Previous |  Up |  Next


Riemannian manifold; homogeneous space; geodesics as orbits
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.
[G] Gordon C.S.: 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
[Ka] Kajzer V.V.: Conjugate points of left-invariant metrics on Lie groups. J. Soviet Math. 34 (1990), 32-44. MR 1106314 | Zbl 0722.53049
[Kp] Kaplan A.: On the geometry of groups of Heisemberg type. Bull. London Math. Soc. 15 (1983), 35-42. MR 0686346
[KoNo] Kobayashi S., Nomizu K.: Foundations of Differential Geometry, I and II. Interscience Publisher, New York, 1963, 1969. MR 0152974
[Kw] Kowalski O.: Generalized symmetric spaces. Lecture Notes in Math. 805, Springer-Verlag, Berlin, Heidelberg, New York, 1980. MR 0579184 | Zbl 0543.53041
[KN] Kowalski O., Nikčević S.: On geodesic graphs of Riemannian g.o. spaces. Archiv der Math. 73 (1999), 223-234. MR 1705019
[KNV] Kowalski O., Nikčević S., Vlášek Z.: Homogeneous geodesics in homogeneous Riemannian manifolds. Examples. Reihe Mathematik, TU Berlin, No. 665/2000 (9 pages). MR 1801906
[KPV] Kowalski O., Prüfer F., Vanhecke L.: D'Atri spaces. in Topics in Geometry, Birkhäuser, Boston, 1996, pp.241-284. MR 1390318 | Zbl 0862.53039
[KS] Kowalski O., Szenthe J.: On the existence of homogeneous geodesics in homogeneous Riemannian manifolds. Geom. Dedicata 81 (2000), 209-214. MR 1772203 | Zbl 0980.53061
[KV] Kowalski O., Vanhecke L.: Riemannian manifolds with homogeneous geodesics. Boll. Un. Mat. Ital. 5 (1991), 189-246. MR 1110676 | Zbl 0731.53046
[M] Milnor J.: Curvatures of left-invariant metrics on Lie groups. Adv. Math. 21 (1976), 293-329. MR 0425012 | Zbl 0341.53030
[TV] Tricerri F., Vanhecke L.: Homogeneous structures on Riemannian manifolds. London Math. Soc. Lecture Note Series 83, Cambridge Univ. Press, Cambridge, 1983. MR 0712664 | Zbl 0641.53047
Partner of
EuDML logo