Akademik Veri Yönetim Sistemi
INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY, cilt.17, sa.2, ss.447-465, 2024 (ESCI, Scopus, TRDizin)
The purpose of this paper is to generalize definitions of Bertrand and Mannheim curves to nonnull framed curves and to non-flat three-dimensional (Riemannian or Lorentzian) space forms. Denote by Mnq (c) the n-dimensional space form of index q = 0, 1 and constant curvature c not equal 0. We introduce two types of framed Bertrand curves and framed Mannheim curves in M3q(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 M3q(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: A kappa + mu tau = 1 for Bertrand curves and A(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.