Weakly symmetric sub-Riemannian manifolds with reductive isometry groups
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In this paper, we introduce the notion of weakly symmetric sub-Riemannian manifolds, which is a natural generalization of Riemannian weakly symmetric spaces. Then we prove that any weakly symmetric sub-Riemannian manifold is isometric to a homogeneous weakly symmetric space endowed with an invariant sub-Riemannian structure. As applications, we obtain a complete classification of connected and simply connected weakly symmetric sub-Riemannian manifolds with reductive isometry groups. Finally, we show that each weakly symmetric sub-Riemannian manifold with reductive isometry group is a geodesic orbit space, i.e., every geodesic is the orbit of a one-parameter subgroup of the full isometry group.
Abstract
In this paper, we introduce the notion of weakly symmetric sub-Riemannian manifolds, which is a natural generalization of Riemannian weakly symmetric spaces. Then we prove that any weakly symmetric sub-Riemannian manifold is isometric to a homogeneous weakly symmetric space endowed with an invariant sub-Riemannian structure. As applications, we obtain a complete classification of connected and simply connected weakly symmetric sub-Riemannian manifolds with reductive isometry groups. Finally, we show that each weakly symmetric sub-Riemannian manifold with reductive isometry group is a geodesic orbit space, i.e., every geodesic is the orbit of a one-parameter subgroup of the full isometry group.