Table Of ContentInternational J.Math. Combin. Vol.4(2016), 01-07
Isotropic Smarandache Curves in Complex Space C3
Su¨ha Yılmaz
(DokuzEylu¨lUniversity,BucaEducationalFaculty,35150,Buca-Izmir,Turkey)
E-mail: [email protected]
Abstract: Aregularcurveincomplexspace,whosepositionvectoriscomposedbyCartan
frame vectors on another regular curve, is called a isotropic Smarandache curve. In this
paper, I examine isotropic Smarandache curve according to Cartan frame in Complex 3-
space and give some differential geometric properties of Smarandache curves. We define
type-1 e e -isotropic Smarandache curves, type-2 e e -isotropic Smarandache curves and
1 3 1 3
e e e -isotropic Smarandachecurvesin Complex space C3.
1 2 3
Key Words: Complex space C3, isotropic Smarandache curves, isotropic cubic.
AMS(2010): 53A05, 53B25, 53B30.
§1. Introduction
It is observe that the imaginary curve in complex space were pioneered by E. Cartan. Cartan
defined his moving frame and his special equations in C3. In [6], the Cartan equations of
isotropic curve is extended to space C4. MoreoverU. Pekmen [2] wrote some characterizations
of minimal curves by means of E. Cartan equations in C3.
A regular curve in Euclidean 3-space, whose position vector is composed by Frenet frame
vectors on another regular curve, is called Smarandache curve. M. Turgut and S. Yilmaz have
defined a special case of such curves and call it Smarandache TB curves in the space E4 [7].
2 1
A.T.AlihasintroducedsomespecialSmarandachecurvesintheEuclideanspace[9]. Moreover,
special Smarandache curves have been investigated by using Bishop frame in Euclidean space
[10]. SpecialSmarandachecurvesaccordingtoSabbanframehavebeenstudiedby[11]. Besides
some special Smarandache curves have been obtained in E3 by [12]. Apart from M. Turgut
1
defined Smarandache breadth curves [8].
ItisgiventhatcomplexelementsandcomplexcurvestorealspaceR3 whicharementioned
byFerruhSemin,see[1]. IncomplexspaceC3 helicesarecharacterizedin[5]. Incomplexspace
C4,S.Yilmazcharacterizedtheisotropiccurveswithconstantpseudocurvaturewhichiscalled
the slant isotropic helix. Yılmaz and Turgut give some characterization of isotropic helices in
C3 [3].
Several authors introduce different types of helices and investigated their properties. For
instance,Barroset. al. studiedgeneralhelicesin3-dimensionalLorentzianspace. Izumiyaand
1ReceivedJanuary6,2016,Accepted November2,2016.
2 Su¨haYılmaz
Takeuchi defined slant helices by the property that principal normal mekes a constant angle
with a fixed direction [14]. Kula and Yayli studied spherical images of tangent and binormal
indicatrices of slant helices and they have shown that spherical images are spherical helix [15].
Ali and Lopez gave some characterizationof slant helices in Minkowski 3-space E3 [13].
1
In this work, using not common vector field know as Cartan frame, I introduce a new
Smarandache curves in C3. Also, Cartan apparatus of Smarandache curves have been formed
by Cartan apparatus of given curve α=α(s).
§2. Preliminaries
Let x be a complex analytic function of a complex variable t. Then the vector function
p
4
−→x (t)= xp(t)−→kp,
p=1
X
is called an imaginary curve, where −→x :C C4, −→kp are standard basis unit vectors of E3 [6].
→
An isotropic curve x = x(s) in C3 is called an isotropic cubic if pseudo curvature of x(s)
is congruent to zero. A direction (b ,b ,b ) is a minimal direction if and only if
1 2 3
3
b2 =0.
p
p=1
X
A vector which has a minimal direction is called an isotropic vector or minimal vector. A
vector −→ϑ is a minimal vector if and only if −→ϑ2 = 0. Common points of a complex plane and
absolutearecalledsiklik pointsoftheplane. Aplanewhichistangenttothe absoluteiscalled
aminimalplane,see[6]. Thecurves,ofwhichthesquareofthedistancebetweenthetwopoints
equaltozero,arecalledminimalorisotropiccurves[3]. Letsdenotepseudoarc-lengthAcurve
is an minimal (isotropic) curve if and only if ([4,5])
p 2
−→x (t) =0 (2.2)
(cid:2) (cid:3)
where dd→−xt = −→xp(t) 6= 0. Let be each point −→x of the isotropic curve. E. Cartan frame is
defined (for well-known complex number i2 = 1) as follows, (see [1,4])
−
−→e1 =−→xp
−→e2 =i−→xpp (2.3)
−→e3 =−β2−→xp+−→xppp
where β = (−→xppp)2, equation (2.3) denote by −→e1,−→e2,−→e3 the moving E. Cartan frame along
{ }
the isotropic curve −→x in the space C3.
IsotropicSmarandacheCurvesinComplexSpaceC3 3
The inner products of these frame vectors are given by
0 if i+j 1,2,3, (mod4)
−→ei·−→ej = 1 if i+j ≡=4 (2.4)
The cross (vectoral) and fixed products of these frame vectors are given by
−→ej −→ek =i−→ej+k 2
∧ − (2.5)
<−→e1,−→e2 −→e3 >=i
∧
t
for j,k = 1,2,3, s = [−→xp(t)]14dt is a pseudo arc length, also invariant with respect to
−
t0
parameter t. Thus the Rvector −→e1 and −→e3 are isotropic vector, −→e2 is real vector E. Cartan
derivative formulas can be deduced from equation (2.3) as follows
p
−→e1 =i−→e2
−→ep2 =i(k−→e1+−→e3) (2.6)
p
−→e3 =ik−→e2
where k = β is called pseudo curvature of isotropic curve x = x(s). These equations can be
2
usedifthe minimalcurveisatleastofclassC4.Here(p)denotesderivativeaccordingtopseudo
arclengths.Inthe restofthe paper,we willsupposepseudocurvatureisnon-vanishingexpect
in the case of an isotropic cubic. Isotropic sphere with center −→m and radius r > 0 in C3 is
defined by
S2 = −→p =(p1,p2,p3) C3 :(−→p −→m)2 =0 .
∈ −
(cid:8) (cid:9)
§3. Type-1 eαeα Isotropic Smarandache Curves
1 3−
Definition 3.1 Let α=α(s) be a unit speed regular isotropic curve in C3 and eα,eα,eα be
{ 1 2 3}
its moving Cartan frame. Type-1 eαeα-isotropic Smarandache curves can be defined by
1 3
1
ϑ(s )= (eα+eα). (3.1)
∗ √2 1 3
Now,wecaninvestigateCartaninvariantsofeαeα-isotropicSmarandachecurvesaccording
1 3
to α=α(s). Differentiating equation (3.1) with respect to pseudo arc length s, we obtain
dϑ ds i
ϑp = ∗ = (1+kα)eα (3.2)
ds ds −√2 2
∗
where
ds (1+kα)i
∗
= . (3.3)
ds √2
4 Su¨haYılmaz
The tangent isotropic vector of curve ϑ can be expressed as follow
eϑ = √1+kαeα (3.4)
1 − 2
Differentiating equation (3.4) with respect to pseudo arc length s, we obtain
eϑ p ds∗ =2(1+kα)ieα+(kα)peα+2(1+kα)ieα. (3.5)
1 ds 1 2 3
(cid:0) (cid:1)
Substituting equation (3.3) into equation (3.5), we find
p
p √2(kα)
eϑ = 2√2kα eα i eα+2√2eα.
1 1 − 1+kα ! 2 3
(cid:0) (cid:1) (cid:16) (cid:17)
p
Since eϑ = ieϑ, the principal vector field of curve ϑ
1 − 2
(cid:0) (cid:1)
√2(kα)p
eϑ = 2√2kα eα ieα+2√2eα. (3.6)
2 1 − 1+kα ! 2 3
(cid:16) (cid:17)
Using Cartan equation (2.6) , we have
3
√2(kα)p
eϑ =i kϑ 2√2kαeα+ eα+2√2ieα ds (3.7)
3 1 1+kα 2 3
" #
Z
and
p
eϑ
kϑ = 3 i. (3.8)
− eϑ
(cid:0) 2(cid:1)
Substituting equations (3.6) and (3.7) into equation (3.8), we obtain
p
√2(kα)p
i kϑ 2√2kαeα+ eα+2√2ieα ds
1 1+kα 2 3
( " # )
kϑ = Z i. (3.9)
2√2kαeα+ √2(kα)peα+2√2ieα
1 1+kα 2 3
Proposition 3.1 If ϑ a isotropic Smarandache curves in C3, then kα = 1.
−
Proof Using equation (3.4) and definition isotropic curves, it is seen straightforwardly.
2
Proposition 3.2 Let α = α(s) be a unit speed regular isotropic curve in C3, If δ a isotropic
cubic in C3, then pseudo curvature of α satisfies eϑ =constant and eϑ =0.
3 2 6
Proof It is seen straightforwardlyfrom definition isotrobic cubic.
2
§4. Type-2 eαeα Isotropic Smarandache Curves
1 3−
Definition 4.1 Let α=α(s) be a unit speed regular isotropic curve in C3 and eα,eα,eα be
{ 1 2 3}
IsotropicSmarandacheCurvesinComplexSpaceC3 5
its moving Cartan frame. Type-2 eαeα-isotropic Smarandache curves can be defined by
1 3
i
δ(s )= (eα eα). (4.1)
∗ √2 1 − 3
Now, we can investigate Cartan invariants of type-2 eαeα-isotropic Smarandache curves
1 3
according to α = α(s). Differentiating equation (4.1) with respect to pseudo arc length s, we
obtain
dδ ds 1
δp = ∗ = (kα 1)eα (4.2)
ds ds −√2 − 2
∗
and
ds 1
eδ ∗ = (kα 1)eα
1 ds −√2 − 2
where
ds √kα 1
∗
= − . (4.3)
ds √2
The tangent isotropic vector of curve δ can be expressed as follow
eδ = √kα 1eα (4.4)
1 − − 2
Differentiating equation (4.4) with respect to pseudo arc length s, we obtain
p
i(kα)
eδ =√kα 1kαeα eα+√kα 1eα. (4.5)
2 − 1 − 2√kα 1 2 − 3
−
Using definition, binormal vector field and pseudo curvature of isotropic Smarandache
curve δ are respectively,
p
i(kα)
eδ =i kδ √kα 1kαeα eα+√kα 1eα ds (4.6)
3 " − 1 − 2√kα 1 2 − 3#
Z −
and
p
p
i(kα)
i kδ √kα 1kαeα eα+√kα 1eα ds
(− " − 1 − 2√kα 1 2 − 3# )
kδ = Z − i. (4.7)
√kα 1kαeα i(kα)p eα+√kα 1eα
− 1 − 2√kα 1 2 − 3
−
Proposition 4.1 If δ a isotropic Smarandache curves in C3, then kα =1.
Proof Using equation (4.4) and definition isotropic curves, it is seen straightforwardly.
2
Proposition 4.2 Let α = α(s) be a unit speed regular isotropic curve in C3, If δ a isotropic
cubic in C3, then pseudo curvature of α satisfies eδ =constant and eδ =0.
3 2 6
Proof It is seen straightforwardlyfrom definition isotrobic cubic.
2
6 Su¨haYılmaz
§5. eαeαeα Isotropic Smarandache Curves
1 2 3−
Definition 5.1 Let α=α(s) be a unit speed regular isotropic curve in C3 and eα,eα,eα be
{ 1 2 3}
its moving Cartan frame. Type-1 eαeα-isotropic Smarandache curves can be defined by
1 3
1
η(s )= (eα+eα+eα). (5.1)
∗ √3 1 2 3
Now, we can investigate Cartan invariants of eαeαeα-isotropic Smarandache curves ac-
1 2 3
cording to α = α(s). Differentiating equation (5.1) with respect to pseudo arc length s, we
have
dη ds 1
ηp = ∗ = [ikαeα i(kα+1)eα+ieα] (5.2)
ds ds √3 1 − 2 3
∗
and
ds 1
ηp =eη ∗ = [ikαeα i(kα+1)eα+ieα]
1 ds √3 1 − 2 3
where
ds √1+kα
∗
= . (5.3)
ds √3
The tangent isotropic vector of curve η can be written as follow:
1
eη = [ikαeα i(kα+1)eα+ieα] (5.4)
1 √1+kα 1 − 2 3
Differentiating equation (5.4) with respect to pseudo arc length s, we obtain
p
eη = √3 [i(kα)p+(kα+1)k] √3 ikα eα
2 1−+kα − 1−+kα 1
(cid:26)(cid:16) (cid:17) (cid:16) (cid:17) p (cid:27)
√3 [ 2kα+(kα+1)p] √3 (kα+1) eα
− 1+kα − − 1−+kα 2
(cid:26)(cid:16) (cid:17)p (cid:16) (cid:17) (cid:27)
√3 + √3 eα
− 1−+kα 1+kα 3
(cid:26)(cid:16) (cid:17) (cid:16) (cid:17)(cid:27)
Using definition, binormal vector field and pseudo curvature of isotropic Smarandache
curve η are respectively
p
eη = i kη −√3 [(kα)p+(kα+1)kα] −√3 ikα eα
3 − { 1+kα! − 1+kα! 1
Z
p
√3 [2kα+(kα+1)p] √3 (kα+1) eα
− 1+kα − 1−+kα 2
(cid:26)(cid:16) (cid:17)p (cid:16) (cid:17) (cid:27)
√3 + √3 eα ds
− 1+kα 1+kα 3}
(cid:26)(cid:16) (cid:17) (cid:16) (cid:17)(cid:27)
Let eη =H(s) and eη =G(s) ın this case, we have
3 2
p
(H(s))
kη = . (5.5)
G(s)
IsotropicSmarandacheCurvesinComplexSpaceC3 7
Proposition 5.1 If η a isotropic Smarandache curves in C3, then kα = 1.
6 −
Proof Using equation (5.4) and definition isotropic curves, it is seen straightforwardly.
2
Proposition 5.2 Let α = α(s) be a unit speed regular isotropic curve in C3, If η a isotropic
cubic in C3, then pseudo curvature of α satisfies eη =constant and eη =0.
3 2 6
Proof It is seen straightforwardlyfrom definition isotrobic cubic.
2
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