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

2009/10/29 by Arcozzi, N., Ferrari, F., Montefalcone, F.
#22E60 #46E35 #49Q15 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0910.5648

Abstract

In this paper we study the main geometric properties of the Carnot-Carathéodory (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 δ\cc from a \cont^k-smooth hypersurface S.

Related

• The Carnot-Carathéodory distance on 2-step groups
• SubRiemannian geodesics for Carnot groups of step 3
• Sub-Riemannian Calculus on Hypersurfaces in Carnot Groups
• Metric Lie Groups. Carnot-Carathéodory spaces from the homogeneous viewpoint
• Sub-Riemannian geometry on some step-two Carnot groups