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This article is cited in 18 scientific papers (total in 18 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.
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
Linking options:
https://www.mathnet.ru/eng/mt290 https://www.mathnet.ru/eng/mt/v18/i2/p3
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Abstract page: | 348 | Full-text PDF : | 123 | References: | 40 | First page: | 7 |
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