Abstract
We study the singular Bertrand curves and Mannheim curves in the 3-dimensional space forms. We introduce the geometrical properties of such special curves. Moreover, we get the relationships between singularities of original curves and torsions of another mate curves.
Highlights
Bertrand curves as special curves have been deeply explored in Euclidean space; see [1]
Riemannian space forms [6] and in non-flat 3-dimensional space forms [7,8]. Mannheim curves as another kind of special curves are broadly concerned
Mannheim curves have been studied in the 3-dimensional Riemannian space forms [10] and in non-flat 3-dimensional space forms [11]
Summary
Bertrand curves as special curves have been deeply explored in Euclidean space; see [1]. Riemannian space forms [6] and in non-flat 3-dimensional space forms [7,8] Mannheim curves as another kind of special curves are broadly concerned. Mannheim curves have been studied in the 3-dimensional Riemannian space forms [10] and in non-flat 3-dimensional space forms [11]. For the regular Bertrand and Mannheim curves, Takahashi and Honda found that the existence condition is not sufficient. In [14], the authors added the non-degenerate condition when proving a regular curve is a Bertrand or Mannheim curve. They discussed a framed curve in R3 , under what conditions, can be either a Bertrand or Mannheim curve. We assume here that all maps and manifolds are C ∞ unless otherwise stated
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