Advertisement

Annals of Global Analysis and Geometry

, Volume 55, Issue 3, pp 575–589 | Cite as

On homogeneous geodesics and weakly symmetric spaces

  • Valeriĭ Nikolaevich Berestovskiĭ
  • Yuriĭ Gennadievich NikonorovEmail author
Article
  • 50 Downloads

Abstract

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 \(\ge 2\) 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.

Keywords

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

Mathematics Subject Classification

53C20 53C25 53C35 

Notes

Acknowledgements

The first author was supported by the Ministry of Education and Science of the Russian Federation (Grant 1.3087.2017/4.6). The authors are indebted to Prof. Andreas Arvanitoyeorgos for helpful discussions concerning this paper. The authors are grateful to the referee for helpful comments and suggestions that improved the presentation of this paper.

References

  1. 1.
    Akhiezer, D.N., Vinberg, È.B.: Weakly symmetric spaces and spherical varieties. Transform. Groups 4, 3–24 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arutyunov, A.V., Zhukovskiy, S.E.: Properties of surjective real quadratic maps. Sb. Math. 207(9), 1187–1214 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arvanitoyeorgos, A.: Homogeneous manifolds whose geodesics are orbits. Recent results and some open problems. Irish Math. Soc. Bull. 79, 5–29 (2017)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bangert, V.: Non-closed isometry-invariant geodesics. Arch. Math. 106(6), 573–580 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berestovskii, V.N., Nikonorov, Y.G.: Killing vector fields of constant length on locally symmetric Riemannian manifolds. Transform. Groups 13(1), 25–45 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berestovskii, V.N., Nikonorov, Y.G.: Killing vector fields of constant length on Riemannian manifolds. Sib. Math. J. 49(3), 395–407 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Berestovskii, V.N., Nikonorov, Y.G.: Regular and quasiregular isometric flows on Riemannian manifolds. Siberian Adv. Math. 18(3), 153–162 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Berndt, J., Vanhecke, L.: Geometry of weakly symmetric spaces. J. Math. Soc. Japan 48(4), 745–760 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berndt, J., Kowalski, O., Vanhecke, L.: Geodesics in weakly symmetric spaces. Ann. Global Anal. Geom. 15(2), 153–156 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  11. 11.
    Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145, 1–49 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cairns, G., Hinić-Galić, A., Nikolayevsky, Y., Tsartsaflis, I.: Geodesic bases for Lie algebras. Linear Multilinear Algebra 63, 1176–1194 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cairns, G., Le, N.T.T., Nielsen, A., Nikolayevsky, Y.: On the existence of orthonormal geodesic bases for Lie algebras. Note Mat. 33(23), 11–18 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Calvaruso, G., Kowalski, O., Marinosci, R.A.: Homogeneous geodesics in solvable Lie groups. Acta Math. Hungar. 101(4), 313–322 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gleason, A.M.: On the structure of locally compact groups. Proc. Natl. Acad. Sci. USA 35, 384–386 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Goto, M.: Orbits of one-parameter groups. III. (Lie group case). J. Math. Soc. Japan 23, 95–102 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grove, K.: Isometry-invariant geodesics. Topology 13, 281–292 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hilgert, J., Neeb, K.-H.: Structure and Geometry of Lie Groups. Springer Monographs in Mathematics. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  19. 19.
    Iwasawa, K.: On some types of topological groups. Ann. Math. 2(50), 507–558 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kaĭzer, V.V.: Conjugate points of left invariant metrics on Lie groups. Sov. Math. 34(11), 32–44 (1990)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. V. II. Interscience Publishers, New York (1969)zbMATHGoogle Scholar
  22. 22.
    Kowalski, O., Szenthe, J.: On the existence of homogeneous geodesics in homogeneous Riemannian manifolds. Geom. Dedicata 81(1–3), 209–214 (2000). (correction: Ibid. 84(1–3), 331–332 (2001)) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics. Boll. Unione Mat. Ital. Ser. B 5(1), 189–246 (1991)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Marinosci, R.A.: Homogeneous geodesics in a three-dimensional Lie group. Comment. Math. Univ. Carolin. 43(2), 261–270 (2002)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Montgomery, D., Zippin, L.: Topological Transformation Goups. Interscience Publishers, New York (1955)zbMATHGoogle Scholar
  26. 26.
    Nikonorov, Y.G.: On the structure of geodesic orbit Riemannian spaces. Ann. Global Anal. Geom. 52(3), 289–311 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rodionov, E.D.: Homogeneous Riemannian Z-manifolds. Sib. Math. J. 22(2), 315–320 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rodionov, E.D.: Homogeneous Riemannian manifolds of rank one. Sib. Math. J. 25(4), 642–644 (1984)CrossRefzbMATHGoogle Scholar
  29. 29.
    Rodionov, E.D.: Homogeneous Riemannian almost P-manifolds. Sib. Math. J. 31(5), 789–794 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces, with applications to Dirichlet series. J. Indian Math. Soc. 20, 47–87 (1956)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Wolf, J.A.: Harmonic Analysis on Commutative Spaces. Mathematical Surveys and Monographs, vol. 142. American Mathematical Society, Providence (2007)CrossRefGoogle Scholar
  32. 32.
    Yakimova, O.S.: Weakly symmetric Riemannian manifolds with a reductive isometry group. Sb. Math. 195(3–4), 599–614 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Yau, S.T.: Remarks on the group of isometries of a Riemannian manifold. Topology 16(3), 239–247 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ziller, W.: Closed geodesics and homogeneous spaces. Math. Z. 152, 67–88 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Ziller, W.: The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces. Comment. Math. Helv. 52, 573–590 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Ziller, W.: Weakly symmetric spaces. In: Gindikin, S. (ed.) Topics in Geometry. Progress in Nonlinear Differential Equations and Their Application, vol. 20, pp. 355–368. Birkhäuser, Boston (1996)Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of SciencesNovosibirskRussia
  2. 2.Mechanics-Mathematical DepartmentNovosibirsk State UniversityNovosibirskRussia
  3. 3.Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of SciencesVladikavkazRussia

Personalised recommendations