Explicit Formulas for Extremals in Sub-Lorentzian Problems on Three-Dimensional Unimodular Lie Groups

This talk considers sub-Lorentzian problems with an arbitrary anti-norm on all three-dimensional unimodular Lie groups: $SU(2)$, $SL(2)$, $SE(2)$, $SH(2)$, $\mathbb{H}_3$, and on the Lobachevsky plane (which corresponds to the group $\mathrm{Aff}_+(\mathbb{R})$). Using the functions $\mathrm{ch}_\Omega$ and $\mathrm{sh}_\Omega$, which are a convenient generalization of the $\mathrm{ch}$ and $\mathrm{sh}$ functions, explicit formulas for the PMP extremals in these problems are obtained. Properties of the anti-norm are also formulated that are sufficient for the normal extremals of the system to be timelike, and for the abnormal ones to be lightlike or abnormal for the distribution.
This talk is based on joint work with L.V. Lokutsievskiy and N.V. Prilepin [1].
[1] E. A. Ladeyshchikov, L. V. Lokutsievskiy, N. V. Prilepin, “Explicit formulas for extremals in sub-Lorentzian and Finsler problems on 2- and 3-dimensional Lie groups”, *Mat. Sb.*, 216:12 (2025)