Abstract
The authors found geodesics, shortest arcs, cut loci, and conjugate sets for some leftinvariant sub-Riemannian metric on the Lie group SL(2) that is right-invariant relative to the Lie subgroup SO(2) ⊂ SL(2) (in other words, for invariant sub-Riemannian metric on weakly symmetric space (SL(2) × SO(2))/SO(2)).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berestovskiĭ V. N., “Universal methods of the search of normal geodesics on Lie groups with left-invariant sub-Riemannian metric,” Sib. Math. J., 55, No. 5, 783–791 (2014).
Boscain U. and Rossi F., “Invariant Carnot–Carátheodory metrics on S 3, SO(3), SL(2) and lens spaces,” SIAM J. Control Optim., 47, No. 4, 1851–1878 (2008).
Berestovskiĭ V. N., “(Locally) shortest arcs of a special sub-Riemannian metric on the Lie group SO 0(2, 1),” St. Petersburg Math. J., 27, No. 1, 1–14 (2016).
Berestovskiĭ V. N. and Zubareva I. A., “Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group SO(3),” Sib. Math. J., 56, No. 4, 602–611 (2015).
Selberg A., “Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series,” J. Indian Math. Soc. (N.S.), 1, No. 4, 47–87 (1956).
Yakimova O. S., “Weakly symmetric Riemannian manifolds with a reductive isometry group,” Sb. Math., 195, No. 3–4, 599–614 (2004).
Berestovskiĭ V. N., “Homogeneous spaces with intrinsic metric. II,” Sib. Math. J., 30, No. 2, 180–191 (1989).
Berestovskii V. N. and Gorbatsevich V. V., “Homogeneous spaces with inner metric and with integrable invariant distributions,” Anal. Math. Phys., 4, No. 4, 263–331 (2014).
Gantmacher F. R., The Theory of Matrices. 2 vols, Taylor and Francis, London (1959).
Vershik A. M. and Gershkovich V. Ya., “Nonholonomic dynamical systems. Geometry of distributions and variational problems,” in: Current Problems in Mathematics. Fundamental Trends. Vol. 16 [in Russian], VINITI, Moscow, 1987, pp.5–85 (Itogi Nauki i Tekhniki).
Gromoll D., Klingenberg W., and Meyer W., Riemannsche Geometrie im Grossen, Springer-Verlag, Berlin; New York (1968) (Lecture Notes in Mathematics, No. 55).
Berestovskii V. N. and Guijarro L., “A metric characterization of Riemannian submersions,” Ann. Global Anal. Geom., 18, No. 6, 577–588 (2000).
Berestovskiĭ V. N. and Zubareva I. A., “Sub-Riemannian distance on Lie groups SU(2) and SO(3),” Mat. Tr., 18, No. 2, 3–21 (2015).
Cohn–Vossen S., “Existenz kürzester Wege,” Compositio Math., 3, 441–452 (1936).
Author information
Authors and Affiliations
Corresponding author
Additional information
Novosibirsk; Omsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 3, pp. 527–542, May–June, 2016; DOI: 10.17377/smzh.2016.57.304.
Rights and permissions
About this article
Cite this article
Berestovskiĭ, V.N., Zubareva, I.A. Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group SL(2). Sib Math J 57, 411–424 (2016). https://doi.org/10.1134/S0037446616030046
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446616030046