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arXiv · 1604.00180

Sub-Riemannian curvature and a Gauss-Bonnet theorem in the Heisenberg group

Zoltán Balogh·Jeremy T. Tyson·Eugenio Vecchi
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Abstract

We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carath\'eodory distance in $\mathbb{H}$ is provided.

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Zoltán Balogh, Jeremy T. Tyson, Eugenio Vecchi. 2016-04-01. Sub-Riemannian curvature and a Gauss-Bonnet theorem in the Heisenberg group. https://arxiv.org/abs/1604.00180

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