Mannheim Curves in Nonflat 3-Dimensional Space Forms

Abstract

We consider the Mannheim curves in nonflat 3-dimensional space forms (Riemannian or Lorentzian) and we give the concept of Mannheim curves. In addition, we investigate the properties of nonnull Mannheim curves and their partner curves. We come to the conclusion that a necessary and sufficient condition is that a linear relationship with constant coefficients will exist between the curvature and the torsion of the given original curves. In the case of null curve, we reveal that there are no null Mannheim curves in the 3-dimensional de Sitter space.

Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere

“…Recently, mathematicians have paid attention to Bertrand and Mannheim curves in other spaces, such as in a three-dimensional sphere and in non-flat space form [9][10][11][12][13][14]. In the three-dimensional sphere, a Bertrand curve is a spherical curve whose principal normal geodesic is the same as the principal normal geodesic of another spherical curve.…”

Alternative partner curves in the Euclidean 3-space

“…These curves are de…ned by Mannheim with the equality 2 + 2 = w 2 =constant. Another characterization can be made as two curves and in E 3 which are called Manneim partner curves if the principal normal vector …elds of coincide with the binormal vector …elds of at the corresponding points of curves [5,6,12,14].…”

Bertrand-B Curves in the Three Dimensional Sphere

“…It satisfies τ = 0 or κ + aτ = b for constants a and b = 0 [7] as a necessary and sufficient condition. Recently, a definition of Mannheim curves in both Riemannian and Lorentzian space forms has also been given [8], [9]. All of the surveys mentioned above are done with reference to the Frenet-Serret frame which was adopted to formulate a space curve in ambient spaces.…”