Framed Bertrand and Mannheim Curves in Three-Dimensional Space Forms of Non-zero Constant Curvatures

Year 2024, Volume: 17 Issue: 2, 447 - 465, 27.10.2024

Orhan Oğulcan Tuncer

https://doi.org/10.36890/iejg.1440270

Abstract

The purpose of this paper is to generalize definitions of Bertrand and Mannheim curves to non-null framed curves and to non-flat three-dimensional (Riemannian or Lorentzian) space forms. Denote by $\mathbb{M}_q^n(c)$ the $n$-dimensional space form of index $q=0,1$ and constant curvature $c\neq 0$. We introduce two types of framed Bertrand curves and framed Mannheim curves in $\mathbb{M}_q^3(c)$ by using two different moving frames: the general moving frame and the Frenet-type frame. We investigate geometric properties of these framed Bertrand and framed Mannheim curves in $\mathbb{M}_q^3(c)$ that may have singularities. We then give characterizations for a non-null framed curve to be a framed Bertrand curve or to be a framed Mannheim curve. We show that in special cases these characterizations reduce to the well-known classical formulas: $\lambda \kappa+\mu \tau=1$ for Bertrand curves and $\lambda(\kappa^2+\tau^2)=\kappa$ for Mannheim curves. We provide several examples to support our results, and we visualize these examples by using the Hopf map, the hyperbolic Hopf map, and the spherical projection.

Keywords

Bertrand curves, Mannheim curves, framed curves, singularity, non-flat space forms, Hopf map

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