Preferred Parameterisations on Homogeneous Curves

This article is motivated by the theory of distinguished curves in parabolic geometries, as developed in [2]. A parabolic geometry is, by definition, modelled on a homogeneous space of the form G/P where G is a real semisimple Lie group and P is a parabolic subgroup. (There is also a complex theory which corresponds to the choices of complex G’s and P ’s with specific curvature restrictions for the holomorphic cases.) The notion of Cartan connection replaces the Maurer-Cartan form on G, viewed as a principal fibre bundle over G/P with structure group P , and much of the geometry of G/P automatically carries over to parabolic geometries in general (see also [4]). In particular, the curves on G/P obtained by exponentiating elements in the Lie algebra g of G have counterparts in general obtained by development under the Cartan connection. These matters are thoroughly discussed in [2] and will not be repeated here. Suffice it to say that results concerning distinguished curves on G/P have immediate consequences for the corresponding general parabolic geometry. Here, we shall discuss only the homogeneous setting G/P .