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Homogeneous geodesics in sub-Riemannian geometry

A. V. Podobryaev

Abstract

We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step $1$ and $2$. Finally, we get a broad condition for existence of at least one homogeneous geodesic.

Homogeneous geodesics in sub-Riemannian geometry

Abstract

We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step and . Finally, we get a broad condition for existence of at least one homogeneous geodesic.
Paper Structure (5 sections, 12 theorems, 43 equations, 2 figures)

This paper contains 5 sections, 12 theorems, 43 equations, 2 figures.

Table of Contents

  1. Sub-Riemannian geodesics
  2. Homogeneous geodesics
  3. Nilpotent geodesic orbit sub-Riemannian manifolds
  4. Some general properties of geodesic orbit sub-Riemannian manifolds
  5. Existence of a homogeneous geodesic

Key Result

Theorem 1

Assume that $\tilde{u} \in L^{\infty}([0, T], \Delta)$ is an optimal control and $\tilde{q} : [0, T] \rightarrow G$ is the corresponding optimal curve of the problem eq-controlproblem, then there exist a Lipschitz curve $\lambda: [0, T] \rightarrow T^*G$ and a number $\nu \leqslant 0$ such that $(1

Figures (2)

  • Figure 1: Vertical subsystem in axisymmetric case.
  • Figure 2: Vertical subsystem in general case.

Theorems & Definitions (52)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1: Pontryagin maximum principle
  • Definition 3
  • Definition 4
  • Remark 2
  • Lemma 1: Geodesic lemma
  • Lemma 2
  • Remark 3
  • ...and 42 more