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Méthodes géométriques et analytiques pour étudierl'application exponentielle, la sphère et le front d'onde en géométriesous-riemannienne dans le cas Martinet

Published online by Cambridge University Press:  15 August 2002

Bernard Bonnard  and
Monique Chyba
Bernard Bonnard
Affiliation:
Université de Bourgogne, Laboratoire de Topologie, UMR 5584 du CNRS, BP. 400, 21004 Dijon Cedex France; bbonnard@u-bourgogne.fr.
Monique Chyba
Affiliation:
Université de Paris VI, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France; chyba@sunny.unige.ch.
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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.

Keywords

Sub-riemannian geometryMartinet caseabnormal geodesicssphere and wave front of small radius. Mots clés : Géométrie sous-riemanniennele cas Martinetgéodésiques anormalessphère et front d'onde de petit rayon.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 245 - 334
Copyright
© EDP Sciences, SMAI, 1999

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