Homogeneous Sub-Riemannian Manifolds Whose Normal Extremals are Orbits

Abstract

In this paper, we study homogeneous sub-Riemannian manifolds whose normal extremals are the orbits of one-parameter subgroups of the group of smooth isometries (abbreviated as sub-Riemannian geodesic orbit manifolds). Following Tóth’s approach, we first obtain a sufficient and necessary condition for a homogeneous sub-Riemannian manifold to be geodesic orbit. Secondly, we study left-invariant sub-Riemannian geodesic orbit metrics on connected and simply connected nilpotent Lie groups. It turns out that every sub-Riemannian geodesic orbit nilmanifold is the restriction of a Riemannian geodesic orbit nilmanifold. Thirdly, we provide a method to construct compact and non-compact sub-Riemannian geodesic orbit manifolds and present a large number of sub-Riemannian geodesic orbit manifolds from Tamaru’s classification of Riemannian geodesic orbit manifolds fibered over irreducible symmetric spaces. Finally, we give a complete description of sub-Riemannian geodesic orbit metrics on spheres, and show that many of sub-Riemannian geodesic orbit manifolds have no abnormal sub-Riemannian geodesics.