The aim of this paper was to obtain Gauss–Bonnet theorems on the Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane. At the same time, the sub-Lorentzian limits of Gaussian curvature for surfaces which are $C^2$-smooth in the Lorentzian Heisenberg group away from characteristic points and signed geodesic curvature for curves which are $C^2$-smooth on surfaces are studied. Using a similar method, we also studied the corresponding contents on Lorentzian group of rigid motions of the Minkowski plane.

中文翻译：

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Minkowski平面的刚性运动的洛伦兹海森堡群和洛伦兹群中的高斯-邦尼定理

本文的目的是获得Minkowski平面的刚性运动的Lorentz Heisenberg群和Lorentzian群上的Gauss-Bonnet定理。同时，高斯曲率的亚洛仑兹极限$C^2$Lorentzian Heisenberg组中的-平滑曲线远离特征点和有符号的测地曲率 $C^2$研究了表面的光滑度。使用类似的方法，我们还研究了Minkowski平面的刚性运动的Lorentzian组上的相应内容。