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

Balogh, Zoltan; Tyson, Jeremy; Vecchi, Eugenio (2017). Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group. Mathematische Zeitschrift, 287(1-2), pp. 1-38. Springer 10.1007/s00209-016-1815-6

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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.

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