Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 
 
 

 

Sub-Riemannian distance on the Lie group $ \operatorname{SO}_0(2,1)$


Authors: V. N. Berestovskiĭ and I. A. Zubareva
Translated by: the authors
Original publication: Algebra i Analiz, tom 28 (2016), nomer 4.
Journal: St. Petersburg Math. J. 28 (2017), 477-489
MSC (2010): Primary 57S20
DOI: https://doi.org/10.1090/spmj/1460
Published electronically: May 4, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A left-invariant sub-Riemannian metric $ d$ on the shortened Lorentz group $ \operatorname {SO}_0(2,1)$ is studied under the condition that $ d$ is right-invariant relative to the orthogonal Lie subgroup $ 1\otimes \operatorname {SO}(2)$. For $ (\operatorname {SO}_0(2,1),d)$, the distance between arbitrary two elements is found, along with the cut locus (as the union of the subgroup $ 1\otimes \operatorname {SO}(2)$ with the set antipodal in the open solid torus $ \operatorname {SO}_0(2,1)$ to the submanifold of symmetric matrices), and the conjugate set for the unit.


References [Enhancements On Off] (What's this?)

  • 1. V. N. Berestovskiĭ, (Locally) shortest arcs of a special sub-Riemannian metric on the Lie group 𝑆𝑂₀(2,1), Algebra i Analiz 27 (2015), no. 1, 3–22 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 27 (2016), no. 1, 1–14. MR 3443263
  • 2. V. N. Berestovskiĭ and I. A. Zubareva, Functions with a (non)timelike gradient in space-time, Mat. Tr. 17 (2014), no. 2, 41–60 (Russian, with Russian summary); English transl., Siberian Adv. Math. 25 (2015), no. 4, 243–254. MR 3330050
  • 3. -, Sub-Riemannian distance on the Lie group $ SL(2)$, Sibirsk Mat. Zh. 58 (2017), no. 1, 22-35; English transl., Siber. Math. J. 58 (2017), no. 1, 16-27.
  • 4. V. N. Berestovskiĭ and I. A. Zubareva, Shapes of spheres of special nonholonomic left-invariant intrinsic metrics on some Lie groups, Sibirsk. Mat. Zh. 42 (2001), no. 4, 731–748, i (Russian, with Russian summary); English transl., Siberian Math. J. 42 (2001), no. 4, 613–628. MR 1865469, https://doi.org/10.1023/A:1010439312070
  • 5. Shigeo Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds. II, Tôhoku Math. J. (2) 14 (1962), 146–155. MR 0145456, https://doi.org/10.2748/tmj/1178244169
  • 6. Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885
  • 7. V. N. Berestovskii and Luis Guijarro, A metric characterization of Riemannian submersions, Ann. Global Anal. Geom. 18 (2000), no. 6, 577–588. MR 1800594, https://doi.org/10.1023/A:1006683922481
  • 8. A. V. Pogorelov, Differential geometry, Translated from the first Russian ed. by L. F. Boron, P. Noordhoff N. V., Groningen, 1959. MR 0114163
    A. V. Pogorelov, \cyr Differentsial′naya geometriya., Fifth edition, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0268787
  • 9. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 0088511
  • 10. A. M. Vershik and V. Ya. Gershkovich, Nonholonomic dynamical systems. Geometry of distributions and variational problems, Current problems in mathematics. Fundamental directions, Vol. 16 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, pp. 5–85, 307 (Russian). MR 922070
  • 11. D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics, No. 55, Springer-Verlag, Berlin-New York, 1968 (German). MR 0229177
  • 12. V. N. Berestovskiĭ and I. A. Zubareva, Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group $ SL(2)$, Sibirsk. Mat. Zh. 57 (2016), no. 3, 527-542; English transl., Sib. Math. J. 57 (2016), no. 3, 411-424.
  • 13. Stefan Cohn-Vossen, Existenz kürzester Wege, Compositio Math. 3 (1936), 441–452 (German). MR 1556958
  • 14. Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 57S20

Retrieve articles in all journals with MSC (2010): 57S20


Additional Information

V. N. Berestovskiĭ
Affiliation: Sobolev Institute of Mathematics, Siberian Division, Russian Academy of Sciences, Acad. Koptyug avenue 4, 630090 Novosibirsk, Russia
Email: vberestov@inbox.ru

I. A. Zubareva
Affiliation: Omsk Branch, Sobolev Institute of Mathematics, Siberian Division, Russian Academy of Sciences, Pevtsova st. 13, 644043 Omsk, Russia
Email: i_gribanova@mail.ru

DOI: https://doi.org/10.1090/spmj/1460
Keywords: Conjugate set, cut locus, distance, geodesic, Lie algebra, Lie group, invariant sub-Riemannian metric, shortest arc
Received by editor(s): July 16, 2015
Published electronically: May 4, 2017
Additional Notes: The work is partially supported by the Russian Foundation for Basic Research (Grant 14-01-00068-a), a grant of the Government of the Russian Federation for the State Support of Scientific Research (Agreement 14.B25.31.0029), and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2263.2014.1)
Article copyright: © Copyright 2017 American Mathematical Society