A note on helices in the Euclidean space and in the hyperbolic space

Rafael López

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arXiv · 2507.12861

A note on helices in the Euclidean space and in the hyperbolic space

Rafael López

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Abstract

We prove that general helices in Euclidean space for Killing vector fields associated to rotations are helices, that is, curves with constant curvature and constant torsion. In hyperbolic space $\h^3$, we obtain the parametrization of helices for the Killing vector fields associated to hyperbolic rotations, spherical rotations and parabolic rotations. It is proved that these helices are geodesics in suitable surfaces of $\h^3$.

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Rafael López. 2025-07-17. A note on helices in the Euclidean space and in the hyperbolic space. https://arxiv.org/abs/2507.12861

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