General properties of Bertrand curves in Riemann–Otsuki space

Yılmaz, Münevver Yıldırım; Bektaş, Mehmet

doi:10.1016/j.na.2007.10.003

Cited by 18 publications

(5 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3-dimensional Euclidean space E 3 , 3-dimensional sphere S 3 and 3-dimensional hyperbolic space H 3 by using the curvature functions of the curve. Moreover, it is done in other spaces, such as in Riemann-Otsuki spaces [34]. In [9], S. Honda and M. Takahashi studied Frenet type framed Bertrand curves in E 3 .…”

Section: Introductionmentioning

confidence: 99%

New approach to timelike Bertrand curves in 3-dimensional Minkowski space

Erdem,

Uçum,

İlarslan

et al. 2023

Carpathian Math. Publ.

View full text Add to dashboard Cite

In the theory of curves in Euclidean $3$-space, it is well known that a curve $\beta $ is said to be a Bertrand curve if for another curve $\beta^{\star}$ there exists a one-to-one correspondence between $\beta $ and $\beta^{\star}$ such that both curves have common principal normal line. These curves have been studied in different spaces over a long period of time and found wide application in different areas. In this article, the conditions for a timelike curve to be Bertrand curve are obtained by using a new approach in contrast to the well-known classical approach for Bertrand curves in Minkowski $3$-space. Related examples that meet these conditions are given. Moreover, thanks to this new approach, timelike, spacelike and Cartan null Bertrand mates of a timelike general helix have been obtained.

show abstract

Section: Introductionmentioning

confidence: 99%

New approach to timelike Bertrand curves in 3-dimensional Minkowski space

Erdem,

Uçum,

İlarslan

et al. 2023

Carpathian Math. Publ.

View full text Add to dashboard Cite

show abstract

“…Bertrand curves and their characterizations were studied by many researchers in Minkowski 3-space and Minkowski space-time (see [1,2,8,10,12,13,22,23] as well as in Euclidean space. In addition, there are some other studies about Bertrand curves such as [9,14,16,19,24].…”

Section: Introductionmentioning

confidence: 99%

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

“…In [14], Pears extended the well-known properties of Bertrand curves in R 3 to curves in the n-dimensional Euclidean space R n . Also, many authors have studied Bertrand curves in other ambient spaces(endoed with Riemannian or pseudo-Riemannian metric): in the three dimensional Lorentz-Minkowski space R 3 1 [3], [4], in Riemann-Otsuki space [16], in the Galilean space [13], in space forms with sectional curvature c [5], and so on.…”

Section: Introductionmentioning

confidence: 99%

Classiffications of special curves in the three-dimensional Lie group

Lee¹,

Lee²

2016

ijma

View full text Add to dashboard Cite

In the present paper, we define a Bertrand curve in the three-dimensional Lie group G with a bi-invariant metric, and we show a Frenet curve α with Frenet curvatures k 1 and k 2 in G is a Bertrand curve if and only if it satisfies Ak 1 + B(k 2 +k 2) = 1, where A and B are some constants andk 2 = 1/2 [V 1 , V 2 ], V 3. Also, we investigate a Bertrand curve using the Frenet curvature conditions of AW(k)-type (k = 1, 2, 3) curves in G.

show abstract

General properties of Bertrand curves in Riemann–Otsuki space

Cited by 18 publications

References 3 publications

New approach to timelike Bertrand curves in 3-dimensional Minkowski space

New approach to timelike Bertrand curves in 3-dimensional Minkowski space

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

Classiffications of special curves in the three-dimensional Lie group

Contact Info

Product

Resources

About