Sub-Riemannian Curvature of Carnot Groups with Rank-Two Distributions

Abstract

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 [2], 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. Acknowledgement 1. The author wishes to thank Andrei Agrachev, Davide Barilari and Luca Rizzi for their constant interest and the many helpful conversations on sub-Riemannian geometry, especially on their paper [2]. The author also would like to thank the anonymous reviewers for their constructive comments, which helped to improve the contents of the paper, especially the content of Theorem 4.2. This work was done while the author was visiting the International School for Advanced Studies (SISSA) at Trieste, Italy.

Sub-Riemannian interpolation inequalities

“…This is the case for all Carnot groups with Goursat-type distribution and dimension n ≥ 4, such as the Cartan and the Engel groups. See [Mun17]. The aim of this appendix is to give a self-contained proof of the fact that geodesics not containing abnormal segments lose minimality after their first conjugate point, following [ABB16b,Sar80].…”

“…This is the case for all Carnot groups with Goursat-type distribution and dimension n ≥ 4, such as the Cartan and the Engel groups. See [Mun17]. and curvature, which are not available in the sub-Riemannian setting.…”

Affine connections and curvature in sub-Riemannian geometry

“…Of applications of this theory, we mention comparison theorems for conjugate points and diameter [8,7], sub-Laplacian and volume comparison theorems [5,30,10], measure contraction properties [31] and interpolation inequalities [11,10]. The concrete examples done so far are given in the codimension one case [32,33], contact geometry [4,2], some Carnot group of rank 2 [36] and 3-Sasakian manifolds [38]. We also mention papers [14,16] which use a Lagrangian approach for Sasakian and H-type manifolds as well, while also relying on a taming metric.…”

Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups

“…The Lagrangian point of view has its roots in the study of Jacobi vector fields initiated in the fundamental works of Agrachev-Li-Zelenko [6,7,141] (ALZ for short) and later developed in [3,32,33,83,84,113]. Besides the numerous applications inspired by some classical results of Riemannian Geometry, see [2,5,31,32,103,126] for example, a deep and powerful byproduct of the Lagrangian approach -in the original spirit of Cordero-Erausquin-McCann-Schmuckenschläger [63] -is a precise control of the distorsion coefficients in the sub-Riemannian interpolation inequalities.…”