Maxwell strata in sub-Riemannian problem on the group of motions of a plane

TL;DR

This paper analyzes the sub-Riemannian problem on the group of motions of a plane, describing geodesics, symmetries, Maxwell points, and providing an upper bound on the cut time.

Contribution

It characterizes Maxwell points and symmetries in the sub-Riemannian problem on the plane motion group, deriving an upper bound on the cut time.

Findings

Jacobi's functions parametrize geodesics.

Discrete symmetries are generated by pendulum reflections.

An upper bound on the cut time is established.

Abstract

The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parametrized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained.