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

TL;DR

This paper investigates the relationship between different notions of Gaussian curvature in the Heisenberg space, using Riemannian approximations and rotation surfaces to establish a Gauss-Bonnet theorem in this sub-Riemannian setting.

Contribution

It compares two curvature definitions in Heisenberg space, proves their limits coincide, and applies this to establish a Gauss-Bonnet theorem via Riemannian approximation schemes.

Findings

Limits of Gaussian curvatures exist in Riemannian approximations

Curvatures defined by different methods do not coincide

Rotation surfaces of constant curvature are classified into three types

Abstract

We verify if Gausssian curvature of surfaces and normal curvature of curves in surfaces introduced by Diniz-Veloso arXiv:1210.7110 and by Balogh-Tyson-Vecchi arXiv:1604.00180 to prove Gauss-Bonnet theorems in Heisenberg space $\mathbb H^1$ are equal. The authors in arXiv:1604.00180 utilize a limit of Gaussian and normal curvatures defined in the Riemannian approximations scheme $(\mathbb R^3,g_L)$ introduced by Gromov to study sub-Riemannian spaces. 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 $(\mathbb R^3,g_L)$ when $L$ goes to infinity. The approach in arXiv:1210.7110 uses an analogous of Gauss map defined in the unitary cylinder and they show that the curvature defined by the Gauss map corresponds to the curvature of covariant derivative which is null on left invariant vector fields of $\mathbb H^1$. The proof of Gauss-Bonnet theorem follows as the classical one. Curvatures do not coincide. For comparison sake we apply the same formalism of arXiv:1210.7110 to get the curvatures of arXiv:1604.00180 . With the obtained formulas, it is possible to prove the Gauss-Bonnet theorem in arXiv:1604.00180 as a straightforward application of Stokes theorem. To exemplify the Gaussian curvature of arXiv:1604.00180 we calculate the rotation surfaces of constant curvature in $\mathbb H^1$, which are only of three types.