On homogeneous geodesics and weakly symmetric spaces

Valeriĭ Nikolaevich Berestovskiĭ, Yuriĭ Gennadievich Nikonorov

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


In this paper, we establish a sufficient condition for a geodesic in a Riemannian manifold to be homogeneous, i.e. an orbit of an 1-parameter isometry group. As an application of this result, we provide a new proof of the fact that every weakly symmetric space is a geodesic orbit manifold, i.e. all its geodesics are homogeneous. We also study general properties of homogeneous geodesics, in particular, the structure of the closure of a given homogeneous geodesic. We present several examples where this closure is a torus of dimension ≥ 2 which is (respectively, is not) totally geodesic in the ambient manifold. Finally, we discuss homogeneous geodesics in Lie groups supplied with left-invariant Riemannian metrics.

Original languageEnglish
Pages (from-to)575-589
Number of pages15
JournalAnnals of Global Analysis and Geometry
Issue number3
Publication statusPublished - 1 Apr 2019


  • Geodesic orbit Riemannian space
  • Homogeneous Riemannian manifold
  • Homogeneous space
  • Quadratic mapping
  • Totally geodesic torus
  • Weakly symmetric space




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