Gauss–Bonnet Theorems in the BCV Spaces and the Twisted Heisenberg Group

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $$C^2$$ -smooth surface in the BCV spaces and the twisted Heisenberg group away from characteristic points and signed geodesic curvature for Euclidean $$C^2$$ -smooth curves on surfaces. We get Gauss–Bonnet theorems in the BCV spaces and the twisted Heisenberg group.