://eudml.org/doc/116572},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Pasillas-Lépine, William
AU - Respondek, Witold
TI - On the Geometry of Goursat Structures
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 119
EP - 181
AB - A Goursat structure on a manifold of dimension n is a rank two distribution Ɗ such that dim Ɗ(i) = i + 2, for 0 ≤ i ≤ n-2, where Ɗ(i) denote the elements of the derived flag of Ɗ, defined by Ɗ(0) = Ɗ and Ɗ(i+1) = Ɗ(i) + [Ɗ(i),Ɗ(i)] . 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.
LA - eng
KW - Goursat structures; Kumpera-Ruiz normal forms; abnormal curves; nonholonomic control systems; trailer systems.; trailer systems
UR - http://eudml.org/doc/116572
ER -

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