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 -
