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

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

Research output: Contribution to journalArticlepeer-review

Abstract

We use a Riemannnian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean C2-smooth surface in the Heisenberg group H away from characteristic points, and a notion of intrinsic 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.

Original languageEnglish (US)
JournalMathematische Zeitschrift
Volume287
Issue number1-2
DOIs
StatePublished - Nov 21 2017

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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