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Constant curvature surfaces in a pseudo-isotropic space. (English) Zbl 1405.53015

Summary: In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space \(\mathbb{I}^3_p\) that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron for spacelike and timelike curves, respectively. The causal character of all admissible surfaces in \(\mathbb{I}^3_p\) has to be timelike up to its absolute. We introduce the formulas of Gaussian and mean curvature for timelike surfaces in \(\mathbb{I}^3_p\). As applications, we describe the surfaces of revolution which are the orbits of a plane curve under a hyperbolic rotation with constant Gaussian and mean curvature.

MSC:

53A35 Non-Euclidean differential geometry
53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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