O. Kowalski and J. Szenthe [KS] proved that every homogeneous Riemannian manifold admits at least one homogeneous geodesic, i.e\. one geodesic which is an orbit of a one-parameter group of isometries. In [KNV] the related two problems were studied and a negative answer was given to both ones: (1) Let $M=K/H$ be a homogeneous Riemannian manifold where $K$ is the largest connected group of isometries and $\dim M\geq 3$. Does $M$ always admit more than one homogeneous geodesic? (2) Suppose that $M=K/H$ admits $m = \dim M$ linearly independent homogeneous geodesics through the origin $o$. Does it admit $m$ mutually orthogonal homogeneous geodesics? In this paper the author continues this study in a three-dimensional connected Lie group $G$ equipped with a left invariant Riemannian metric and investigates the set of all homogeneous geodesics.
@article{119318,
author = {Rosa Anna Marinosci},
title = {Homogeneous geodesics in a three-dimensional Lie group},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {43},
year = {2002},
pages = {261-270},
zbl = {1090.53038},
mrnumber = {1922126},
language = {en},
url = {http://dml.mathdoc.fr/item/119318}
}
Marinosci, Rosa Anna. Homogeneous geodesics in a three-dimensional Lie group. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 261-270. http://gdmltest.u-ga.fr/item/119318/
Homogeneous Riemannian manifolds whose geodesics are orbits, Topics in Geometry, in Memory of Joseph D'Atri, 1996, pp.155-174. | MR 1390313 | Zbl 0861.53052
Conjugate points of left-invariant metrics on Lie groups, J. Soviet Math. 34 (1990), 32-44. (1990) | MR 1106314 | Zbl 0722.53049
On the geometry of groups of Heisemberg type, Bull. London Math. Soc. 15 (1983), 35-42. (1983) | MR 0686346
Foundations of Differential Geometry, I and II, Interscience Publisher, New York, 1963, 1969. | MR 0152974
Generalized symmetric spaces, Lecture Notes in Math. 805, Springer-Verlag, Berlin, Heidelberg, New York, 1980. | MR 0579184 | Zbl 0543.53041
On geodesic graphs of Riemannian g.o. spaces, Archiv der Math. 73 (1999), 223-234. (1999) | MR 1705019
Homogeneous geodesics in homogeneous Riemannian manifolds. Examples, Reihe Mathematik, TU Berlin, No. 665/2000 (9 pages). | MR 1801906
D'Atri spaces, in Topics in Geometry, Birkhäuser, Boston, 1996, pp.241-284. | MR 1390318 | Zbl 0862.53039
On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata 81 (2000), 209-214. (2000) | MR 1772203 | Zbl 0980.53061
Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. 5 (1991), 189-246. (1991) | MR 1110676 | Zbl 0731.53046
Curvatures of left-invariant metrics on Lie groups, Adv. Math. 21 (1976), 293-329. (1976) | MR 0425012 | Zbl 0341.53030
Homogeneous structures on Riemannian manifolds, London Math. Soc. Lecture Note Series 83, Cambridge Univ. Press, Cambridge, 1983. | MR 0712664 | Zbl 0641.53047