On a Certain Sub-Riemannian Geodesic Flow on the Heisenberg Group

S. V. Agapov, M. R. Borchashvili

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Under study is an integrable geodesic flow of a left-invariant sub-Riemannian metric for a right-invariant distribution on the Heisenberg group. We obtain the classification of the trajectories of this flow. There are a few examples of trajectories in the paper which correspond to various values of the first integrals. These trajectories are obtained by numerical integration of the Hamiltonian equations. It is shown that for some values of the first integrals we can obtain explicit formulae for geodesics by inverting the corresponding Legendre elliptic integrals.

Original languageEnglish
Pages (from-to)943-951
Number of pages9
JournalSiberian Mathematical Journal
Issue number6
Publication statusPublished - 1 Nov 2017


  • geodesic flow
  • left-invariant metric
  • sub-Riemannian geometry


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