@article{f826d6f1f7b84d73bb788478f8ccb084,
title = "Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group",
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{\textquoteright}s formula for the Carnot–Carath{\'e}odory distance in H is provided.",
keywords = "Gauss–Bonnet theorem, Heisenberg group, Riemannian approximation, Steiner formula, Sub-Riemannian geometry",
author = "Balogh, {Zolt{\'a}n M.} and Tyson, {Jeremy T.} and Eugenio Vecchi",
note = "Funding Information: Zolt{\'a}n M. Balogh and Eugenio Vecchi were supported by the Swiss National Science Foundation Grant No. 200020-146477, and have also received funding from the People Programme (Marie Curie Actions) of the European Union{\textquoteright}s Seventh Framework Programme FP7/2007-2013/ under REA Grant Agreement No. 607643 (ERC Grant MaNET {\textquoteleft}Metric Analysis for Emergent Technologies{\textquoteright}). Jeremy T. Tyson acknowledges support from U.S. National Science Foundation Grants DMS-1201875 and DMS-1600650 and Simons Foundation Collaboration Grant 353627. Funding Information: Research for this paper was conducted during visits of the second and third authors to the University of Bern in 2015 and 2016. The hospitality of the Institute of Mathematics of the University of Bern is gratefully acknowledged. The authors would also like to thank Luca Capogna for many valuable conversations on these topics and for helpful remarks concerning the proof of Theorem 1.1. The authors would also like to thank the referee for a careful reading of the paper and for the numerous useful comments, ideas and suggestions which have improved the paper. Zolt{\'a}n M. Balogh and Eugenio Vecchi were supported by the Swiss National Science Foundation Grant No. 200020-146477, and have also received funding from the People Programme (Marie Curie Actions) of the European Union{\textquoteright}s Seventh Framework Programme FP7/2007-2013/ under REA Grant Agreement No. 607643 (ERC Grant MaNET {\textquoteleft}Metric Analysis for Emergent Technologies{\textquoteright}). Jeremy T. Tyson acknowledges support from U.S. National Science Foundation Grants DMS-1201875 and DMS-1600650 and Simons Foundation Collaboration Grant 353627. Publisher Copyright: {\textcopyright} 2016, Springer-Verlag Berlin Heidelberg.",
year = "2017",
month = nov,
day = "21",
doi = "10.1007/s00209-016-1815-6",
language = "English (US)",
volume = "287",
pages = "1--38",
journal = "Mathematische Zeitschrift",
issn = "0025-5874",
publisher = "Springer",
number = "1-2",
}