Abstract
It is proved that every homogeneous Riemannian manifold admits a geodesic which is an orbit of a one-parameter group of isometries.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gordon, C. S.: Homogeneous Riemannian manifolds whose geodesics are orbits, In: Topics in Geometry, in Memory of Joseph D'Atri, BirkhÌuser, Basel, 1996, pp. 155–174.
Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.
Kajzer, V. V.: Conjugate points of left-invariant metrics on Lie groups, Soviet Math. 34 (1990), 32–44; transl. from Izv. Vyssh Uchebn. Zaved. Mat. 342 (1990), 27–37.
Kaplan, A.: On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35–42.
Kobyashi, S. and Nomizu, K.: Foundations of Differential Geometry I, II, Interscience Publ., New York, 1963, 1969.
Kowalski, O. and Nikčević, S. Ž.: On geodesic graphs of Riemannian g.o. spaces, Arch. Math. 73 (1999), 223–234.
Kowalski, O., Prüfer, F. and Vanhecke, L.: D'Atri spaces, In: Topics in Geometry, in Memory of Joseph D'Atri, Birkhäuser, Basel, 1996, pp. 241–282.
Kowalski, O. and Vanhecke, L.: Riemannian manifolds with homogeneous geodesics, Bull. Un. Mat. Ital. 5 (1991), 189–246.
Nomizu, K.: Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33–65.
Wolf, J.: The geometry and structure of isotropy irreducible homogeneous spaces, Acta Math. 120 (1968), 59–148.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kowalski, O., Szenthe, J. On the Existence of Homogeneous Geodesics in Homogeneous Riemannian Manifolds. Geometriae Dedicata 81, 209–214 (2000). https://doi.org/10.1023/A:1005287907806
Issue Date:
DOI: https://doi.org/10.1023/A:1005287907806