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

TL;DR

This paper investigates the limits of Gaussian and normal curvatures on surfaces in sub-Riemannian 3D manifolds using Riemannian approximation schemes, establishing a Gauss-Bonnet theorem in this context.

Contribution

It generalizes previous Riemannian approximation methods to broader sub-Riemannian 3D manifolds and proves an analogous Gauss-Bonnet theorem.

Findings

Limits of Gaussian and normal curvatures exist in the approximation scheme.

Gauss-Bonnet theorem holds as a limit in sub-Riemannian 3D manifolds.

Extension of previous results to new classes of manifolds.

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 $L\rightarrow\infty$. 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.