Bertrand Curves in Riemannian Space

Pears, L. R.

doi:10.1112/jlms/s1-10.2.180

Cited by 20 publications

(14 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When we investigate the properties of Bertrand curves in Euclidean n -space, it is easy to see that either k 2 or k 3 is zero, which means that Bertrand curves in E n (n > 3) are degenerate curves [20]. This result was restated by Matsuda and Yorozu [17].…”

Section: Introductionmentioning

confidence: 87%

Generalized Bertrand Curves with Spacelike $\left(1,3\right) $-Normal Plane in Minkowski Space-Time

Uçum¹,

Keçilioğlu²,

İlarslan³

2016

Turk J Math

View full text Add to dashboard Cite

Abstract:In this paper, we reconsider the (1, 3) -Bertrand curves with respect to the casual characters of a (1, 3) -normal plane that is a plane spanned by the principal normal and the second binormal vector fields of the given curve. Here, we restrict our investigation of (1, 3) -Bertrand curves to the spacelike (1, 3) -normal plane in Minkowski space-time. We obtain the necessary and sufficient conditions for the curves with spacelike (1, 3) -normal plane to be (1, 3) -Bertrand curves and we give the related examples for these curves.

show abstract

Section: Introductionmentioning

confidence: 87%

Generalized Bertrand Curves with Spacelike $\left(1,3\right) $-Normal Plane in Minkowski Space-Time

Uçum¹,

Keçilioğlu²,

İlarslan³

2016

Turk J Math

View full text Add to dashboard Cite

show abstract

“…In [10], Pears studied this problem for curves in the n-dimensional Euclidean space E n , n > 3, and showed that a Bertrand curve in E n must belong to a three-dimensional subspace E 3 ⊂ E n . This result is restated by Matsuda and Yorozu [9].…”

Section: Introductionmentioning

confidence: 99%

On Timelike Bertrand Curves in Minkowski 3-Space

Uçum¹,

İlarslan²

2016

Honam Mathematical Journal

View full text Add to dashboard Cite

Abstract. In this paper, we study the timelike Bertrand curves in Minkowski 3-space. Since the principal normal vector of a timelike curve is spacelike, the Bertrand mate curve of this curve can be a timelike curve, a spacelike curve with spacelike principal normal or a Cartan null curve, respectively. Thus, by considering these three cases, we get the necessary and sufficient conditions for a timelike curve to be a Bertrand curve. Also we give the related examples.

show abstract

“…Pears [21] studied this problem for curves in the n-dimensional Euclidean space R n , n > 3, and showed that a Bertrand curve in R n must belong to a three-dimensional subspace R 3 ⊂ R n (see also [1, p. 176] and [18]). Many authors have studied Bertrand curves in other ambient spaces ( [8], [10], [11], [13], [16], [20], [23]).…”

Section: Introductionmentioning

confidence: 99%

Bertrand Curves in Non-Flat 3-Dimensional (Riemannian or Lorentzian) Space Forms

Lucas¹,

Ortega-Yagües²

2013

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. Let M 3 q (c) denote the 3-dimensional space form of index q = 0, 1, and constant curvature c = 0. A curve α immersed in M 3 q (c) is said to be a Bertrand curve if there exists another curve β and a one-to-one correspondence between α and β such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: nonnull Bertrand curves in M 3 q (c) correspond with curves for which there exist two constants λ = 0 and µ such that λκ + µτ = 1, where κ and τ stand for the curvature and torsion of the curve. As a consequence, non-null helices in M 3 q (c) are the only twisted curves in M 3 q (c) having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.

show abstract

Bertrand Curves in Riemannian Space

Cited by 20 publications

References 0 publications

Generalized Bertrand Curves with Spacelike $\left(1,3\right) $-Normal Plane in Minkowski Space-Time

Generalized Bertrand Curves with Spacelike $\left(1,3\right) $-Normal Plane in Minkowski Space-Time

On Timelike Bertrand Curves in Minkowski 3-Space

Bertrand Curves in Non-Flat 3-Dimensional (Riemannian or Lorentzian) Space Forms

Contact Info

Product

Resources

About