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Mat. Tr., 2015, Volume 18, Number 2, Pages 3–21 (Mi mt290)  

This article is cited in 9 scientific papers (total in 9 papers)

Sub-Riemannian distance in the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$

V. N. Berestovskiia, I. A. Zubarevab

a Sobolev Institute of Mathematics, Novosibirsk, Russia
b Sobolev Institute of Mathematics, Omsk Division, Omsk, Russia

Abstract: 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.

Key words: Lie algebra, geodesic, Lie group, invariant sub-Riemannian metric, shortest arc, distance.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 14.B25.31.0029
НШ-2263.2014.10
Russian Foundation for Basic Research 14-01-00068-a


DOI: https://doi.org/10.17377/mattrudy.2015.18.201

Full text: PDF file (264 kB)
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English version:
Siberian Advances in Mathematics, 2016, 26:2, 77–89

Bibliographic databases:

Document Type: Article
UDC: 519.46+514.763+512.81+519.9+517.911
Received: 18.11.2014

Citation: V. N. Berestovskii, I. A. Zubareva, “Sub-Riemannian distance in the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$”, Mat. Tr., 18:2 (2015), 3–21; Siberian Adv. Math., 26:2 (2016), 77–89

Citation in format AMSBIB
\Bibitem{BerZub15}
\by V.~N.~Berestovskii, I.~A.~Zubareva
\paper Sub-Riemannian distance in the Lie groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$
\jour Mat. Tr.
\yr 2015
\vol 18
\issue 2
\pages 3--21
\mathnet{http://mi.mathnet.ru/mt290}
\crossref{https://doi.org/10.17377/mattrudy.2015.18.201}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3588288}
\elib{http://elibrary.ru/item.asp?id=24639775}
\transl
\jour Siberian Adv. Math.
\yr 2016
\vol 26
\issue 2
\pages 77--89
\crossref{https://doi.org/10.3103/S1055134416020012}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. N. Berestovskiǐ, I. A. Zubareva, “Sub-Riemannian distance on the Lie group $SO_0(2,1)$”, St. Petersburg Math. J., 28:4 (2017), 477–489  mathnet  crossref  mathscinet  isi  elib
    2. V. N. Berestovskiǐ, I. A. Zubareva, “Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group $SL(2)$”, Siberian Math. J., 57:3 (2016), 411–424  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. V. N. Berestovskii, I. A. Zubareva, “Locally isometric coverings of the Lie group $\mathrm{SO}_0(2,1)$ with special sub-Riemannian metric”, Sb. Math., 207:9 (2016), 1215–1235  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. A. Mashtakov, R. Duits, “A cortical based model for contour completion on the retinal sphere”, Programmnye sistemy: teoriya i prilozheniya, 7:4 (2016), 231–247  mathnet  elib
    5. A. Mashtakov, R. Duits, Yu. Sachkov, E. J. Bekkers, I. Beschastnyi, “Tracking of lines in spherical images via sub-Riemannian geodesics in $\mathrm{SO}(3)$”, J. Math. Imaging Vis., 58:2 (2017), 239–264  crossref  mathscinet  zmath  isi  scopus
    6. A. P. Mashtakov, R. Duits, Yu. L. Sachkov, E. Bekkers, I. Yu. Beschastnyi, “Sub-Riemannian geodesics in $\mathrm{SO}(3)$ with application to vessel tracking in spherical images of retina”, Dokl. Math., 95:2 (2017), 168–171  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    7. V. N. Berestovskiǐ, I. A. Zubareva, “Sub-Riemannian distance on the Lie group $\operatorname{SL}(2)$”, Siberian Math. J., 58:1 (2017), 16–27  mathnet  crossref  crossref  isi  elib  elib
    8. V. N. Berestovskii, “Geodesics and curvatures of special sub-Riemannian metrics on Lie groups”, Siberian Math. J., 59:1 (2018), 31–42  mathnet  crossref  crossref  isi  elib
    9. M. V. Tryamkin, “Geodezicheskie subrimanovoi metriki na gruppe poluaffinnykh preobrazovanii evklidovoi ploskosti”, Sib. matem. zhurn., 60:1 (2019), 214–228  mathnet  crossref
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