Gauss-Bonnet theorem in sub-Riemannian Heisenberg space H¹

2012/10/26 by José M. M. Veloso, Veloso, José M. M., Marcos M. Diniz +1 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #53C17 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Topological and Geometric Data Analysis #math.DG #msc:53C17

paper · pdf · doi:10.48550/arxiv.1210.7110

12 pages

arxiv created 2012/10/26 · openalex publication_date 2012/10/26 · arxiv updated 2012/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space H¹. The sub-Riemannian distance makes H¹ a metric space and consenquently with a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transverse to the tangent space of S. If all points of S have this property, we prove a Gauss-Bonnet formula and for compact surfaces (which are topologically a torus) we obtain ∫_S K = 0.

Cited by

• Curvature measures and the sub-Riemannian Gauss-Bonnet theorem
• Sub-Riemannian curvature and a Gauss-Bonnet theorem in the Heisenberg group

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