[Paper Review] CC-distance and metric normal of smooth hypersurfaces in sub-Riemannian Carnot groups

Nicola Arcozzi, Fausto Ferrari+1 more|ArXiv.org|Oct 29, 2009

Geometric Analysis and Curvature Flows34 references7 citations

TL;DR

This paper establishes a sub-Riemannian version of the Gauss Lemma and proves the existence of the metric normal for smooth non-characteristic hypersurfaces in sub-Riemannian Carnot groups. It derives variational formulae and Jacobi-type equations for normal CC-geodesics, and shows that the CC-distance function from a $ C^k $-smooth hypersurface in a 2-step Carnot group is $ C^k $-smooth in a neighborhood of the hypersurface, excluding the characteristic set.

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 $S$.

Motivation & Objective

• To establish a sub-Riemannian analog of the classical Gauss Lemma in Carnot groups.
• To prove the existence of the metric normal for smooth non-characteristic hypersurfaces in sub-Riemannian Carnot groups.
• To analyze the regularity of the CC-distance function from a $ C^k $-smooth hypersurface in 2-step Carnot groups.
• To provide explicit integration of normal CC-geodesic equations in 2-step Carnot groups.

Proposed method

• Derives variational formulae and Jacobi-type equations for normal CC-geodesics using the sub-Riemannian exponential map.
• Introduces and analyzes the sub-Riemannian exponential map $ \mathrm{exp}_{\mathcal{SR}} $ for 2-step Carnot groups.
• Uses the concept of normal geodesics and their iterative integration to characterize the CC-distance.
• Applies the notion of metric normal via the gradient of the CC-distance function and its relation to the Riemannian normal on CC-spheres.
• Establishes the unique nearest point property in a neighborhood of the hypersurface, excluding the characteristic set.
• Relies on the structure of $ k $-step Carnot groups and the explicit form of CC-geodesics in the 2-step case to derive smoothness results.

Experimental results

Research questions

• RQ1 Does a sub-Riemannian version of the Gauss Lemma hold in Carnot groups?
• RQ2 Can the metric normal be defined for smooth non-characteristic hypersurfaces in sub-Riemannian Carnot groups?
• RQ3 Is the CC-distance function from a $ C^k $-smooth hypersurface in a 2-step Carnot group itself $ C^k $-smooth?
• RQ4 How can the system of normal CC-geodesic equations be integrated step by step in 2-step Carnot groups?
• RQ5 What is the regularity of the projection map onto the nearest point on a smooth hypersurface in the sub-Riemannian setting?

Key findings

• The paper proves a sub-Riemannian version of the Gauss Lemma, establishing that the gradient of the CC-distance function at a point is the metric normal vector.
• The metric normal exists and is well-defined for smooth non-characteristic hypersurfaces in Carnot groups.
• In 2-step Carnot groups, the CC-distance function from a $ C^k $-smooth hypersurface is $ C^k $-smooth in a neighborhood of the hypersurface, excluding the characteristic set.
• The sub-Riemannian exponential map $ \mathrm{exp}_{\mathcal{SR}} $ is explicitly computed for 2-step Carnot groups.
• The gradient of the CC-distance function is shown to be $ C^{k-1} $-smooth, implying the distance function is $ C^k $-smooth.
• The unique nearest point property holds in a neighborhood of the hypersurface, ensuring local uniqueness of the closest point on the hypersurface.