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

2016/04/01 by Zoltán M. Balogh, Balogh, Zoltán, Jeremy T. Tyson +3
Mathematics · Medicine · Physics and Astronomy · #52A39 #53A35 #53C17 #Advanced Differential Geometry Research #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1604.00180

openalex publication_date 2016/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use a Riemannnian approximation scheme to define a notion of sub-Riemannian Gaussian curvature for a Euclidean C²-smooth surface in the Heisenberg group ℍ away from characteristic points, and a notion of sub-Riemannian signed geodesic curvature for Euclidean C²-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-Carathéodory distance in ℍ is provided.

Citations

• The fundamental theorems for curves and surfaces in 3d Heisenberg group
• Gauss-Bonnet theorem in sub-Riemannian Heisenberg space H¹

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