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.
Title: Geodesics in the Engel group with a sub-Lorentzian metric
Authors: Qihui Cai, Tiren Huang, Yu. L. Sachkov, Xiaoping Yang
Journal title: Journal of Dynamical and Control Systems
ArXiv ID (ENG): 1506.06127