(Locally) shortest arcs of a special sub-Riemannian metric on the Lie group \(\mathrm{SO}_0(2,1)\) (Q2786966)
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scientific article; zbMATH DE number 6545124
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | (Locally) shortest arcs of a special sub-Riemannian metric on the Lie group \(\mathrm{SO}_0(2,1)\) |
scientific article; zbMATH DE number 6545124 |
Statements
24 February 2016
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geodesic
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Lie algebra
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Lie group \(\mathrm{SO}(2,1)\)
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left-invariant sub-Riemannian metric
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shortest arc
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cut locus
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conjugate set
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(Locally) shortest arcs of a special sub-Riemannian metric on the Lie group \(\mathrm{SO}_0(2,1)\) (English)
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The author finds the geodesics and shortest arcs for the left-invariant and \(\mathrm{SO}(2)\)-right-invariant sub-Riemannian metric on the Lie-Lorentz group \(\mathrm{SO}_0(2,1)\) (where \(\mathrm{SO}(2) \subset\mathrm{SO}_0(2,1)\)).NEWLINENEWLINENEWLINEIt is proved that any geodesic \(\gamma\) in \(\mathrm{SO}_0(2, 1)\) parameterized by the arc length and with initial condition \(\gamma (0) = e\) is a product of two special 1-parameter subgroups. Also an explicit matrix presentations of such geodesics are given.NEWLINENEWLINENEWLINEThe author also calculates the cut locus and the conjugate set for above mentioned metric on \(\mathrm{SO}_0(2,1)\).
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