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arXiv · 1210.7110

Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$

José M. M. Veloso·Marcos M. Diniz
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Abstract

We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$. The sub-Riemannian distance makes $H^1$ 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 $\int_S K = 0$.

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José M. M. Veloso, Marcos M. Diniz. 2012-10-26. Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$. https://arxiv.org/abs/1210.7110

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