Sub-Riemannian curvature of Carnot groups with rank-two distributions

TL;DR

This paper explores the generalized curvature of Carnot groups with rank-two distributions, focusing on specific groups like the Cartan and Goursat types, and analyzes geodesic properties and curvature dependencies.

Contribution

It characterizes the curvature and geodesic regularity in Carnot groups with rank-two distributions, especially Goursat-type, Engel, and Cartan groups, extending curvature concepts beyond Riemannian geometry.

Findings

Curvature depends only on the Engel part in Goursat-type groups.

Existence of initial covectors with infinitely many non-equiregular geodesics.

Characterization of ample and equiregular geodesics in specific Carnot groups.

Abstract

The notion of curvature discussed in this paper is a far going generalization of the Riemannian sectional curvature. It was first introduced by Agrachev, Barilari and Rizzi in arXiv:1306.5318, and it is defined for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler, and sub-Finsler structures. In this work we study the generalized sectional curvature of Carnot groups with rank-two distributions. In particular, we consider the Cartan group and Carnot groups with horizontal distribution of Goursat-type. In these Carnot groups we characterize ample and equiregular geodesics. For Carnot groups with horizontal Goursat distribution we show that their generalized sectional curvatures depend only on the Engel part of the distribution. This family of Carnot groups contains naturally the three-dimensional Heisenberg group, as well as the Engel group. Moreover, we also show that in the Engel and Cartan groups there exist initial covectors for which there is an infinite discrete set of times at which the corresponding ample geodesics are not equiregular.