# On the geometry of Goursat structures

William Pasillas-Lépine; Witold Respondek

ESAIM: Control, Optimisation and Calculus of Variations (2001)

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

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topPasillas-Lépine, William, and Respondek, Witold. "On the geometry of Goursat structures." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 119-181. <http://eudml.org/doc/90587>.

@article{Pasillas2001,

abstract = {A Goursat structure on a manifold of dimension $n$ is a rank two distribution $\mathcal \{D\}$ such that dim $\mathcal \{D\}^\{(i)\}=i+2$, for $0 \le i \le n-2$, where $\mathcal \{D\}^\{(i)\}$ denote the elements of the derived flag of $\mathcal \{D\}$, defined by $\mathcal \{D\}^\{(0)\}=\mathcal \{D\}$ and $\mathcal \{D\}^\{(i+1)\}=\mathcal \{D\}^\{(i)\}+[\mathcal \{D\}^\{(i)\},\mathcal \{D\}^\{(i)\}]$. 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 $n \ge 6$, nor abnormal curves of all elements of the derived flag, if $n \ge 9$, 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},

language = {eng},

pages = {119-181},

publisher = {EDP-Sciences},

title = {On the geometry of Goursat structures},

url = {http://eudml.org/doc/90587},

volume = {6},

year = {2001},

}

TY - JOUR

AU - Pasillas-Lépine, William

AU - Respondek, Witold

TI - On the geometry of Goursat structures

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2001

PB - EDP-Sciences

VL - 6

SP - 119

EP - 181

AB - A Goursat structure on a manifold of dimension $n$ is a rank two distribution $\mathcal {D}$ such that dim $\mathcal {D}^{(i)}=i+2$, for $0 \le i \le n-2$, where $\mathcal {D}^{(i)}$ denote the elements of the derived flag of $\mathcal {D}$, defined by $\mathcal {D}^{(0)}=\mathcal {D}$ and $\mathcal {D}^{(i+1)}=\mathcal {D}^{(i)}+[\mathcal {D}^{(i)},\mathcal {D}^{(i)}]$. 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 $n \ge 6$, nor abnormal curves of all elements of the derived flag, if $n \ge 9$, 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

UR - http://eudml.org/doc/90587

ER -

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