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

Abstract

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.