Framed Bertrand and Mannheim curves in three-dimensional space forms of non-zero constant curvatures

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 Mqⁿ (c) the n-dimensional space form of index q=0, 1 and constant curvature c 0. We introduce two types of framed Bertrand curves and framed Mannheim curves in Mq³ (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 Mq³ (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: + =1 for Bertrand curves and (²+²) = 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.