On Some Sub-riemannian Objects in Hypersurfaces of Sub-riemannian Manifolds

We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then given two points, there exists at least one piecewise smooth horizontal curve in this hypersurface connecting them. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry: there exists a unique nonholonomic connection which is completely determined by the subRiemannian structure and a complement of the horizontal bundle, is “symmetric” and is compatible with the sub-Riemannian metric. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces.