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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
Published electronically: May 4, 2017
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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.

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Additional Information

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

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

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