Abstract
We define a generalized lightlike Bertrand curve pair and a generalized non-lightlike Bertrand curve pair, discuss their properties and prove the necessary and sufficient condition of a curve which is a generalized lightlike or a generalized non-lightlike Bertrand curve. Moreover, we study the relationship between slant helices and generalized Bertrand curves.
Highlights
IntroductionThe Bertrand curve is widely studied by many mathematicians in classical differential geometry
The Bertrand curve is widely studied by many mathematicians in classical differential geometry.It was first proposed by Bertrand who supported that a curve is a Bertrand curve in R3 if and only if the curvature κ and the torsion τ satisfy ηκ + μτ = 1, where η 6= 0, μ are constants
For non-flat space, such as the 3-dimensional sphere S3, Lucas and Yagües defined a new Bertrand curve [7,8]. They considered the correspondence of the principal normal geodesics by using the tools of connection, and gave the relationship between (1, 3)-type Bertrand curve in R4 and the Bertrand curve on 3-dimensional sphere
Summary
The Bertrand curve is widely studied by many mathematicians in classical differential geometry. It was first proposed by Bertrand who supported that a curve is a Bertrand curve in R3 if and only if the curvature κ and the torsion τ satisfy ηκ + μτ = 1, where η 6= 0, μ are constants. In [4], some properties of the non-lightlike curve in 3-dimensional Lorentz space were given. For non-flat space, such as the 3-dimensional sphere S3 , Lucas and Yagües defined a new Bertrand curve [7,8]. Sun and some others introduced the properties of non-lightlike curves in Minkowski 3-space. We mainly study the the generalization of Bertrand curves in Minkowski 3-space. We suppose here that all manifolds and maps are smooth
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