Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet

Bernard Bonnard; Monique Chyba

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

  • Volume: 4, page 245-334
  • ISSN: 1292-8119

Abstract

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Consider a sub-riemannian geometry(U,D,g) where U is a neighborhood of 0 in R3, D is a Martinet type distribution identified to ker ω, ω being the 1-form: ω = d z - y 2 2 d x , q=(x,y,z) and g is a metric on D which can be taken in the normal form: g = a ( q ) d x 2 + c ( q ) d y 2 , a=1+yF(q), c=1+G(q), G | x = y = 0 = 0 . In a previous article we analyze the flat case: a=c=1; we describe the conjugate and cut loci, the sphere and the wave front. The objectif of this article is to provide a geometric and computational framework to analyze the general case. This frame is obtained by analysing three one parameter deformations of the flat case which clarify the role of the three parameters α , β , γ in the gradated normal form of order 0 where: a = ( 1 + α y ) 2 , c = ( 1 + β x + γ y ) 2 . More generally this analysis provides an explanation of the role of abnormal minimizers in SR-geometry.

How to cite

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Bonnard, Bernard, and Chyba, Monique. "Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 245-334. <http://eudml.org/doc/197273>.

@article{Bonnard2010,
abstract = { Consider a sub-riemannian geometry(U,D,g) where U is a neighborhood of 0 in R3, D is a Martinet type distribution identified to ker ω, ω being the 1-form: $\omega=dz-\frac\{y^2\}\{2\}dx$, q=(x,y,z) and g is a metric on D which can be taken in the normal form: $g=a(q)dx^2+c(q)dy^2$, a=1+yF(q), c=1+G(q), $G_\{|_\{x=y=0\}\}=0$. In a previous article we analyze the flat case: a=c=1; we describe the conjugate and cut loci, the sphere and the wave front. The objectif of this article is to provide a geometric and computational framework to analyze the general case. This frame is obtained by analysing three one parameter deformations of the flat case which clarify the role of the three parameters $\alpha,\beta,\gamma$ in the gradated normal form of order 0 where: $a=(1+\alpha y)^2$, $c=(1+\beta x+\gamma y)^2$. More generally this analysis provides an explanation of the role of abnormal minimizers in SR-geometry. },
author = {Bonnard, Bernard, Chyba, Monique},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Sub-riemannian geometry; Martinet case; abnormal geodesics; sphere and wave front of small radius. Mots clés : Géométrie sous-riemannienne; le cas Martinet; géodésiques anormales; sphère et front d'onde de petit rayon.; sub-Riemannian geometry; abnormal geodesics; sphere and wavefront of small radius; one parameter deformations},
language = {fre},
month = {3},
pages = {245-334},
publisher = {EDP Sciences},
title = {Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet},
url = {http://eudml.org/doc/197273},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Bonnard, Bernard
AU - Chyba, Monique
TI - Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 245
EP - 334
AB - Consider a sub-riemannian geometry(U,D,g) where U is a neighborhood of 0 in R3, D is a Martinet type distribution identified to ker ω, ω being the 1-form: $\omega=dz-\frac{y^2}{2}dx$, q=(x,y,z) and g is a metric on D which can be taken in the normal form: $g=a(q)dx^2+c(q)dy^2$, a=1+yF(q), c=1+G(q), $G_{|_{x=y=0}}=0$. In a previous article we analyze the flat case: a=c=1; we describe the conjugate and cut loci, the sphere and the wave front. The objectif of this article is to provide a geometric and computational framework to analyze the general case. This frame is obtained by analysing three one parameter deformations of the flat case which clarify the role of the three parameters $\alpha,\beta,\gamma$ in the gradated normal form of order 0 where: $a=(1+\alpha y)^2$, $c=(1+\beta x+\gamma y)^2$. More generally this analysis provides an explanation of the role of abnormal minimizers in SR-geometry.
LA - fre
KW - Sub-riemannian geometry; Martinet case; abnormal geodesics; sphere and wave front of small radius. Mots clés : Géométrie sous-riemannienne; le cas Martinet; géodésiques anormales; sphère et front d'onde de petit rayon.; sub-Riemannian geometry; abnormal geodesics; sphere and wavefront of small radius; one parameter deformations
UR - http://eudml.org/doc/197273
ER -

References

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