A Note on Special Smarandache Curves in The Null Cone Q3

2018·0.22 MB·

by  F. Almaz

Acta Universitatis Apulensis No. 56/2018
ISSN: 1582-5329 pp. 111-124
http://www.uab.ro/auajournal/ doi: 10.17114/j.aua.2018.56.10
A NOTE ON SPECIAL SMARANDACHE CURVES IN THE NULL
CONE Q3
F. Almaz, M.A. Ku¨lahci
Abstract. As it is well-known, the geometry of curve in three-dimensions is
actually characterized by Frenet vectors. In this paper, we obtain Smarandache
curves by using cone frame formulas in null cone Q3 . Also, we give an example
related to these curves.
2010 Mathematics Subject Classification: 53A40, 53A35.
Keywords: smarandache curve, asymptotic orthonormal frame, null cone.
1. Introduction
Human being were bewitched by curves and curved shapes long before they took
into account them as mathematical objects. But the greatest effect in the research
of curves was, of course, the discovery of the calculus. Geometry before calculus
includes only the simplest curves.
Smarandache geometry is a geometry which has at least one Smarandachely
denied axiom [4]. An axiom is said to be Smarandachely denied, if it behaves in at
least two different ways within the same space. Smarandache curve is defined as a
regular curve whose position vector is composed by Frenet frame vectors of another
regular curve.
The popularity of Smarandache curves in various ambient spaces have been
classfied in [1]-[7], [13]-[16]. In this study, we define special Smarandache curves
such as xαβ,xβy,xαy,αβy-Smarandache curves according to asymptotic orthonor-
mal frame in the null cone Q3 and we examine the curvatures and the asymptotic
orthonormalframe’svectorsofSmarandachecurves. Wealsogiveanexamplerelated
to these curves.
111
F. Almaz, M.A. Ku¨lahci – A Note on Special Smarandache Curves ...
2. Preliminaries
Some basics of the curves in the null cone are provided from, [8]-[9].
Let E4 be the 4−dimensional pseudo-Euclidean space with the
1
g(X,Y) = (cid:104)X,Y(cid:105) = x y +x y +x y −x y
(cid:101) 1 1 2 2 3 3 4 4
for all X = (x ,x ,x ,x ) Y = (y ,y ,y ,y ) ∈ E4. E4 is a flat pseudo-Riemannian
1 2 3 4 1 2 3 4 1 1
manifold of signature (3,1).
Let M be a submanifold of E4. If the pseudo-Riemannian metric g of E4 in-
(cid:101)
1 1
duces a pseudo-Riemannian metric g(respectively, a Riemannian metric, a degener-
ate quadratic form) on M, then M is called a timelike( respectively, spacelike, de-
generate) submanifold of E3. Let c be a fixed point in E4. The pseudo-Riemannian
1 1
lightlike cone(quadric cone ) is defined by
Q3(c) = (cid:8)x ∈ E4 : g(x−c,x−c) = 0(cid:9),
1 1
where the point c is called the center of Q3(c). When c = 0, we simply denote Q3(0)
1 1
by Q3 be and call it the null cone.
Let E4 be 4-dimensional Minkowski space and Q3 the lightlike cone in E4. A
1 1
vector V (cid:54)= 0 in E4 is called spacelike, timelike or lightlike, if (cid:104)V,V(cid:105) > 0, (cid:104)V,V(cid:105) < 0
1 (cid:112)
or (cid:104)V,V(cid:105) = 0, respectively. The norm of a vector x ∈ E4 is given by (cid:107)x(cid:107) = (cid:104)x,x(cid:105),
1
[12].
We assume that curve x : I → Q3 ⊂ E4 is a regular curve in Q3 for t ∈ I. In the
1
following, we always assume that the curve is regular.
A frame field {x,α,β,y} on E4 is called an asymptotic orthonormal frame field,
1
if
(cid:104)x,x(cid:105) = (cid:104)y,y(cid:105) = (cid:104)x,α(cid:105) = (cid:104)y,α(cid:105) = (cid:104)β,α(cid:105) = (cid:104)y,β(cid:105) = (cid:104)x,β(cid:105) = 0,
(cid:104)x,y(cid:105) = (cid:104)α,α(cid:105) = (cid:104)β,β(cid:105) = 1.
Using x(cid:48)(s) = α(s) we know that {x(s),α(s),β(s),y(s)} from an asymptotic
orthonormal frame along the curve x(s) and the cone frenet formulas of x(s) are
given by
x(cid:48)(s) = α(s)
α(cid:48)(s) = κ(s)x(s)−y(s) (2.1)
β(cid:48)(s) = τ(s)x(s)
y(cid:48)(s) = −κ(s)α(s)−τ(s)β(s)
where the functions κ(s) and τ(s) are called cone curvature functions of the curve
x(s), [10].
112
F. Almaz, M.A. Ku¨lahci – A Note on Special Smarandache Curves ...
Let x : I → Q3 ⊂ E4 be a spacelike curve in Q3 with an arc length parameter
1
s. Then x = x(s) = (x ,x ,x ,x ) can be written as
1 2 3 4
1
x(s) = (2f,2g,1−f2−g2,1+f2+g2) (2.2)
(cid:112)
2 f2+g2
s s
for some non constant function f(s) and g(s), [11].
3. Smarandache Curves in The Null Cone Q3
In this section, we define binary Smarandache curves according to the asymptotic
orthonormal frame in Q3. Also, we obtain the asymptotic orthonormal frame and
cone curvature functions of the Smarandache partners lying on Q3 using cone frenet
formulas.
Smarandache curve γ = γ(s∗(s)) of the curve x is a regular unit speed curve
lying fully on Q3. Let {x,α,β,y} and (cid:8)γ,α ,β ,y (cid:9) be the moving asymptotic
γ γ γ
orthonormal frames of x and γ, respectively.
Definition 1. Let x be unit speed spacelike curve lying on Q3 with the moving
asymptotic orthonormal frame {x,α,β,y}. Then, xαβ− Smarandache curve of x is
defined by
1
γ (s∗) = √ (ax(s)+bα(s)+cβ(s)), (3.1)
xαβ
b2+c2
where a,b,c ∈ R+.
0
Theorem 1. Let x be unit speed spacelike curve in Q3 with the moving asymptotic
orthonormal frame {x,α,β,y} and cone curvatures κ(s),τ(s) and let γ be xαβ−
xαβ
(cid:110) (cid:111)
Smarandache curve with asymptotic orthonormal frame γ ,α ,β ,y .
xαβ xαβ xαβ xαβ
Then the following relations hold:
(cid:110) (cid:111)
i)Theasymptoticorthonormalframe γ ,α ,β ,y ofthexαβ-Smaran-
xαβ xαβ xαβ xαβ
dache curve γ is given as
xαβ
γ  √ a √ b √ c 0 x
xαβ b2+c2 b2+c2 b2+c2
α   bκ+cτ a 0 −bα
 xαβ =  Ψ Ψ Ψ  , (3.2)
β   B B B B β
xαβ 1 2 3 4
y Υ Υ Υ Υ y
xαβ 1 2 3 4
113
F. Almaz, M.A. Ku¨lahci – A Note on Special Smarandache Curves ...
where
(cid:112) Ψ
Ψ = a2−2b(bκ+cτ),w = √ (3.3)
b2+c2
bκ+cτ a 1
A = ,A = ,A = ; (3.4)
1 Ψ 2 Ψ 3 w
and
√
b2+c2
B = (Ψ(cid:48)(bκ+cτ)+Ψ(bκ(cid:48)+cτ(cid:48))+aΨκ), (3.5)
1 Ψ3
√
b2+c2 (cid:112)
B = (Ψ(bκ+cτ)−a−Ψκ b2+c2),
2 Ψ3
√
−τ(b2+c2) b2+c2 (cid:112)
B = ,B = (−aΨ+ b2+c2);
3 Ψ2 4 Ψ3
aD bD cD
Υ = −B − √ ,Υ = −B − √ ,Υ = −B − √ ,
1 1 2 b2+c2 2 2 2 b2+c2 3 3 2 b2+c2
Υ = −B
4 4
and
(A(cid:48) +κA )(−A +A(cid:48)) 1
D = 2 1 2 2 3 + (A +A(cid:48) −κA )2+(τA )2
w2 w2 1 2 3 3
or
(b2+c2) (cid:112)
D = (−2(Ψ(cid:48)(bκ+cτ)+Ψ(bκ(cid:48)+cτ(cid:48))+aΨκ)(aΨ+ b2+c2)
Ψ6
(cid:112) τ2(b2+c2)
+(Ψ(bκ+cτ)−a−Ψκ b2+c2)2)+
Ψ2
ii) The cone curvatures κ (s∗) and τ (s∗) of the curve γ is given by
γ γ
xαβ xαβ xαβ
D
κ (s∗) = −
γ
xαβ 2
(cid:113)
τ (s∗) = 2(Υ −κ(cid:48))Υ +(Υ −κ)2+Υ2−κ2 , (3.6)
γxαβ 1 4 2 3 γxαβ
where
(cid:90)
1 (cid:112)
s∗ = √ a2−2b(bκ+cτ)ds.
b2+c2
114
F. Almaz, M.A. Ku¨lahci – A Note on Special Smarandache Curves ...
Proof. i) We assume that the curve x is a unit speed spacelike curve with the
asymptotic orthonormal frame {x,α,β,y} and cone curvatures κ,τ. Differentiating
the equation (3.1) with respect to s and considering (2.1), we have
(cid:18) (cid:19)
(cid:16)a(cid:17) bκ+cτ 1
γ(cid:48) (s∗) = α(s)+( )x(s)+ y(s), (3.7)
xαβ Ψ Ψ w
where
ds∗ 1 (cid:112)
w = = √ a2−2b(bκ+cτ), (3.8)
ds b2+c2
(cid:112)
Ψ = a2−2b(bκ+cτ). (3.9)
It can be easily seen that the tangent vector γ(cid:48) (s∗) = α (s∗) is a unit space-
xαβ xαβ
like vector.
Differentiating (3.7), we obtain equation as follows
γ(cid:48)(cid:48) (s∗) = B x(s)+B α(s)+B β(s)+B y(s), (3.10)
xαβ 1 2 3 4
where B = A(cid:48)1+κA2,B = A1+A(cid:48)2−κA3,B = −τA3,B = −A2+A(cid:48)3.
1 w 2 w 3 w 4 w
1 (cid:68) (cid:69)
y (s∗) = −γ(cid:48)(cid:48) − γ(cid:48)(cid:48) ,γ(cid:48)(cid:48) γ (3.11)
xαβ
xαβ 2 xαβ xαβ xαβ
By the help of previous equation (3.11), we obtain
y (s∗) = Υ x(s)+Υ α(s)+Υ β(s)+Υ y(s), (3.12)
xαβ 1 2 3 4
where Υ = −B − √aD ,Υ = −B − √bD ,Υ = −B − √cD ,Υ = −B .
1 1 2 b2+c2 2 2 2 b2+c2 3 3 2 b2+c2 4 4
(cid:68) (cid:69)
ii) Using equations κ (s∗) = −1 γ(cid:48)(cid:48) ,γ(cid:48)(cid:48) and
γxαβ 2 xαβ xαβ
τ2 (s∗) = (cid:104)x(cid:48)(cid:48)(cid:48)−κα−κ(cid:48)x,x(cid:48)(cid:48)(cid:48)−κα−κ(cid:48)x(cid:105)−κ2 (s∗).
γ γ
xαβ xαβ
The curvatures κ (s∗) and τ (s∗) of the γ (s∗) are explicity obtained by
γ γ
xαβ xαβ xαβ
1
κ (s∗) = − D
γ
xαβ 2
τ2 (s∗) = 2(Υ −κ(cid:48))Υ +(Υ −κ)2+Υ2−κ2 . (3.13)
γxαβ 1 4 2 3 γxαβ
Thus, the theorem is proved.
115
F. Almaz, M.A. Ku¨lahci – A Note on Special Smarandache Curves ...
Definition 2. Let x be unit speed spacelike curve lying on Q3 with the moving
asymptotic orthonormal frame {x,α,β,y}. Then, xβy−smarandache curve of x is
defined by
1
γ (s∗) = √ (ax(s)+bβ(s)+cy(s)), (3.14)
xβy 2ac+b2
where a,b.c ∈ R+.
0
Theorem 2. Let x be unit speed spacelike curve in Q3 with the moving asymptotic
orthonormal frame {x,α,β,y} and cone curvatures κ(s),τ(s) and let γ be xβy−
xβy
(cid:110) (cid:111)
Smarandache curve with asymptotic orthonormal frame γ ,α ,β ,y .
xβy xβy xβy xβy
Then the following relations hold:
(cid:110) (cid:111)
i)Theasymptoticorthonormalframe γ ,α ,β ,y ofthexβy−Smaran-
xβy xβy xβy xβy
dache curve γ is given as
xβy
 
γ  √ a 0 √ b √ c x
xβy 2ac+b2 2ac+b2 2ac+b2
α   bτ a−cκ −cτ 0 α
 xβy =  η η η  , (3.15)
β   B∗ B∗ B∗ B∗ β
xβy  1 2 3 4 
y ω ω ω ω y
xβy 1 2 3 4
where
(cid:112) η ds∗
η = a2−2ac+c2(κ2+τ2),ω = √ = ;
2ac+b2 ds
bτ a−cκ −cτ
A = ,A = ,A = ;
1 η 2 η 3 η
A(cid:48) +κA +τA A +A(cid:48) A(cid:48) −A
B∗ = 1 2 3,B∗ = 1 2,B∗ = 3,B∗ = 2
1 w 2 w 3 w 4 w
and
aT bT
ω = −B∗− √ ,ω = −B∗,ω = −B∗− √ ,
1 1 2 2ac+b2 2 2 3 3 2 2ac+b2
cT
ω = −B∗− √ ,
4 4 2 2ac+b2
T = 1 (cid:0)−2A (A(cid:48) +κA +τA )+(A +A(cid:48))2+A(cid:48)2(cid:1).
ω2 2 1 2 3 1 2 3
or
 
2ac+b2 2(cκ−a)(b(τ(cid:48)η−τη(cid:48))+η(cid:0)κa−c(cid:0)κ2+τ2(cid:1)(cid:1))
η3
T = η  + 1 (cid:16)(η(bτ −cκ)+η(cid:48)(a−cκ))2+ c2(τη(cid:48)−τ(cid:48)η)2(cid:17) .
η4 η
116
F. Almaz, M.A. Ku¨lahci – A Note on Special Smarandache Curves ...
ii) The cone curvature κ (s∗) and τ (s∗) of the curve γ is given by
γ γ
xβy xβy xβy
−T
κ (s∗) = ,
γ
2
xβy
τ2 (s∗) = 2(ω −κ(cid:48))ω +(ω −κ)2+ω2−κ2 (3.16)
γ 1 4 2 3 γ
xβy xβy
where
(cid:90)
1 (cid:112)
s∗ = √ a2−2ac+c2(κ2+τ2)ds. (3.17)
2ac+b2
.
Proof. i) We assume that the curve x is a unit speed spacelike curve with the
asymptotic orthonormal frame {x,α,β,y} and cone curvatures κ,τ. Differentiating
the equation (3.14) with respect to s and considering (2.1), we have
γ(cid:48) (s∗) = bτ→−x(s)+ a−cκ→−α(s)− cτ→−β(s) (3.18)
xβy η η η
or
γ(cid:48) (s∗) = A →−x +A →−α +A →−β.
xβy 1 2 3
By considering (3.17), we get
γ(cid:48) (s∗) = α(s) = α . (3.19)
xβy xβy
→−
Here,itcanbeeasilyseenthatthetangentvector α isaunitspacelikevector.
xβy
Differentiating (3.19) and using (3.17), we obtain
γ(cid:48)(cid:48) (s∗) = B∗→−x +B∗→−α +B∗→−β −B∗→−y, (3.20)
xβy 1 2 3 4
where B∗ = A(cid:48)1+κA2+τA3,B∗ = A1+A(cid:48)2,B∗ = A(cid:48)3,B∗ = −A2.
1 w 2 w 3 w (cid:68)4 w (cid:69)
By the help of equation y (s∗) = −γ(cid:48)(cid:48) − 1 γ(cid:48)(cid:48) ,γ(cid:48)(cid:48) γ , we write
xβy xβy 2 xβy xβy xβy
y (s∗) = ω x(s)+ω α(s)+ω β(s)+ω y(s), (3.21)
xβy 1 2 3 4
where
aT
ω = −B∗− √ ,ω = −B∗,
1 1 2 2ac+b2 2 2
bT cT
ω = −B∗− √ ,ω = −B∗− √ .
3 3 2 2ac+b2 4 4 2 2ac+b2
ii)
1 (cid:68) (cid:69)
κ (s∗) = − γ(cid:48)(cid:48) ,γ(cid:48)(cid:48) ,
γ
xβy 2 xβy xβy
117
F. Almaz, M.A. Ku¨lahci – A Note on Special Smarandache Curves ...
τ2 (s∗) = (cid:104)β −κα−κ(cid:48)x,β−κα−κ(cid:48)x(cid:105)−κ2 . (3.22)
γ γ
xβy xβy
By using (3.22), the curvatures κ (s∗) and τ (s∗) of the γ (s∗) are explicity
γ γ
xβy xβy xβy
obtained
1 (cid:68) (cid:69) −T
κ (s∗) = − γ(cid:48)(cid:48) ,γ(cid:48)(cid:48) = ,
γ
xβy 2 xβy xβy 2
τ2 (s∗) = 2(ω −κ(cid:48))ω +(ω −κ)2+ω2−κ2
γ 1 4 2 3 γ
xβα xα
where
 
2ac+b2 2(cκ−a)(b(τ(cid:48)η−τη(cid:48))+η(cid:0)κa−c(cid:0)κ2+τ2(cid:1)(cid:1))
η3
T = η  + 1 (cid:16)(η(bτ −cκ)+η(cid:48)(a−cκ))2+ c2(τη(cid:48)−τ(cid:48)η)2(cid:17) .
η4 η
Definition 3. Let x be unit speed spacelike curve lying on Q3 with the moving
asymptotic orthonormal frame {x,α,β,y}. Then, xαy− Smarandache curve of x is
defined by
1
γ (s∗) = √ (ax(s)+bα(s)+cy(s)), (3.23)
xαy 2ac+b2
where a,b,c ∈ R+.
0
Theorem 3. Let x be unit speed spacelike curve in Q3 with the moving asymp-
totic orthonormal frame {x,α,β,y} and cone curvatures κ,τ and let γ be xαy−
xαy
(cid:110) (cid:111)
Smarandachecurvewithasymptoticorthonormalframe γ ,α ,β ,y .Then
xαy xαy xαy xαy
the following relations hold:
(cid:110) (cid:111)
i)Theasymptoticorthonormalframe γ ,α ,β ,y ofthexαy−Smaran-
xαy xαy xαy xαy
dache curve γ is given as
xαy
 
γ  √2aac+b2 √2abc+b2 0 √2acc+b2 x
αxαy  ρ ρ ρ ρ α
 xαy =  ρ(cid:48)+κρ1+ρ τ ρ(cid:48)+ρ 2−κρ ρ(cid:48)−3τρ −ρ +4ρ(cid:48)  ,
β   1 2 3 2 1 4 3 4 2 4 β
xαy  M M M M 
y ρ(cid:48)+κρ +ρ τ ρ(cid:48)+ρ −κρ ρ(cid:48)+τρ −ρ +ρ(cid:48) y
xαy − 1 2 3 −aφ − 2 1 4 −bφ − 3 4 − 2 4 −cφ
M M M M
(3.24)
where
bκ a−cκ −cτ −b
ρ = ,ρ = ,ρ = ,ρ =
1 Ω 2 Ω 3 Ω 4 Ω
(cid:112) Ω
Ω = a2−2(ac+b2)κ+c2(κ2+τ2);M = √ , (3.25)
2ac+b2
118
F. Almaz, M.A. Ku¨lahci – A Note on Special Smarandache Curves ...
1
L = (2(ρ(cid:48) +κρ +ρ τ)(−ρ +ρ(cid:48))+(ρ(cid:48) +ρ −κρ )2+(ρ(cid:48) −τρ )2),
M2 1 2 3 2 4 2 1 4 3 4
L
φ = √ .
2 2ac+b2
ii) The cone curvatures κ (s∗) and τ (s∗) of the curve γ is given by
γ γ
xαy xαy xαy
L
κ (s∗) = − , (3.26)
γ
xαy 2
ρ(cid:48) +κρ +ρ τ −ρ +ρ(cid:48)
τ2 (s∗) = 2( 1 2 3 +aφ+κ(cid:48))( 2 4 +cφ)
γxαy M M
ρ(cid:48) +ρ −κρ ρ(cid:48) +τρ L2
+( 2 1 4 +bφ+κ)2+( 3 4)2− (3.27)
M M 4
where
(cid:90)
1 (cid:112)
s∗ = √ a2−2(ac+b2)κ+c2(κ2+τ2)ds. (3.28)
2ac+b2
Proof. i) Let the curve x be a unit speed spacelike curve with the asymptotic or-
thonormal frame {x,α,β,y} and cone curvatures κ,τ. Differentiating the equation
(3.23) with respect to s and considering (2.1), we find
ds∗ 1 (cid:16) −−→ −−→ −−→ −−→(cid:17)
γ(cid:48) (s∗) = √ bκx(s)+(a−cκ)α(s)−cτβ(s)−by(s) .
xαy ds 2ac+b2
This can be written as following
bκ−−→ a−cκ−−→ cτ−−→ b−−→
α (s∗) = x(s)+ α(s)− β(s)− y(s), (3.29)
xαy Ω Ω Ω Ω
where
(cid:112)
Ω = a2−2(ac+b2)κ+c2(κ2+τ2). (3.30)
Differentiating (3.29) and using (3.30), we get
ρ(cid:48) +κρ +ρ τ ρ(cid:48) +ρ −κρ
γ(cid:48)(cid:48) = ( 1 2 3 )x+( 2 1 4)α
xαy M M
ρ(cid:48) −τρ −ρ +ρ(cid:48)
+( 3 4)β +( 2 4)y, (3.31)
M M
where ρ = bκ,ρ = a−cκ,ρ = −cτ,ρ = −b.
1 Ω 2 Ω 3 Ω 4 Ω
1 (cid:68) (cid:69) (cid:68) (cid:69)
y (s∗) = −γ(cid:48)(cid:48) − γ(cid:48)(cid:48) ,γ(cid:48)(cid:48) γ , and γ(cid:48)(cid:48) ,γ(cid:48)(cid:48) = L. (3.32)
xαy xαy 2 xαy xαy xαy xαy xαy
119
F. Almaz, M.A. Ku¨lahci – A Note on Special Smarandache Curves ...
By the help of equation (3.32), we obtain
ρ(cid:48) +κρ +ρ τ ρ(cid:48) +ρ −κρ
y (s∗) = (− 1 2 3 −aφ)x(s)+(− 2 1 4 −bφ)α(s)
xαy M M
ρ(cid:48) +τρ −ρ +ρ(cid:48)
+(− 3 4)β(s)+(− 2 4 −cφ)y(s), (3.33)
M M
where
L 1
φ = √ ,L = (2(ρ(cid:48) +κρ +ρ τ)(−ρ +ρ(cid:48))
2 2ac+b2 M2 1 2 3 2 4
+(ρ(cid:48) +ρ −κρ )2+(ρ(cid:48) −τρ )2),
2 1 4 3 4
ii) Using (3.22), we have (3.26) and (3.27).
Definition 4. Let x be unit speed spacelike curve lying on Q3 with the moving
asymptotic orthonormal frame {x,α,β,y}. Then, αβy−Smarandache curve of x is
defined by
1
γ (s∗) = √ (aα(s)+bβ(s)+cy(s)), (3.34)
αβy a2+b2
where a,b,c ∈ R+.
0
Theorem 4. Let x be unit speed spacelike curve in Q3 with the moving asymp-
totic orthonormal frame {x,α,β,y} and cone curvatures κ,τ and let γ be αβy−
αβy
(cid:110) (cid:111)
Smarandache curve with asymptotic orthonormal frame γ ,α ,β ,y .
αβy αβy αβy αβy
Then the following relations hold:
(cid:110) (cid:111)
i)Theasymptoticorthonormalframe γ ,α ,β ,y oftheαβy−Smaran-
αβy αβy αβy αβy
dache curve γ is given as
αβy
γ   0 √ a √ b √ c x
αβy a2+b2 a2+b2 a2+b2
α   A A A A α
 αβy =  1 2 3 4  ,
β   C C C C β
 αβy  1 2 3 4 
y −C −C − √aM −C − √bM −C − √cM y
αβy 1 2 2 a2+b2 3 2 a2+b2 4 2 a2+b2
(3.35)
where
(cid:112) ξ ds∗
ξ = c2τ2+(c2−2a2)κ2−2abτ;w = √ =
a2+b2 ds
120