ON THE SET OF GEODESIC VECTORS OF A LEFT-INVARIANT METRIC

János SZENTHE

If (M, <, >) is a Riemannian manifold its geodesic γ : R → M is said to be homogeneous if there is a 1-parameter group of isometries Φ : R × M → M of the Riemannian manifold such that

γ(τ) = Φ(τ, γ(0)), τ ∈ ℝ

holds. As the comprehensive paper of C. S. Gordon shows the existence of homogeneous geodesics in homogeneous Riemannian manifolds has essential geometric consequences [G]. Several results have been obtained recently concerning the existence of homogeneous geodesics. First it has been shown by V. V. Kajzer that if G is a Lie group and <, > a left-invariant Riemannian metric on G then the Riemannian manifold (G, <, >) has at least 1 homogeneous geodesic [Ka]. Generalizing this result of Kajzer it has been shown by O. Kowalski and J. Szenthe that if M = G/H is a homogeneous manifold and <, > an invariant metric on G/H then the homogeneous Riemannian manifold (G/H, <, >) has at least 1 homogeneous geodesic [Ko-Sz]. Moreover, it has been shown by Szenthe that if G is a compact semi-simple Lie group of rank ≥ 2 and <, > is a left-invariant Riemannian metric on G then the Riemannian manifold has infinitely many homogeneous geodesics [Sz]. In the study of the set of homogenous

geodesics of a homogeneous Riemannian manifold (G/H, <, >) the concept of geodesic vector proved to be convenient [Ko-V]. Let Φ : G × (G/H) → G/H be the canonical action, g the Lie algebra of G and Exp : g → G its exponential map. Put o = H ∈ G/H, fix a tangent vector v ∈ T_{o}(G/H) and consider the geodesic γ : R → G/H defined by v = γ̇(0). It is said that v is a geodesic vector if γ is a homogeneous geodesic of (G/H, <, >); in other words if

γ(τ) = Φ(Exp(τX), o), τ ∈ ℝ

holds with some X ∈ g. The study of the set of homogeneous geodesics of a homogenous Riemannian manifold is obviously reducible to the study of the set of its homogeneous vectors. However, it seems that the set of the geodesic vectors

of a homogeneous Riemannian manifold does not admit a simple description in general. Namely, O. Kowalski, S. Nikčević and Z. Vlašek in a joint paper have given several examples where the set of geodesic vectors have essentially different structure. In the lecture results are presented concerning the set of geodesic vectors of a homogeneous Riemannian manifold (G, <, >), where G is a compact semi-simple Lie group and <, > is a left-invariant Riemannian metric on G.

References

[ G] Gordon, C. S., Homogeneous Riemannian manifolds whose geodesics are orbits, Topics in Geometry in Memory of Joseph D’Atri, 1996, pp155-174.

[ Ka] Kajzer, V. V., Conjugate points of left-invariant metrics on Lie groups, Sov. Math. 34 (1990), translation from Izv. Vyssh. Uchebn. Zaved. Mat. 342 (1990), pp 27-37.

[ Ko-N-V] Kowalski, O., Nikčević, S., Vlášek, Z., Homogeneous geodesics in homogenous Riemannian manifolds, to appear.

[ Ko-Sz] Kowalski, O., Szenthe, J., On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata 81 (2000), pp 209-214.

[ Ko-V] Kowalski, O., Vanhecke, L., Homogeneous Riemannian manifolds with homogeneous geodesics, Bull. Un. Mat. Ital., 5 (1991), pp 189-246.

[ Sz] Szenthe, J., Homogeneous geodesics of left-invariant metrics, Univ. Iagel. Acta Math. 38 (2000), pp 99-103.

Address: J. Szenthe, Dept. of Geometry, Eötvös Univ. Kecskeméti u. 10-12, Budapest, H-1053, Hungary, e-mail: szenthe@ludens.elte.hu