CC-distance and metric normal of smooth hypersurfaces in sub-Riemannian Carnot groups

Abstract

In this paper we study the main geometric properties of the Carnot-Carath\'eodory (abbreviated CC) distance $\dc$ in the setting of $k$-step sub-Riemannian Carnot groups from many different points of view. An extensive study of the so-called normal CC-geodesics is given. We state and prove some related variational formulae and we find suitable Jacobi-type equations for normal CC-geodesics. One of our main results is a sub-Riemannian version of the Gauss Lemma. We show the existence of the metric normal for smooth non-characteristic hypersurfaces. We also compute the sub-Riemannian exponential map $\exp\sr$ for the case of 2-step Carnot groups. Other features of normal CC-geodesics are then studied. We show how the system of normal CC-geodesic equations can be integrated step by step. Finally, we show a regularity property of the CC-distance function $\delta\cc$ from a $\cont^k$-smooth hypersurface $S$.