Abstract
We apply the notions of Gaussian curvature and mean curvature to rotation surfaces in sub-Riemannian Heisenberg space \(\mathbb H^1\). In Diniz and Veloso (J Dyn Control Syst 22(4):807–820, 2016) we introduced a notion of Gaussian curvature, and here we classify rotation surfaces which are of constant Gaussian curvature. We study these surfaces rotating a horizontal curve \(\gamma \) around the z-axis. They have a resemblance to rotation surfaces of constant curvature in Euclidean space \(\mathbb R^3\). The rotation surfaces of constant mean curvature in \(\mathbb H^1\) are well known. The mean curvature of a rotation surface S in \(\mathbb H^1\) is the curvature of the curve in \(\mathbb R^2\) which is the projection of \(\gamma \), and we use this property to classify them.
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Veloso, J. . Rotation surfaces of constant Gaussian curvature and mean curvature in sub-Riemannian Heisenberg space \(\mathbb H^1\). J. Geom. 113, 38 (2022). https://doi.org/10.1007/s00022-022-00652-4
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DOI: https://doi.org/10.1007/s00022-022-00652-4
Keywords
- Sub-Riemannian geometry
- Heinsenberg space
- Gaussian curvature
- Mean curvature
- Rotation surfaces
- Surfaces of constant
- Gaussian curvature
- Surfaces of constant mean curvature