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

Sub-Riemannian Curvature of Carnot Groups with Rank-Two Distributions The notion of curvature discussed in this paper is a far-going generalization of the Riemannian sectional curvature. It was first introduced by Agrachev et al. ([2015]), 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Dynamical and Control Systems Springer Journals

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

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media New York
Subject
Engineering; Vibration, Dynamical Systems, Control; Calculus of Variations and Optimal Control; Optimization; Analysis; Applications of Mathematics; Systems Theory, Control
ISSN
1079-2724
eISSN
1573-8698
D.O.I.
10.1007/s10883-017-9365-8
Publisher site
See Article on Publisher Site

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 et al. ([2015]), 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.

Journal

Journal of Dynamical and Control SystemsSpringer Journals

Published: Mar 31, 2017

References

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