A study on null cartan curve in Minkowski 3-space

Muhammad Abubakar Isah 1  and Mihriban Alyamaç Külahcı 1
  • 1 Department of Mathematics, Elazig, Turkey
Muhammad Abubakar Isah and Mihriban Alyamaç Külahcı

Abstract

Null cartan curves have been studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curves are not considered. In this paper, we study weak AW (k) – type and AW (k) – type null cartan curve in Minkowski 3-space E13 . We define helix according to Bishop frame in E13 . Furthermore, the necessary and sufficient conditions for the helices in Minkowski 3-space are obtained.

1 Introduction

Curves are one of the basic structures of differential geometry. It is safe to report that G. Monge initiated the many important results in Euclidean 3-space curve theory and G. Darboux pioneered the idea of a moving frame.

The curve theory has been one of the most studied subjects because of its many applications area from geometry to the various branch of science. Especially the characteristics of curvature and torsion play an important role in special curve types such as so-called helices. In Euclidean 3-space E3, a general helix or a constant slope curve is defined in such a way that the tangent makes a constant angle with a fixed direction. A classical result stated by M. A Lancret in 1802 and first demonstrated by B. de Saint Venant in 1845 [5,6,11,12]. For nature’s helical structures, helices arise in nano-springs, carbon nano-tubes, helices, DNA double and collagen triple helix, lipid bilayers, bacterial flagella in salmonella and escherichia coli, aerial hyphae in actinomycetes, bacterial shape in spirochetes, horns, tendrils, vines, screws, springs, helical staircases and shells of the sea [1,2,9]. In fractal geometry, helical structures are used.

In Minkowski 3-space, null Cartan curves are known as the curves whose and Cartan frame contains two null (lightlike) vector fields (see [7] for more information). Null curves of AW(k)-type are studied in the 3-dimensional Lorentzian space by M. Külahcı [8].

In [4] B. Bukcu and M. K. Karacan defined a slant helix according to Bishop frame of the timelike curve and they have given some necessary and sufficent conditions for the slant helix. Ahmad T. Ali and Rafael Lopez gave characterizations of slant helices in terms of the curvature and torsion and discussed the tangent and binormal indicatrices of slant curves in E13 [3,10,13].

F. Gökçelik and I.Gök defined a new kind of slant helix called W-slant helix in 3-dimensional Minkowski space as a curve whose binormal lines make a constant angle with a fixed direction [14].

2 Preliminaries

Definition 1

The Minkowski 3-spaceE13is the real vector space E3which is endowed with the standard indefinite flat metric 〈.,.〉 defined by

u,v=u1v1+u2v2+u3v3,
for any two vectors u = (u1, u2, u3) and v = (v1, v2, v3) inE13 . Since 〈.,.〉 is an indefinite metric, an arbitrary vectoruE13\{0}can have one of three properties:
  1. i)it can be space-like, ifu, u1 > 0,
  2. ii)time-like, ifu, u1 < 0 or
  3. iii)light-like or isotropic or null vector, ifu, u1 = 0, but u ≠ 0.

In particular, the norm (length) of a non lightlike vectoruE13is given by

u=|u,u|.

Given a regular curveα:IE13can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors α(t) satisfyα(t), α(t) 〉1 > 0, 〈 α(t), α(t) 〉1 < 0 orα(t), α(t) 〉1 =0, respectively, at any tɛI, whereα(t)=dαdt. .

Definition 2

A curveα:IE13is called a null curve, if its tangent vector α = T is a null vector. A null curve α = α(s) is called a null Cartan curve, if it is parameterized by the pseudo-arc function s defined by

s(t)=0tα(u)du.

There exists a unique Cartan frame {T,N,B} along a non-geodesic null Cartan curve satisfying the Cartan equations

[TNB]=[0k0τ0k0τ0][TNB],
where the curvature k(s) = 1 and the torsion τ(s) is an arbitrary function in pseudo-arc parameter s. If τ(s) = 0, the null Cartan curve is called a null Cartan cubic. The Cartan’s frame vectors satisfy the relations
N,N=1,T,T=B,B=0,T,B=1,T,N=N,B=0
and
T×N=T,N×B=B,B×T=N.

Cartan frame {T,N,B} is positively oriented, if det(T,N,B) = [T,N,B] = 1.

The Frenet frame is created for the non-degenerated curves of three times continuously differentiable. But, at some points on the curve, curvature may vanish. In this case, we need an alternative frame in E3. Bishop introduced a new frame called Bishop frame or parallel transport frame, which is well defined even if the curve has a vanishing second derivative [7].

2.1 The Bishop Frame

The Bishop frame or relatively parallel adapted frame {T,N1,N2} of a regular curve in Euclidean 3-space is introduced by R.L. Bishop. It contains the tangential vector field T and two normal vector fields N1 and N2, which can be obtained by rotating the Frenet vectors N and B in the normal plane T of the curve, in such a way that they become relatively parallel. This means that their derivatives N1 and N2 with respect to the arc-length parameter s of the curve are collinear with the tangential vector field T [7].

Remark 1

We can also define N1and N2to be relatively parallel, if the normal componentT1=span{N1,N2}of their derivativesN1 and N2is zero, which implies that the mentioned derivatives are collinear with T1.

2.2 The Bishop frame of a null Cartan curve in E13

The Bishop frame {T1,N1,N2} of a non-geodesic null Cartan curve in E13 is positively oriented pseudo-orthonormal frame consisting of the tangential vector field T1, relatively parallel spacelike normal vector field N1 and relatively parallel lightlike transversal vector field N2.

Theorem 1

Let α be a null Cartan curve inE13parameterized by pseudo-arc s with the curvature k(s) = 1 and the torsion τ(s). Then the Bishop frame {T1,N1,N2} and the Cartan frame {T,N,B} of α are related by

[T1N1N2]=[100k210k222k21][TNB],
and the Cartan equations of according to the Bishop frame read
[T1N1N2]=[k2k1000k100k2][T1N1N2],
where the first Bishop curvature k1(s) = 1 and the second Bishop curvature satisfies Riccati differential equation
k2(s)=12k22(s)τ(s),
which satisfies the conditions
N1,N1=1,T1,T1=N2,N2=0,T1,N2=1,T1,N1=N1,N2=0[7].

3 Curves of AW(k)- type

Proposition 1

Let α be a Frenet curve of osculating order 3, by using the cartan equations of α according to the Bishop frame(2.7), then we have

α(s)=T1(s),α(s)=T1(s)=k2T1+k1N1,α′″(s)=(k2+k22)T1+(k1+k1k2)N1+k12N2,α″″(s)=(k2+3k2k2+k23)T1+(k1+2k1k2+k1k2+k1k22)N1+3k1k1N2.

Notation 1

Let us write

M1(s)=k1N1,
M2(s)=(k1+k1k2)N1+k12N2,
M3(s)=(k1+2k1k2+k1k2+k1k22)N1+3k1k1N2.

Corollary 1

α (s), α (s), α″′ (s) and α″″ (s) are linearly dependent if and only if M1 (s), M2 (s) and M3 (s) are linearly dependent.

Definition 3

Frenet curves of osculating order 3 are of

  1. i)type weak AW (2) if they satisfy
    M3(s)=M3(s),M2(s)M2(s),
  2. ii)type weak AW (3) if they satisfy
    M3(s)=M3(s),M1(s)M1(s),
    where
    M1(s)=M1(s)M1(s),
    M2(s)=M2(s)M2(s),M1(s)M1(s)M2(s)M2(s),M1(s)M1(s),
  3. iii)AW (1)-type, if they satisfy
    M3(s)=0,
  4. iv)AW (2)-type, if they satisfy
    M2(s)2M3(s)=M3(s),M2(s)M2(s),
  5. v)AW (3)-type, if they satisfy
    M1(s)2M3(s)=M3(s),M1(s)M1(s).

Proposition 2

Let α be a Frenet curve of osculating order 3, then α is AW (1)-type if and only if i) k1is a constant function, and ii)

k1+2k1k2+k1k2+k1k22=0.

Proof

Since α is a curve of type AW (1), then α must satisfy (3.8)

M3(s)=(k1+2k1k2+k1k2+k1k22)N1+3k1k1N2,0=(k1+2k1k2+k1k2+k1k22)N1+3k1k1N2.

Since N1 and N2 are linearly independent, therefore

3k1k1=0,
k1 is a constant function, and
k1+2k1k2+k1k2+k1k22=0.

Hence the proposition is proved.

Proposition 3

Let α be a Frenet curve of osculating order 3. Then, if α are of AW (2)–type, we have

3(k1)2+3k1k1k2=k1k1+2k12k2+k1k1k2+k12k22.

Proof

Suppose that α is a Frenet curve of osculating order 3. From (3.2) and (3.3) we can write

M2(s)=β(s)N1+γ(s)N2,M3(s)=δ(s)N1+η(s)N2,
where β (s), γ (s), δ (s) and η (s) are differential functions. Since M2 (s) and M3 (s) are linearly dependent, then the determinant of the coefficients of N1 and N2 is equal to zero, hence one can write
|β(s)γ(s)δ(s)η(s)|=0,
where
β(s)=k1+k1k2,γ(s)=k12,δ(s)=k1+2k1k2+k1k2+k1k22andη(s)=3k1k1.

By considering (3.14) in (3.13) we get

3(k1)2+3k1k1k2=k1k1+2k12k2+k1k1k2+k12k22.

Hence the proposition is proved.

Proposition 4

Let α be a Frenet curve of osculating order 3, then α is of type AW (3) if and only if k1 (s) is a constant function

Proof

Suppose that α is a Frenet curve of order 3. From (3.1) and (3.3) we can write

M1(s)=β(s)N1+γ(s)N2,M3(s)=δ(s)N1+η(s)N2,
where β (s), γ (s), δ (s) and η (s) are differential functions. Since M2 (s) and M3 (s) are linearly dependent, then the determinant of the coefficients of N1 and N2 is equal to zero, one can write
|β(s)γ(s)δ(s)η(s)|=0,
where
β(s)=k1,γ(s)=0,δ(s)=k1+2k1k2+k1k2+k1k22andη(s)=3k1k1.

By substituting (3.16) in (3.15) we get

k12k1=0.

For k12k1 to be zero, k1 (s) has to be a constant function. Hence the proposition is proved.

Proposition 5

Let α be a Frenet curve of osculating order 3, then α is of weak AW (2)–type if and only if

  1. i)k1 (s) is a constant function,
  2. ii)
    k1+2k1k2+k1k2+k1k22=0andk2(s)=2s2c,
    where s and c are arc-length parameter and constant respectively.

Proof

Since α is of weak AW (2)–type, it must satisfy (3.4), by using (3.4), (3.6), (3.7) and (3.1)

M1(s)=M1(s)M1(s)=k1N1k12=N1,
M2(s)=M2(s)M2(s),M1(s)M1(s)M2(s)M2(s),M1(s)M1(s),M2(s)=(k1+k1k2)N1+k12N2(k1+k1k2)N1(k1+k1k2)N1+k12N2(k1+k1k2)N1,M2(s)=N2.

Since α is of weak AW (2)–type, then it must satisfy

M3(s)=M3(s),M2(s)M2(s),=(k1+2k1k2+k1k2+k1k22)N1+3k1k1N2,N2N2,=0.

Therefore

(k1+2k1k2+k1k2+k1k22)N1+3k1k1N2=0,
then
k1+2k1k2+k1k2+k1k22=0and3k1k1=0.

For the k12k1 to be zero, k1 (s) has to be a constant function.

Then (3.20) turns to

2k1k2+k1k22=0,
by integrating the above equation
2k2+k22=0,k2k22=12,k2k22ds=12ds,
let
k2(s)=u,
k2ds=du,
therefore (3.22) turns to
duu2=12ds,1u=12+c,
by simplifying the above equation and using (3.23) we get
k2(s)=2s2c.

Hence the proposition is proved.

Proposition 6

Let α be a Frenet curve of osculating order 3, then α is of weak AW (3)–type if and only if k1 (s) is a constant function.

Proof

Since α is of weak AW (3)–type, by using (3.3) and (3.18)

M3(s)=M3(s),M1(s)M1(s),=(k1+2k1k2+k1k2+k1k22)N1

Therefore

(k1+2k1k2+k1k2+k1k22)N1+3k1k1N2=(k1+2k1k2+k1k2+k1k22)N1k1k1=0.

For the k12k1 to be zero, k1 (s) has to be a constant function. Hence the proposition is proved.

4 The helices according to Bishop frame of null cartan curve in Minkowski 3-space

Definition 4

Helix can be defined as a curve whose tangent lines make a constant angle with a fixed direction. Helices are characterized by the fact that the ratiok1k2is constant along the curve.

Theorem 2

Let α be a null cartan curve inE13 , then α is a general helix if and only ifk1k2is constant.

Proof

Let α be a general helix in E13 and 〈T, U〉 is constant, then α is a general helix, from the definition we have

T,U=ccisconstant,
by differentiating the above equation
T,U+T,U=0,T,U=0,k2cosθ+k1sinθ=0,k1k2=cotθ(constant),
as disered.

Theorem 3

Suppose that α is a null cartan curve inE13 , then α is a general helix if and only if

det(T1,T1,T1′″)=k12(k1k2k2k1).

Proof

(⇒) Let k1k2 be constant. We have equalities as

T1(s)=k2T1+k1N1,T1(s)=(k2+k22)T1+(k1+k1k2)N1+k12N2,T1′″(s)=(k2+3k2k2+k23)T1+(k1+2k1k2+k1k2+k1k22)N1+3k1k1N2.

So we get

det(T1,T1,T1′″)=|k2k10(k2+k22)(k1+k1k2)k12(k2+3k2k2+k23)(k1+2k1k2+k1k2+k1k22)3k1k1|,det(T1,T1,T1′″)=k12k23(k1k2)+3k1k1k22(k1k2)+k12(k1k2k2k1).

Since α is a general helix, and k1k2 is constant. Hence, we have

det(T1,T1,T1′″)=k12(k1k2k2k1),butk20.

(⇐) Suppose that det(T1,T1,T1′″)=k12(k1k2k2k1) , then it is clear that the k1k2 is constant, since (k1k2) is zero. Hence the theorem is proved.

Theorem 4

Let α be a null cartan curve inE13 , then α is a general helix if and only if

det(N1,N1,N1′″)=0.

Proof

(⇒) Suppose that k1k2 be constant. We have equalities as

N1=k1N2,N1=(k1k1k2)N2,N1′″=(k12k1k2k1k2+k1k22)N2.

So we get

det(N1,N1,N1′″)=|00k100k1200(k12k1k2k1k2+k1k22)|,det(N1,N1,N1′″)=0.

(⇐) Suppose that det(N1,N1,N1′″)=0 , then it is clear that the k1k2 is constant, since (k1k2) is zero. Hence the theorem is proved.

Theorem 5

Let α be a null cartan curve inE13 , then α is a general helix if and only if

det(N2,N2,N2′″)=0.

From (2.7)

α(s)=T,DTT=k2T1+k1N1,DTN1=k1N2,DTN2=k2N2.

Theorem 6

Letα:IE13be a unit speed null cartan curve with the cartan frame apparatus {T, N2, N2, k1, k2}, then α is a general helix if and only if

DT(DTDTN1)+k1k2DTN2=(k13k1k2)1k1DTN1.

Proof

(⇒) Suppose that α is a general helix. Then, from (4.6), we have

DTN1=k1N2,DT(DTN1)=k1N2k1k2N2,
DT(DTDTN1)=k1N2(k1k2+k1k2)N2k1k2N2k1k2DTN2.

Since α is a general helix

k1k2=ccisconstant,
by differentiating (4.10) we get
(k1k2)=2k1k2,
but
DTN1=k1N2,N2=1k1DTN1.

By substituting (4.11) and (4.12) in (4.9) we get

DT(DTDTN1)=(k13k1k2)(1k1DTN1)k1k2DTN2,DT(DTDTN1)+k1k2DTN2=(k13k1k2)(1k1DTN1).
(⇐) We will show that null cartan curve α is a general helix. By differentianting (4.12) covariently
N2=1k1DTN1,DTN2=k1k12DTN1+1k1DTDTN1,
DTDTN2=(k1k12)DTN12k1k12DTDTN1+1k1DTDTDTN1.

By substituting (4.8) and (4.13) in (4.15) we get

DTDTN2=[(k1k12)+(k13k1k2)1k12]DTN12(k1)2k12N2(2k1k1+k2)DTN2.

From (4.6)

DTN2=k2N2,DT(DTN2)=k2N2k2DTN2.
By comparing (4.16) and (4.17)
(2k1k1+k2)=k2,
by integrating the above equation we get
k1=1,
to find k2, by comparing (4.16) and (4.17) we have
2(k1)2k12=k2.

But k1 = 1, therefore

k2=0,
which means k2 is a constant function.
k1k2is constant.
Hence α is a general helix.

Theorem 7

Letα:IE13be a unit speed null cartan curve with the cartan frame apparatus {T, N2, N2, k1, k2}, then α is a general helix if and only if

DT(DTDTN2)=(k23k2k2+k23)(1k2DTN2).

The above theorem can be proven analogously, so we skip its proof.

References

  • [1]

    A.A. Lucas, P. Lambin, Diffraction by DNA, carbon nanotubes and other helical nanostructures, Rep. Prog. Phys., 1181–1249, 68 (2005).

  • [2]

    Ahmad T. Ali, Position vectors of general helices in Euclidean 3-space, Bulletin of Mathematical Analysis and Applications, Volume 3, 2, Pages 198–205 (2011).

  • [3]

    Ahmad T. Ali and Rafael Lopez, Slant Helices in Minkowski Space E 1 3 E_1^3 , J. Korean Math. Soc. Vol. 48, No. 1, pp. 159–167, (2011).

  • [4]

    B. Bukcu, M.K. Karacan, On the slant helices according to Bishop frame of the timelike curve in Lorentzian space, Tamkang J. Math. 39 (3), 255–262, (2008).

  • [5]

    C. Özgür and F. Gezgin, “On some curves of AW(k)–type,” Differential Geometry—Dynamical Systems, vol. 7, pp. 74–80, 2005.

  • [6]

    L. Kula, N. Ekmekci, Y. Yaylı and K. Ilarslan, Characterizations of slant helices in Euclidean 3-space. Turk J Math 34, 261 – 273 (2010).

  • [7]

    Milica Grbović and Emilija Nesovic, “On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space,” Journal of Mathematical Analysis and Applications, Vol. 461, pp. 219–233, 2018.

  • [8]

    M. Külahcı, M. Bektaş and M. Ergüt, On Harmonic Curvatures of Null Curves of the AW(k)-Type in Lorentzian Space Z. Naturforsch. 63a, 248 – 252 (2008).

  • [9]

    N. Chouaieb, A. Goriely, J.H. Maddocks, Helices, PANS, 103, 9398–9403, (2006).

  • [10]

    Yildirim Yilmaz M. and Bektaş M., Slant helices of (k, m)-type in E 4 Acta Univ. Sapientiae, Mathematica, 10, pp 395–401, 2 (2018).

  • [11]

    Yildirim Yilmaz M. and Bektaş M., Helices of the 3- dimensional Finsler Manifold, Journal of Adv. Math. Studies 2(1), 107–113, 2009.

  • [12]

    Yildirim Yilmaz M., Biharmonic General Helices in Three Dimensional Finsler Manifold F 3, Karaelmas Fen ve Müh. Derg. 7(1):1–4, 2017.

  • [13]

    Yildirim Yilmaz M., Külahcı M. and Öğrenmiş A. O., A Slant Helix Characterization in Riemam- Otsuki Space, Mathematica moravica vol. 16–2, 99–106, (2012).

  • [14]

    F. Gökçelik and I. Gök, Null W-slant helices in E 1 3 E_1^3 J. Math. Anal. Appl. 420, 222–241, (2014).

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    A.A. Lucas, P. Lambin, Diffraction by DNA, carbon nanotubes and other helical nanostructures, Rep. Prog. Phys., 1181–1249, 68 (2005).

  • [2]

    Ahmad T. Ali, Position vectors of general helices in Euclidean 3-space, Bulletin of Mathematical Analysis and Applications, Volume 3, 2, Pages 198–205 (2011).

  • [3]

    Ahmad T. Ali and Rafael Lopez, Slant Helices in Minkowski Space E 1 3 E_1^3 , J. Korean Math. Soc. Vol. 48, No. 1, pp. 159–167, (2011).

  • [4]

    B. Bukcu, M.K. Karacan, On the slant helices according to Bishop frame of the timelike curve in Lorentzian space, Tamkang J. Math. 39 (3), 255–262, (2008).

  • [5]

    C. Özgür and F. Gezgin, “On some curves of AW(k)–type,” Differential Geometry—Dynamical Systems, vol. 7, pp. 74–80, 2005.

  • [6]

    L. Kula, N. Ekmekci, Y. Yaylı and K. Ilarslan, Characterizations of slant helices in Euclidean 3-space. Turk J Math 34, 261 – 273 (2010).

  • [7]

    Milica Grbović and Emilija Nesovic, “On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space,” Journal of Mathematical Analysis and Applications, Vol. 461, pp. 219–233, 2018.

  • [8]

    M. Külahcı, M. Bektaş and M. Ergüt, On Harmonic Curvatures of Null Curves of the AW(k)-Type in Lorentzian Space Z. Naturforsch. 63a, 248 – 252 (2008).

  • [9]

    N. Chouaieb, A. Goriely, J.H. Maddocks, Helices, PANS, 103, 9398–9403, (2006).

  • [10]

    Yildirim Yilmaz M. and Bektaş M., Slant helices of (k, m)-type in E 4 Acta Univ. Sapientiae, Mathematica, 10, pp 395–401, 2 (2018).

  • [11]

    Yildirim Yilmaz M. and Bektaş M., Helices of the 3- dimensional Finsler Manifold, Journal of Adv. Math. Studies 2(1), 107–113, 2009.

  • [12]

    Yildirim Yilmaz M., Biharmonic General Helices in Three Dimensional Finsler Manifold F 3, Karaelmas Fen ve Müh. Derg. 7(1):1–4, 2017.

  • [13]

    Yildirim Yilmaz M., Külahcı M. and Öğrenmiş A. O., A Slant Helix Characterization in Riemam- Otsuki Space, Mathematica moravica vol. 16–2, 99–106, (2012).

  • [14]

    F. Gökçelik and I. Gök, Null W-slant helices in E 1 3 E_1^3 J. Math. Anal. Appl. 420, 222–241, (2014).

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