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arXiv · 0910.5648

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

N. Arcozzi·F. Ferrari·F. Montefalcone
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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 $S$.

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N. Arcozzi, F. Ferrari, F. Montefalcone. 2009-10-29. CC-distance and metric normal of smooth hypersurfaces in sub-Riemannian Carnot groups. https://arxiv.org/abs/0910.5648

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