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.
Similar content being viewed by others
References
Akhiezer, D.N., Vinberg, È.B.: Weakly symmetric spaces and spherical varieties. Transform. Groups 4, 3–24 (1999)
Arutyunov, A.V., Zhukovskiy, S.E.: Properties of surjective real quadratic maps. Sb. Math. 207(9), 1187–1214 (2016)
Arvanitoyeorgos, A.: Homogeneous manifolds whose geodesics are orbits. Recent results and some open problems. Irish Math. Soc. Bull. 79, 5–29 (2017)
Bangert, V.: Non-closed isometry-invariant geodesics. Arch. Math. 106(6), 573–580 (2016)
Berestovskii, V.N., Nikonorov, Y.G.: Killing vector fields of constant length on locally symmetric Riemannian manifolds. Transform. Groups 13(1), 25–45 (2008)
Berestovskii, V.N., Nikonorov, Y.G.: Killing vector fields of constant length on Riemannian manifolds. Sib. Math. J. 49(3), 395–407 (2008)
Berestovskii, V.N., Nikonorov, Y.G.: Regular and quasiregular isometric flows on Riemannian manifolds. Siberian Adv. Math. 18(3), 153–162 (2008)
Berndt, J., Vanhecke, L.: Geometry of weakly symmetric spaces. J. Math. Soc. Japan 48(4), 745–760 (1996)
Berndt, J., Kowalski, O., Vanhecke, L.: Geodesics in weakly symmetric spaces. Ann. Global Anal. Geom. 15(2), 153–156 (1997)
Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)
Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145, 1–49 (1969)
Cairns, G., Hinić-Galić, A., Nikolayevsky, Y., Tsartsaflis, I.: Geodesic bases for Lie algebras. Linear Multilinear Algebra 63, 1176–1194 (2015)
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)
Calvaruso, G., Kowalski, O., Marinosci, R.A.: Homogeneous geodesics in solvable Lie groups. Acta Math. Hungar. 101(4), 313–322 (2003)
Gleason, A.M.: On the structure of locally compact groups. Proc. Natl. Acad. Sci. USA 35, 384–386 (1949)
Goto, M.: Orbits of one-parameter groups. III. (Lie group case). J. Math. Soc. Japan 23, 95–102 (1971)
Grove, K.: Isometry-invariant geodesics. Topology 13, 281–292 (1974)
Hilgert, J., Neeb, K.-H.: Structure and Geometry of Lie Groups. Springer Monographs in Mathematics. Springer, New York (2012)
Iwasawa, K.: On some types of topological groups. Ann. Math. 2(50), 507–558 (1949)
Kaĭzer, V.V.: Conjugate points of left invariant metrics on Lie groups. Sov. Math. 34(11), 32–44 (1990)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. V. II. Interscience Publishers, New York (1969)
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))
Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics. Boll. Unione Mat. Ital. Ser. B 5(1), 189–246 (1991)
Marinosci, R.A.: Homogeneous geodesics in a three-dimensional Lie group. Comment. Math. Univ. Carolin. 43(2), 261–270 (2002)
Montgomery, D., Zippin, L.: Topological Transformation Goups. Interscience Publishers, New York (1955)
Nikonorov, Y.G.: On the structure of geodesic orbit Riemannian spaces. Ann. Global Anal. Geom. 52(3), 289–311 (2017)
Rodionov, E.D.: Homogeneous Riemannian Z-manifolds. Sib. Math. J. 22(2), 315–320 (1981)
Rodionov, E.D.: Homogeneous Riemannian manifolds of rank one. Sib. Math. J. 25(4), 642–644 (1984)
Rodionov, E.D.: Homogeneous Riemannian almost P-manifolds. Sib. Math. J. 31(5), 789–794 (1990)
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)
Wolf, J.A.: Harmonic Analysis on Commutative Spaces. Mathematical Surveys and Monographs, vol. 142. American Mathematical Society, Providence (2007)
Yakimova, O.S.: Weakly symmetric Riemannian manifolds with a reductive isometry group. Sb. Math. 195(3–4), 599–614 (2004)
Yau, S.T.: Remarks on the group of isometries of a Riemannian manifold. Topology 16(3), 239–247 (1977)
Ziller, W.: Closed geodesics and homogeneous spaces. Math. Z. 152, 67–88 (1976)
Ziller, W.: The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces. Comment. Math. Helv. 52, 573–590 (1977)
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)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Berestovskiĭ, V.N., Nikonorov, Y.G. On homogeneous geodesics and weakly symmetric spaces. Ann Glob Anal Geom 55, 575–589 (2019). https://doi.org/10.1007/s10455-018-9641-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-018-9641-1
Keywords
- Geodesic orbit Riemannian space
- Homogeneous Riemannian manifold
- Homogeneous space
- Quadratic mapping
- Totally geodesic torus
- Weakly symmetric space