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The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannianapproximations scheme $(\mathbb H^1,<,>_L)$, in the Heisenberg group,introduced by Gromov, to calculate the limits of Gaussian and normal curvaturesdefined on surfaces of $\mathbb H^1$ when $L\rightarrow\infty$. They show thatthese limits exist (unlike the limit of Riemannian surface area form or lengthform), and they obtain Gauss-Bonnet theorem in $\mathbb H^1$ as limit ofGauss-Bonnet theorems in $(\mathbb H^1,<,>_L)$ when $L$ goes to infinity. Thisconstruction was extended by Wang-Wei in arXiv:1912.00302 to the affine groupand the group of rigid motions of the Minkowski plane. We generalizeconstructions of both papers to surfaces in sub-Riemannian three dimensionalmanifolds following the approach of arXiv:1909.13341, and prove analogousGauss-Bonnet theorem.

Limit of Gaussian and normal curvatures of surfaces in Riemannian  approximation scheme for sub-Riemannian three dimensional manifolds and  Gauss-Bonnet theorem