Sub-Riemannian Geodesics in the Octonionic H-type Group

In this chapter, we study sub-Riemannian geodesics in the octonionic H-type group G(7)(1), which is a nilpotent group of step 2 and, as a manifold, diffeomorphic to R-15. The Lie group structure of G(7)(1), obtained via the Cayley-Dickson construction of real division algebras, induces a natural Riemannian metric and a bracket-generating distribution H of rank eight and step 2 on G(7)(1). Restricting the metric to H we obtain a sub-Riemannian structure on G(7)(1). The class of curves we are interested in are horizontal with respect to H and, most importantly, critical points of the natural sub-Riemannian length functional. We present a characterization of these critical points via a differential equation, similar to the geodesic equation in Riemannian geometry, which states that for critical points of the length functional the intrinsic acceleration del(gamma)gamma is a linear combination with constant coefficients of some special rotations of the velocity gamma.