Geodesics in the Engel group with a sub-Lorentzian metric

TL;DR

This paper investigates the properties of geodesics in the Engel group equipped with a sub-Lorentzian metric, establishing their optimality and providing explicit formulas for certain classes of geodesics.

Contribution

It characterizes time-like normal geodesics as local maximizers and derives explicit expressions for non-space-like geodesics in the Engel group with a sub-Lorentzian metric.

Findings

Time-like geodesics are locally maximizing.

Explicit formulas for non-space-like geodesics.

Properties of horizontal curves in the Engel group.

Abstract

Let E be the Engel group and D be a rank 2 bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first study some properties of horizontal curves on E. Second, we prove that time-like normal geodesics are locally maximizers in the Engel group, and calculate the explicit expression of non-space-like geodesics.