# Sub-Riemannian sphere in Martinet flat case

A. Agrachev; B. Bonnard; M. Chyba; I. Kupka

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

- Volume: 2, page 377-448
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topAgrachev, A., et al. "Sub-Riemannian sphere in Martinet flat case." ESAIM: Control, Optimisation and Calculus of Variations 2 (2010): 377-448. <http://eudml.org/doc/116567>.

@article{Agrachev2010,

abstract = {
This article deals with the local sub-Riemannian geometry on ℜ3,
(D,g) where D is the distribution ker ω, ω being the Martinet
one-form : dz - ½y2dxand g is a Riemannian metric on D. We prove that we can take
g as a sum of squares adx2 + cd2. Then we analyze the flat case where a = c = 1. We parametrize
the set of geodesics using elliptic integrals. This allows to compute
the exponential mapping, the wave front, the conjugate and cut loci
and the sub-Riemannian sphere. A direct consequence of our computations
is to show that the sphere and the distance function are not sub-analytic.
Some of these computations are generalized to a one parameter deformation
of the flat case.
},

author = {Agrachev, A., Bonnard, B., Chyba, M., Kupka, I.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Sub-Riemannian geometry / abnormal geodesics /
conjugate and cut loci / sub-Riemannian sphere.; sub-Riemannian geometry; Martinet distribution; cut locus; conjugate loci},

language = {eng},

month = {3},

pages = {377-448},

publisher = {EDP Sciences},

title = {Sub-Riemannian sphere in Martinet flat case},

url = {http://eudml.org/doc/116567},

volume = {2},

year = {2010},

}

TY - JOUR

AU - Agrachev, A.

AU - Bonnard, B.

AU - Chyba, M.

AU - Kupka, I.

TI - Sub-Riemannian sphere in Martinet flat case

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 2

SP - 377

EP - 448

AB -
This article deals with the local sub-Riemannian geometry on ℜ3,
(D,g) where D is the distribution ker ω, ω being the Martinet
one-form : dz - ½y2dxand g is a Riemannian metric on D. We prove that we can take
g as a sum of squares adx2 + cd2. Then we analyze the flat case where a = c = 1. We parametrize
the set of geodesics using elliptic integrals. This allows to compute
the exponential mapping, the wave front, the conjugate and cut loci
and the sub-Riemannian sphere. A direct consequence of our computations
is to show that the sphere and the distance function are not sub-analytic.
Some of these computations are generalized to a one parameter deformation
of the flat case.

LA - eng

KW - Sub-Riemannian geometry / abnormal geodesics /
conjugate and cut loci / sub-Riemannian sphere.; sub-Riemannian geometry; Martinet distribution; cut locus; conjugate loci

UR - http://eudml.org/doc/116567

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.