Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups

V. N. Berestovskii

Research output: Contribution to journalArticlepeer-review

Abstract

Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S3 = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S3 of all geodesics on (G, d) with respect to p.

Original languageEnglish
Pages (from-to)31-42
Number of pages12
JournalSiberian Mathematical Journal
Volume59
Issue number1
DOIs
Publication statusPublished - 1 Jan 2018

Keywords

  • geodesic orbit space
  • left invariant sub-Riemannian metric
  • Lie algebra
  • Lie group
  • normal geodesic
  • Riemannian symmetric space
  • DISTANCE
  • GROUP SO0(2,1)
  • MANIFOLDS
  • GROUP SL(2)

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