Curves in homogeneous spaces and their contact with 1-dimensional orbits

Let α be a C ^∞ curve in a homogeneous space G/H. For each point x on the curve, we consider the subspace ({S^{alpha}_k}) of the Lie algebra ({mathcal{G}}) of G consisting of the vectors generating a one parameter subgroup whose orbit through x has contact of order k with α. In this paper, we give various important properties of the sequence of subspaces ({mathcal{G} supset S^{alpha}_1 supset S^{alpha}_2 supset S^{alpha}_3 supset cdots}) . In particular, we give a stabilization property for certain well-behaved curves. We also describe its relationship to the isotropy subgroup with respect to the contact element of order k associated with α.