Sub-Riemannian sphere in Martinet flat case

Agrachev, 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 -