On null and pseudo null Mannheim curves in Minkowski 3-space

Grbović, Milica; İlarslan, Kazım; Nešović, Emilija

doi:10.1007/s00022-013-0205-z

Cited by 14 publications

(16 citation statements)

References 2 publications

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“…Then the Frenet formulas are

([], \begin{matrix} array \dot{t} \\ array \dot{n} \\ array \dot{b} \end{matrix}) = ([], \begin{matrix} array 0 & array κ & array 0 \\ array - ϵ κ & array 0 & array τ \\ array 0 & array τ & array 0 \end{matrix}) ([], \begin{matrix} array t \\ array n \\ array b \end{matrix}),

where g ( t , t )=1, g ( n , n )= ϵ =±1, g ( b , b )=− ϵ , g ( t , n )= g ( t , b )= g ( n , b )=0, κ and τ are the curvature and the torsion of α , respectively, and

^{.} = \frac{d}{d s}

. () Let α be a spacelike curve with a lightlike (null) principal normal n . Then the Frenet formulas are

([], \begin{matrix} array \dot{t} \\ array \dot{n} \\ array \dot{b} \end{matrix}) = ([], \begin{matrix} array 0 & array 1 & array 0 \\ array 0 & array τ & array 0 \\ array - 1 & array 0 & array - τ \end{matrix}) ([], \begin{matrix} array t \\ array n \\ array b \end{matrix}),

where g ( t , t )= g ( n , b )=1 and g ( b , b )= g ( t , n )= g ( t , b )= g ( n , b )=0.…”

Section: Preliminariesmentioning

confidence: 99%

“…In the last case, κ has only 2 values: κ =0, when α is a straight line and κ =1, in all other cases. () …”

Section: Preliminariesmentioning

confidence: 99%

“…Curves in Minkowski space have been studied by many mathematicians such as those in the literature. () In Minkowski space

E_{1}^{3}

, it is well known that for any unit speed curve

α : I \subset R \to E_{1}^{3}

with at least 4 continuous derivatives, one can associate 3 orthonormal Frenet frames { t , n , b }, consisting of the tangent, the principal normal, and the binormal vector fields, respectively. But sometimes the principal normal vectors are not defined at some points, which have zero curvature.…”

Section: Introductionmentioning

confidence: 99%

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Equiform spacelike normal curves according to equiform‐Bishop frame in

El-Sayied

Elzawy

Elsharkawy

2017

Math Methods in App Sciences

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In this article, we present the equiform parameter and define the equiform-Bishop frame in Minkowski 3-space E 3 1 . Additionally, we investigate the equiform-Bishop formulas of the equiform spacelike case in Minkowski 3-space.Furthermore, some results of equiform spacelike normal curves according to the equiform-Bishop frame in E 3 1 are considered. KEYWORDS equiform-Bishop frame, equiform curvatures, Minkowski Space, normal curves PRELIMINARIESThe Minkowski space E 3 1 is the space R 3 , equipped with the metric g, where g is given bywhere (x 1 , x 2 , x 3 ) is a coordinate system of E 3 1 . Let v be any vector in E 3 1 , then the vector v is spacelike, timelike, or null (lightlike) if g(v, v) > 0 or v = 0, g(v, v) < 0 or g(v, v) = 0, and v ≠ 0. The causal character of a vector in Minkowski space is the property to be spacelike, timelike, or null (lightlike).

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“…Then the Frenet formulas are

([], \begin{matrix} array \dot{t} \\ array \dot{n} \\ array \dot{b} \end{matrix}) = ([], \begin{matrix} array 0 & array κ & array 0 \\ array - ϵ κ & array 0 & array τ \\ array 0 & array τ & array 0 \end{matrix}) ([], \begin{matrix} array t \\ array n \\ array b \end{matrix}),

where g ( t , t )=1, g ( n , n )= ϵ =±1, g ( b , b )=− ϵ , g ( t , n )= g ( t , b )= g ( n , b )=0, κ and τ are the curvature and the torsion of α , respectively, and

^{.} = \frac{d}{d s}

. () Let α be a spacelike curve with a lightlike (null) principal normal n . Then the Frenet formulas are

([], \begin{matrix} array \dot{t} \\ array \dot{n} \\ array \dot{b} \end{matrix}) = ([], \begin{matrix} array 0 & array 1 & array 0 \\ array 0 & array τ & array 0 \\ array - 1 & array 0 & array - τ \end{matrix}) ([], \begin{matrix} array t \\ array n \\ array b \end{matrix}),

where g ( t , t )= g ( n , b )=1 and g ( b , b )= g ( t , n )= g ( t , b )= g ( n , b )=0.…”

Section: Preliminariesmentioning

confidence: 99%

“…In the last case, κ has only 2 values: κ =0, when α is a straight line and κ =1, in all other cases. () …”

Section: Preliminariesmentioning

confidence: 99%

“…Curves in Minkowski space have been studied by many mathematicians such as those in the literature. () In Minkowski space

E_{1}^{3}

, it is well known that for any unit speed curve

α : I \subset R \to E_{1}^{3}

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Equiform spacelike normal curves according to equiform‐Bishop frame in

El-Sayied

Elzawy

Elsharkawy

2017

Math Methods in App Sciences

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show abstract

“…1 .For an arbitrary curve α(s) in the space E 3 1 the following Frenet formulae are given in [2][3][4][5][6][7]. If α is timelike curve, then the Frenet formulae read If α is pseudo null curve, the Frenet formulas have the form…”

Section: Preliminariesmentioning

confidence: 99%

On The Modified Orthogonal Frame with Curvature and Torsion in 3-Space

Bükcü¹,

Karacan²

2016

Mathematical Sciences and Applications E-Notes

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“…Meanwhile, as is known, the Mannheim curves in nonflat space forms have two cases: one case is the nonnull Mannheim curves; the other case is the null Mannheim curves which are mainly considered in 3-dimensional de Sitter space S 3 1 . Grbović et al discussed the null Mannheim curves in 3dimensional Minkowski space [12]. We know that the case of null curve that is immersed in a three-dimensional de Sitter space is more sophisticated and interesting than nonnull curve in de Sitter space.…”

Section: Introductionmentioning

confidence: 99%

Mannheim Curves in Nonflat 3-Dimensional Space Forms

Zhao¹,

Pei

Cao³

2015

Advances in Mathematical Physics

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We consider the Mannheim curves in nonflat 3-dimensional space forms (Riemannian or Lorentzian) and we give the concept of Mannheim curves. In addition, we investigate the properties of nonnull Mannheim curves and their partner curves. We come to the conclusion that a necessary and sufficient condition is that a linear relationship with constant coefficients will exist between the curvature and the torsion of the given original curves. In the case of null curve, we reveal that there are no null Mannheim curves in the 3-dimensional de Sitter space.

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On null and pseudo null Mannheim curves in Minkowski 3-space

Cited by 14 publications

References 2 publications

Equiform spacelike normal curves according to equiform‐Bishop frame in

Equiform spacelike normal curves according to equiform‐Bishop frame in

On The Modified Orthogonal Frame with Curvature and Torsion in 3-Space

Mannheim Curves in Nonflat 3-Dimensional Space Forms

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