On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds

Abstract

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 there exists at least a piecewise smooth horizontal curve in this hypersurface connecting any two given points in it. 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 sub-Riemannian structure and is “symmetric” and compatible with the sub-Riemannian metric. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces.