Abstract
Based on the fundamental theories of null curves in Minkowski 3-space, the null Darboux mate curves of a null curve are defined which can be regarded as a kind of extension for Bertrand curves and Mannheim curves in Minkowski 3-space. The relationships of null Darboux curve pairs are explored and their expression forms are presented explicitly.
Highlights
The associated curves or the curve pairs, i.e., two curves related to each other at the corresponding points, play important roles in the curve theory of differential geometry
Taking Euclidean 3-space as an example, a Bertrand curve shares its normal line with another curve and its curvature κ, torsion τ satisfy λκ + μτ = 1 for some constants λ and μ [1]; the principal normal line of a Mannheim curve coincides with the binormal line of another curve and its curvature κ, torsion τ satisfy κ = λ(κ2 + τ2) for some constant λ [2]
Many mathematicians extended the notions of curve pairs, such as Bertrand curve, Mannheim curve, evolute and involute and so on from Euclidean space to Lorentz–Minkowski space [3,4,5]
Summary
The associated curves or the curve pairs, i.e., two curves related to each other at the corresponding points, play important roles in the curve theory of differential geometry. The most fascinating examples are Bertrand curves and Mannheim curves in three-dimensional space. The Darboux vector comes to mind naturally when we consider the fact that most curve pairs are proposed from the frame of a space curve. For a curve r(s) framed by {T(s), N(s), B(s)} in three-dimensional space, the Darboux vector D(s) is the axis around which the Frenet frame rotates when r(s) does real-time spirals, and D(s) satisfies Darboux equations as follows. Motivated by the definitions of Bertrand curve and Mannheim curve, we can consider another kind of associated curve by setting a condition that two space curves share the same Darboux vector field at the corresponding points in Minkowski 3-space. The null Darboux curve pairs in three-dimensional Minkowski space are investigated. All geometric objects are smooth and regular unless otherwise stated
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