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

Abstract

The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannian approximations scheme $({\mathbb{H}}^1,<,{>}_L)$, in the Heisenberg group, introduced by Gromov, to calculate the limits of Gaussian and normal curvatures defined on surfaces of ${\mathbb{H}}^1$ when $L\to \infty$. They show that these limits exist (unlike the limit of Riemannian surface area form or length form), and they obtain Gauss-Bonnet theorem in ${\mathbb{H}}^1$ as limit of Gauss-Bonnet theorems in $({\mathbb{H}}^1,<,{>}_L)$ when $L$ goes to infinity. This construction was extended by Wang-Wei in arXiv:1912.00302 to the affine group and the group of rigid motions of the Minkowski plane. We generalize constructions of both papers to surfaces in sub-Riemannian three dimensional manifolds following the approach of arXiv:1909.13341, and prove analogous Gauss-Bonnet theorem.

arXiv admin note: text overlap with arXiv:1909.13341

References in corpus (1)

• Rotation surfaces of constant Gaussian curvature as Riemannian approximation scheme in sub-Riemannian Heisenberg space ${\mathbb{H}}^1$