Geodesic
Sub-Riemannian distance in the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$
Matematičeskie trudy, Tome 18 (2015) no. 2, pp. 3-21.

Voir la notice de l'article provenant de la source Math-Net.Ru

We calculate distances between arbitrary elements of the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$ for special left-invariant sub-Riemannian metrics $\rho$ and $d$. In computing distances for the second metric, we substantially use the fact that the canonical two-sheeted covering epimorphism $\Omega$ of $\mathrm{SU(2)}$ onto $\mathrm{SO(3)}$ is a submetry and a local isometry in the metrics $\rho$ and $d$. Despite the fact that the proof uses previously known formulas for geodesics starting at the unity, F. Klein's formula for $\Omega$, trigonometric functions, and the conventional differential calculus of functions of one real variable, we focus attention on a careful application of these simple tools in order to avoid the mistakes made in previously published mathematical works in this area. 
@article{MT_2015_18_2_a0,
     author = {V. N. Berestovskii and I. A. Zubareva},
     title = {Sub-Riemannian distance in the {Lie} groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$},
     journal = {Matemati\v{c}eskie trudy},
     pages = {3--21},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2015_18_2_a0/}
}
TY  - JOUR
AU  - V. N. Berestovskii
AU  - I. A. Zubareva
TI  - Sub-Riemannian distance in the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$
JO  - Matematičeskie trudy
PY  - 2015
SP  - 3
EP  - 21
VL  - 18
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2015_18_2_a0/
LA  - ru
ID  - MT_2015_18_2_a0
ER  - 
%0 Journal Article
%A V. N. Berestovskii
%A I. A. Zubareva
%T Sub-Riemannian distance in the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$
%J Matematičeskie trudy
%D 2015
%P 3-21
%V 18
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2015_18_2_a0/
%G ru
%F MT_2015_18_2_a0
V. N. Berestovskii; I. A. Zubareva. Sub-Riemannian distance in the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$. Matematičeskie trudy, Tome 18 (2015) no. 2, pp. 3-21. http://geodesic.mathdoc.fr/item/MT_2015_18_2_a0/

[1] Agrachev A. A., Sachkov Yu. L., Geometricheskaya teoriya upravleniya, Fizmatlit, M., 2005

[2] Berestovskii V. N., “Universalnye metody poiska normalnykh geodezicheskikh na gruppakh Li s levoinvariantnoi subrimanovoi metrikoi”, Sib. matem. zhurn., 55:5 (2014), 959–970

[3] Berestovskii V. N., “(Lokalno) kratchaishie spetsialnoi subrimanovoi metriki na gruppe Li $\mathrm{SO}_0(2,1)$”, Algebra i analiz, 27:1 (2015), 3–22

[4] Berestovskii V. N., Zubareva I. A., “Formy sfer spetsialnykh negolonomnykh levoinvariantnykh vnutrennikh metrik na nekotorykh gruppakh Li”, Sib. matem. zhurn., 42:4 (2001), 731–748

[5] Berestovskii V. N., Zubareva I. A., “(Lokalno) kratchaishie spetsialnoi subrimanovoi metriki na gruppe Li $\mathrm{SO}(3)$”, Sib. matem. zhurn., 2015 (to appear)

[6] Klein F., Elementarnaya matematika s tochki zreniya vysshei, v. 1, Arifmetika, algebra, analiz, Nauka, M., 1974

[7] Postnikov M. M., Gruppy i algebry Li, Nauka, M., 1982

[8] Berestovskii V. N., Guijarro L., “A metric characterization of Riemannian submersions”, Ann. Global Anal. Geom., 18:6 (2000), 577–588 | DOI

[9] Boscain U. and Rossi F., “Invariant Carnot–Carathéodory metrics on $S^3$, $\mathrm{SO}(3)$, $\mathrm{SL}(2)$, and lens spaces”, SIAM J. Control Optim., 47:4 (2008), 1851–1878 | DOI | Zbl

[10] Boscain U. and Rossi F., “Shortest paths on 3-D simple Lie groups with nonholonomic constraint”, Proc. of the 47th IEEE Conf. on Decision and Control (Cancun, Mexico, Dec., 9–11, 2008), 1280–1284