Sub-Riemannian distance on the Lie group 𝑆𝑂₀(2,1)

Berestovskiĭ, V.; Zubareva, I.

doi:10.1090/spmj/1460

Sub-Riemannian distance on the Lie group $\operatorname {SO}_0(2,1)$
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by V. N. Berestovskiĭ and I. A. Zubareva
Translated by: the authors

St. Petersburg Math. J. 28 (2017), 477-489

DOI: https://doi.org/10.1090/spmj/1460

Published electronically: May 4, 2017

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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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