Gaussian and mean curvatures of two-dimensional subgroups of three-dimensional unimodular Lorentzian Lie groups (Q1921785)

The Gaussian and mean curvatures of two-dimensional subgroups, of three-dimensional Lie groups, endowed with induced metrics were studied by \textit{V. V. Kajzer} [Sov. Math. 28, 92-96 (1984); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1984, No. 3(262), 68-71 (1984; Zbl 0567.53036)], where the case of left-invariant Riemannian metrics was under consideration. In the present article we consider left-invariant Lorentzian metrics. A Lie group endowed with a left-invariant Lorentzian metric will be referred to as a Lorentzian Lie group. For Riemannian metrics on unimodular Lie groups, there is a single computational formula, while the case of Lorentzian Lie groups is more diverse. Our purpose is to derive some general computational formula and apply it to the calculation of the Gaussian curvature for the two-dimensional subgroups of the Heisenberg group; all three-dimensional nilpotent Lie groups are locally isomorphic to this group. The formula makes it possible to prove some result concerning the three-dimensional unimodular Lie groups in general.