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

2020/02/17 by Veloso, Jose
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2002.07177

Abstract

The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannian approximations scheme (\mathbb H¹,_L), in the Heisenberg group, introduced by Gromov, to calculate the limits of Gaussian and normal curvatures defined on surfaces of \mathbb H¹ when L→∞. 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 H¹ as limit of Gauss-Bonnet theorems in (\mathbb H¹,_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.

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