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

In this paper we study the main geometric properties of the Carnot-Carath'eodory (abbreviated CC) distance $\text{dc}$ 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 $\exp \text{sr}$ 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 $\delta \text{cc}$ from a ${\text{cont}}^k$-smooth hypersurface $S$.