Gauss–Bonnet theorems and the Lorentzian Heisenberg group

Abstract

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a C$^{2}$-smooth surface in the Lorentzian Heisenberg group for the second Lorentzian metric and the third Lorentzian metric and signed geodesic curvature for C$^{2}$-smooth curves on surfaces. We get Gauss-Bonnet theorems in the Lorentzian Heisenberg group for the second Lorentzian metric and the third Lorentzian metric.