Abstract: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.
Keywords: Riemannian manifold, homogeneous space, geodesics as orbits
AMS Subject Classification: 53C20, 53C22, 53C30