Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group

  • Zoltán M. Balogh
  • Jeremy T. Tyson
  • Eugenio Vecchi


We use a Riemannnian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean \(C^{2}\)-smooth surface in the Heisenberg group \(\mathbb {H}\) away from characteristic points, and a notion of intrinsic 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–Carathéodory distance in \(\mathbb {H}\) is provided.


Heisenberg group Sub-Riemannian geometry Riemannian approximation Gauss–Bonnet theorem Steiner formula 

Mathematics Subject Classification

Primary 53C17 Secondary 53A35 52A39 

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Zoltán M. Balogh
    • 1
  • Jeremy T. Tyson
    • 2
  • Eugenio Vecchi
    • 3
  1. 1.Mathematisches InstitutUniversität BernBernSwitzerland
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA
  3. 3.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

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