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Open Physics

formerly Central European Journal of Physics

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Volume 15, Issue 1

Issues

Volume 13 (2015)

Investigation of a curve using Frenet frame in the lightlike cone

Mihriban Kulahci
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  • Department of Mathematics, Firat University, 23119 Elaziğ, Turkey, Tel: +90 424 2370000/3659; Fax: +90 424 2330062
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Published Online: 2017-04-17 | DOI: https://doi.org/10.1515/phys-2017-0018

Abstract

Often times the language of mathematics is used to formulate physical theories. For example, as in this paper, while Minkowski space or the theory of special relativity were studying, their formulation was given by means of mathematical methods. In this manuscript, we study spacelike normal curves lying entirely in the 2-dimensional and 3-dimensional lightlike cone. In particular, some related theorems and definitions are also given. The study of representations of spacelike normal curves in the lightlike cone has led to the existence of different areas of mathematics and physics.

Keywords: Special relativity; spacetime; spacelike curve; lightlike cone

PACS: 02.40.Ma; 03.30.+p

1 Introduction

Physics and mathematics are very close disciplines. Physicists express many events in a geometric space, e.g. geometric space in Newtonian physics is the 3-dimensional Euclidean space E3. In special relativity, this space is referred to as spacetime and is basically dissimilar from the E3 space of Newtonian Physics [1, 2].

From the entirety of curves in spacetime there are three kinds of curves which are used in Special Relativity. Timelike curves, null curves and spacelike curves: timelike curves, which are the world lines of massive particles and which come with a preferred parameter, namely proper time; null geodesics, which are null rays or world lines of massless particles, and spacelike curves, which are everywhere neither timelike nor null. The set of all null geodesics through a point p generates a null (lightlike) cone, N(p): all timelike curves through p fill the inside of N(p), and all spacelike curves fill the outside of N(p). A tangent vector is a timelike, a null or a spacelike according to whether it is tangent to a timelike curve, a null curve, or a spacelike curve, respectively, [3, 4]. More precisely, if the lightlike cone structure of the spacetime is known, then an exhaustive catalog of which pairs of events can casually interact with one another in the spacetime can be constructed [5]. Therefore, lightlike cones are important for both mathematicians and physicists. Many studies have been done on curves in the lightlike cone by many mathematicians. For example, in [6], Liu considered curves in the null cone and in [7], Liu and Mong gave some formulas of curves in Q2 and Q3. Furthermore, Sun and Pei studied the null curves on Q3 and unit semi-Euclidean 3-spheres in [8].

We suppose that a spacelike curve stands for the field lines of diverse dynamical vector fields, e.g., the magnetic field. The analysis of the geometry of spacelike curves can be led to rewrite the dynamic equations of these fields in connection with geometry and then consider geometrical methods to overcome the physical problems [3, 4].

In terms of mathematics, curve theory has been an attraction for differential geometers and so it has been a widely studied topic. One of the most important tools used to analyze a curve is the Frenet frame. Frenet frames are a centrical structure in modern differential geometry, in which construction is explained with regards to an object of interest rather than with regards to exterior coordinate systems. Many studies on curves using frenet frames have been reported by many mathematicians [6, 7, 921].

In curve theory, there are some special curves such as rectifying, normal and osculating curves. In three dimensional Euclidean space, there exist three types of curves, so-called rectifying, normal and osculating curves satisfying the planes spanned by {T,B}, {N,B} and {T,N} besides to each unit speed curve α: IIRE3 whose orthogonal unit vector fields T, N, B, called respectively the tangent, the principal normal and the binormal vector fields. As it is given in [9], if position vector of a curve always lies in its rectifying plane, this curve can be called a rectifying curve in E3. More recently, besides in Euclidean space, many studies have been made in semi-Euclidean space: in [11] if the position vector of a curve always lies in its normal plane, this curve can be called a normal curve in Minkowski 3-space E13. Many studies related to spacelike, timelike and null normal curves, lying fully in the Minkowski 3-space, are given in [913]. Further, in [21], Yu and others define framed curves on Euclidean 2-sphere.

If the position vector of a curve always lies in its normal plane, this curve can be called a normal curve. The position vector field also plays an important role in physics, in particular, in mechanics. In any equation of motion, the position vector x (t) is usually the most sought-after quantity as the position vector field defines the motion of a particle (i.e. a point mass) from its location at some time variable, t. The first and the second derivatives of the position vector field with respect to time, t, give the velocity and acceleration of the particle [22].

In this paper, spacelike normal curves lying entirely in the 2-dimensional and 3-dimensional lightlike cone are considered. Furthermore, differential equations related to these curves are obtained and solved.

2 Spacelike Curves in the Lightlike Cone Qn+1

In the following, the notations and concepts from [6, 7] are used unless otherwise stated.

Denote Eqm as the m-dimensional pseudo-Euclidean space with the following metric G(x,y)=<X,Y>=i=1mqxiyij=mq+1mxjyj(2.1) where X = (x1,x2,…,xm),Y = (y1,y2,…,ym) ∈ Eqm,Eqm is a flat pseudo Riemannian manifold of signature (mq,q).

Assume that M is a submanifold of Eqm. If the pseudo Riemannian metric G of Eqm induces a pseudo-Riemannian metric G (respectively, a Riemannian metric, a degenerate quadratic form) on M, then M is a timelike (respectively, spacelike, degenerate) submanifold of Eqm.

Let c be a fixed point in Eqm r > 0 be an arbitrary constant. The pseudo -Riemannian sphere is defined as follows Sqn(c,r)={xEqn+1:G(xc,xc)=r2}; the pseudo-Riemannian hyperbolic space is defined as follows Hqn(c,r)={xEqn+1:G(xc,xc)=r2}; the pseudo-Riemannian null cone (quadratic cone) is defined as follows Qqn(c,r)={xEqn+1:G(xc,xc)=0}.

It is known that Sqn (c,r) is an entire pseudo Riemannian hypersurface of signature (nq,q), q ≥ 1 in Eqn+1 with a constant sectional curvature r− 2; Hqn (c,r) is a complete pseudo-Riemannian hypersurface of signature (nq,q), q ≥ 1 in Eqn+1 with a constant sectional curvature − r− 2; Qqn (c) is a degenerate hypersurface in Eqn+1. The spaces Eqn, Sqn(c,r), Hqn(c,r) and Qqn(c) are called pseudo-Riemannian space form. The point c is the center of Sqn(c,r), Hqn(c,r) and Qqn(c). When c = 0 and q = 1, we denote Q1n(0) by Qn and call it the lightlike or null cone.

Let E1n+2 be the (n + 2)-dimensional Minkowski space and Qn + 1 the lightlike cone in E1n+2. A vector α ≠ 0 in E1n+2 is spacelike, timelike or lightlike (null), if 〈α,α〉〉0, 〈α,α〉〈0 or 〈α,α〉 = 0, respectively. A frame field {e1,e2,…,en + 1,en + 2 on E1n+2 is an asymptotic orthonormal frame field, if en+1,en+1=en+2,en+2=0,en+1,en+2=1,en+1,ei=en+2,ei=0,ei,ej=δij,i,j=1,2,...,n.

We assume that the curve x: IQn + 1E1n+1, tx(t) ∈ Qn + 1, is a regular curve in Qn + 1. Henceforth, we always suppose that the curve is regular and x(t)=dx(t)dt, for all tI ⊂ ℝ.

Definition 2.1

A curve x(t) in E1n+2 is called a Frenet curve, if for all tI, the vector fields x(t),x′(t),x″(t),…,x(n)(t),x(n + 1)(t) are linearly independent and the vector fields x(t),x′(t),x″(t),…,x(n)(t),x(n + 1)(t),x(n + 2)(t) are linearly dependent, where x(n)(t)=dnx(t)dtn. Since 〈x,x〉 = 0 and 〈x, dx(t)dt = 0, dx(t)dt is spacelike, then the induced arc length(or simply the arc length), s, of the curve x(t) can be written as follows ds2=dx(t)dt,dx(t)dt.

If we consider the arc length, s, of the curve x(t) as the parameter and indicate x(t) = x(t(s)), then x(s)=dxds is a spacelike unit tangent vector field of the curve x(s). At present, we take the vector y(s), the spacelike normal space Vn − 1 of the curve x(s) such that they fulfill the following circumstances: x(s),y(s)=1,x(s),x(s)=y(s),y(s)=x(s),y(s)=0,Vn1={spanR{x,y,x}},SpanR{x,y,x,Vn1}=E1n+2, The functions κ and τ are defined as κ=12<x,x>(2.2) τ2=<x,x>4κ)2.(2.3)

The vectors x and y satisfy y=x12<x,x>x(2.4) we have <y,y>=<x,x>=<y,x>=0,<x,y>=1.(2.5)

3 Spacelike Normal Curves in the Lightlike Cone Q2

It is known that to each unit speed spacelike curve x = x(s): IQ2E13, one can associate a pseudo orthonormal frame {x,α,y}. In this position, the Frenet Frame for unit speed spacelike curve x = x(s): IQ2E13 are given as x(s)=α(s)α(s)=κ(s)x(s)y(s)y(s)=κ(s)α(s),(3.1) where s is an arc length parameter of the curve and x(s), y(s), α(s) satisfy <x,x>=<y,y>=<x,α>=<y,α>=0,<x,y>=<α,α>=1.(3.2)

By definition, for a spacelike normal curve in the 2-dimensional lightlike cone, the position vector x satisfies x(s)=λ(s)x+μ(s)α(3.3) for some differentiable functions λ and μ.

Theorem 3.1

Denote x as a unit speed spacelike normal curve in Q2. Then the coefficients of x and α vector fields of curve x are, respectively, λ=1μ=0 or λ(s)=1c1KeKs+c2KeKsμ(s)=c1eKs+c2eKs(3.4) or λ(s)=1aKsinh(Ks)+bKcosh(Ks)μ(s)=acosh(Ks)+bsinh(Ks)(3.5) where c1=a+b2,c2=ab2, a,b,c1,c2 ϵ IR.

Proof

Let x be a unit speed spacelike normal curve in Q2. Differentiating (3.3) with respect to s and using (3.1) finds α=(λ+μκ)x+(λ+μ)αμy.(3.6)

Exposing the inner product y, α, x to the both side of (3.6) respectively, finds λ+μκ=0λ+μ=1μ=0.(3.7)

From (3.7), we have μ = 0 and λ = 1. Furthermore, making necessary calculations in (3.7), we can write μμκ=0.(3.8)

Solving (3.8), we get μ=c1eKs+c2eKs.(3.9)

Using (3.7), we have λ=1c1KeKs+c2KeKs.(3.10)

Additionally, considering the definition of hyperbolic functions sinh s and cosh s, we obtain (3.5). Thus the theorem is proved. □

Theorem 3.2

Denote x as a unit speed spacelike normal curve in Q2. If the following statements are provided, x is a normal curve so that the curvature of x curve is κ = 1.

  1. the distance function ρ satisfies ρ2(x(s)) = λ2(s) − 2λ(s) + d, d ϵ IR and for λ = 1, ρ is constant.

  2. the components of vector fields x and α of curve are respectively given by x(s),y(s)=λ(s)=1c1es+c2esx(s),α(s)=μ(s)=c1es+c2es.(3.11)

  3. the normal component xN of x is constant.

Proof

Suppose that x is a unit speed spacelike normal curve in Q2 and x is a normal curve such that the curvature of x curve is κ = 1. The position vector x satisfies (3.3). If λ and μ hold (3.9) and (3.10), also using (3.2), then we have ρ2 (x(s)) = ∥x(s)∥2 = μ2 and multiplying with μ the both sides of second equation in (3.7), we get μ2(s)=λ2(s)2λ(s)+d.(3.12)

Considering (3.2) in (3.3), we can obtain λ = 1. From (3.12), we have ρ2(x(s))=d1.

Since dIR, ρ = constant. Thus (1) is proved.

Conversely, suppose that (1) holds. So, let ρ = constant. Differentiating twice ρ2(s) = ∥x(s)∥2 = constant with respect to s, we obtain κ(s)〈x(s),x(s)〉 = 0. Since κ(s) ≠ 0, 〈x(s),x(s)〉 = 0. Thus x is a normal curve.

For the proof of (2), from (3.3), (3.9) and (3.10), respectively, we have x(s),y(s)=λ(s)=1c1es+c2esx(s),α(s)=μ(s)=c1es+c2es.

Conversely, differentiating twice 〈x(s),y(s)〉 = λ(s) with respect to s and considering (3.1), we get κ2(s)〈x(s),x(s)〉 = 0. Since κ2(s) ≠ 0, 〈x(s),x(s)〉 = 0. Thus x is a normal curve. Taking the derivative 〈x(s),α(s)〉 = μ(s), we see κ(s)〈x(s),x(s)〉 = 0. Since κ(s) ≠ 0, 〈x(s),x(s)〉 = 0 and so x is a normal curve. Thus the proof of (2) is completed.

For the proof of (3), the normal component xN of x curve is μα. Hence xN = μ. From (3.12), for λ = 1 we have μ2 = d − 1. Since dIR, xN = constant. Thus (3) is proved.

Conversely, suppose that (3) holds. From (3.3), we can write x(s),x(s)(1λ)=μ(s)x(s),α(s).

Considering (3.2) in (3.3), we can obtain λ = 1. Then we have 0=μ(s)x(s),α(s).

Further, from the second equation of (3.11), for 〈x(s),α(s)〉 = μ(s) we have 0=x(s),α(s)2.

By differentiating the above equation, we find 0=2κ(s)x(s),α(s)x(s),x(s).

Again taking the derivative, we obtain 0=2κ2(s)x(s),x(s)2.

Hence 〈x(s),x(s)〉 = 0. This means that x is a normal curve.     □

Theorem 3.3

Denote x as a unit speed spacelike normal curve in Q2. Then x lies on a lightlike cone Q2(m) with a vertex at m if and only if x is congruous to a normal curve by λ(s)1x(s)+μ(s)α(s)=0(3.13) or aKsinh(Ks)+bKcosh(Ks)x(s)+acosh(Ks)+bsinh(Ks)α(s)=0.(3.14)

Proof

From (3.3), let’s write m=xλxμα.(3.15)

By differentiating (3.15), we get m=1λμαλ+κμxμy.

From (3.7), m′ = 0. Thus m = constant. This means that x is congruous to a normal curve.

Further xm,xm=λx+μα,λx+μαxm,α=0.(3.16)

Hence x lies precisely on a lightlike cone Q2 and since 〈xm,α〉 = 0, xmspan{α}, which means that x is a normal curve.

Furthermore, from 〈xm,xm〉′ = 0, we have ∥xm∥ = cons., which means that x lies on lightlike cone Q2 with vertex at m.

Let’s consider a pseudo orthonormal frame {x,α,y,} in Q2. Then we can write xm=ax+bα+cy,a,b,cIR0+.(3.17)

Exposing the inner product with x,α,y to (3.17), we find xm,x=c,xm,α=b,xm,y=a.

From (3.16), we have 1+κxm,xxm,y=0.(3.18)

Exposing the inner product with x,α,y to both sides of (3.15), we get xm,y=λ,xm,α=μ.

Since 〈xm,α〉 = 0, we obtain μ = 0 and 〈xm,y〉 = 1 . In that case, a = 1, b = 0, c = 0. Hence m = 0 and finally 〈xm,xm〉 = 0, which means that x lies on a lightlike cone Q2.

Further, considering m = 0 in (3.15), we find (3.13) and (3.14). This completes the proof. □

4 Spacelike Normal Curves in the Lightlike Cone Q3

It is known that to each unit speed spacelike curve x = x(s): IQ3E14, one can associate a pseudo orthonormal frame {x,α,y,β}. In this condition, the Frenet Frame for unit speed spacelike curve x = x(s): IQ3E14 are given as x=αα=κxyβ=τxy=κατβ.(4.1)

Let α = x′ and choose β such that det(x,α,β,y)=1.(4.2)

From [6], for any asymptotic orthonormal frame {x,α, y,β} of the unit speed spacelike curve x = x(s): IQ3E14 with <x,α>=<x,β>=<y,α>=<y,β>=<α,β>=0,<α,α>=<β,β>=x,y=1.(4.3)

The frame field {x,α,y,β}} is the cone frenet frame of the curve x(s).

Definition 4.1

Let the functions κ and τ in (4.1) be the (first) cone curvature and cone torsion of the spacelike curve x in Q3E14.

By definition, for a spacelike normal curve in the 3-dimensional lightlike cone, the position vector x satisfies x(s)=λ(s)x(s)+μ(s)α(s)+γ(s)β(s)(4.4) for some differentiable functions λ, μ and β.

Theorem 4.1

Denote x be a unit speed spacelike normal curve with curvatures κ,τ ≠ 0 in Q3. Then the coefficients of curve x is as follows: γ=cμ=0λ=1, or λ=1c1KeKs+c2KeKsμ=c1eKs+c2eKscτκγ=c(4.5) or λ=1aKsinh(Ks)+bKcosh(Ks)μ=acosh(Ks)+bsinh(Ks)cτκγ=c(4.6) where c1=a+b2,c2=ab2, a,b,c,c1,c2 ϵ IR.

Proof

Let x be a unit speed spacelike normal curve in Q3 and κ,τ be cone curvature functions of x. Differentiating (4.4) with respect to s and using (4.1), we have α=λ+μκ+γτxμy+λ+μα+γβ.(4.7)

Exposing the inner product y,x,α,β to the both sides of (4.7), respectively, we find λ+μκ+γτ=0μ=0λ+μ=1γ=0.(4.8)

Making the necessary mathematical operations, we get γ=cμ=0λ=1,(4.9) where cIR.

Again making necessary calculations in (4.8), we can write the following differential equation: μμκ=cτ.(4.10)

Solving (4.10), we obtain (4.5) and considering the definition of hyperbolic functions sinh s and cosh s, we have (4.6). Thus the proof is completed.     □

Theorem 4.2

Denote x be a unit speed spacelike normal curve in Q3. If the following statements are provided, x is a normal curve so that the curvature of curve x is κ, τ ≠ 0.

  1. the distance function d = constant.

  2. the components of vector fields x, α and β of curve are respectively given by x(s),y(s)=λ(s)=1c1KeKs+c2KeKsx(s),α(s)=μ(s)=c1eKs+c2eKscτκx(s),β(s)=γ(s)=c.(4.11)

  3. the normal component xN of x is constant.

Proof

Suppose that x is a unit speed spacelike normal curve in Q3 and x is a normal curve such that its curvature is κ, τ ≠ 0. The position vector, x, satisfies (4.4). If λ and μ hold (4.5), then we have d2(s)=xs2=μ2+γ2.

From (4.8), since μ = 0, γ = c, c ∈ ℝ0 then d = c = cons.

Furthermore, multiplying with μ′ and γ both sides of the second and fourth equations of (4.8) respectively, we get μ2=m2γ2=n2,(4.12) where m ∈ ℝ and n ∈ ℝ0. From (4.12), we find d2(x(s))=xs2=μ2+γ2=m2+n2,(4.13) which means that d2 = m2 + n2 = cons. Thus (1) is proved.

Conversely, suppose that (1) holds. Then be d is constant. Differentiating the second time d2(x(s)) = ∥x(s)∥2 = constant with respect to s, we obtain κx(s),x(s)〉 = 0. Since κ ≠ 0, 〈x(s),x(s)〉 = 0. This means that x is a normal curve.

For the proof of (2), from (4.6) and (4.4), respectively, we have (4.11).

Conversely, differentiating the second time 〈x(s),y(s)〉 = λ(s) and using (4.1), we find (κ2+τ2)〈x(s),x(s)〉 = 0. For κ,τ ≠ 0, since κ2 + τ2 ≠ 0, 〈x(s),x(s) 〉 = 0. This means that x is a normal curve.

Again differentiating 〈x(s),β(s)〉 = γ(s), we have 〈x(s),x(s)〉 = 0. Hence x is a normal curve. Thus (2) is proved.

For the proof of (3), the normal component xN of curve x is μα + γβ. Hence xN2 = μ2 + γ2. From (4.13), we have μ2 + γ2 = m2 + n2. Since m ∈ ℝ and n ∈ ℝ 0, xN2 = m2+n2 = constant. Thus (3) is proved.

Conversely, suppose that (3) holds. From (4.4), we can give xs=λsxs+xNsxs=λsxs+μsαs+γsβsxs,xs1λs=μsxs,αs+γsxs,βsxs,xs1λs=xNs,xs.

Using (4.9), we get 0=0+cxs,βs.(4.14)

Differentiating (4.14), we have 0 = cτx(s),x(s)〉. Since cτ ≠ 0, 〈x(s),x(s)〉 = 0. This means that x is a normal curve. Thus the theorem is proved.     □

Theorem 4.3

Denote x be a unit speed spacelike normal curve in Q3. Then x lies on lightlike cone Q3(m) with vertex at m if and only if x is congruous to a normal curve by λs1xs+μsαs+γsβs=0.(4.15)

Proof

From (4.4), let’s write m=xsλsxsμsαsγsβs,(4.16) differentiating (4.16), we obtain m=1λsμsαsγsβsλs+κsμs+τsγsxsμsys, from (4.8), m′ = 0. Thus m = constant, which means that x is congruous to a normal curve.

Further xsm,xsm=0xsm,αs=0.(4.17)

Hence x lies precisely on a lightlike cone Q3 and since 〈x(s) − m,α(s)〉 = 0, x(s) − mspan{α} , which means that x is a normal curve.

Furthermore, from 〈x(s) − m,x(s) − m〉′ = 0, we have ∥x(s) − m∥ = cons., which means that x lies on lightlike cone Q3 with a vertex at m.

Let’s consider a pseudo orthonormal frame {x,α,β,y} in Q3. Then we can write xm=ax+bα+cy+dβ,a,b,c,dIR0+.(4.18)

Exposing the inner product with y,α,x,β to (4.18), we have xm,y=axm,α=bxm,β=dxm,x=c.

Differentiating 〈xm,α〉 = b, we obtain 1+κxm,xxm,y=0.(4.19)

Exposing the inner product with y,α,β to both sides of (4.16), we get xm,y=λ=1xm,α=μ=0xm,β=γ=c.(4.20)

Substituting (4.19) in (4.16), we find xm=x+cβ.(4.21)

Therefore, considering the first equation of (4.20) in (4.19), we have κxm,x〉 = 0. Since κ ≠ 0, 〈xm,x〉 = 0 = c. Since x normal curve lies on a lightlike cone Q3(m), we can write 〈x(s) − m,x(s) − m〉 = 0 and since 〈x(s) − m,x(s) − m〉 = 〈cβ,cβ〉 = c2, we get c = 0. Hence we have xm = x, therefore we get m = 0. Thus the theorem is proved.     □

5 Conclusions

In summary, we examine the conditions when spacelike curves are normal curves in the lightlike cone. Spacelike curves play an elementary role in special relativity and the search for a spacelike curve in the lightlike cone will continue to receive attention in geometrical methods for sometime.This investigation can be considered for other kinds of curves in the lightlike cone. Also, it is valuable to learn more about curves and their related differential equations, we expect that these results can be useful in future studies. Attempts to comprehend the exact relation between theories in mathematics and physics has led to a great deal of investigation.

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About the article

Received: 2016-09-06

Accepted: 2017-02-01

Published Online: 2017-04-17


Citation Information: Open Physics, Volume 15, Issue 1, Pages 175–181, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0018.

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© 2017 Mihriban Kulahci. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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