b−Smarandache m1m2 Curves of Biharmontic New Type b−Slant Helices According to Bishop Frame in the Sol Space Sol3

2012·0.17 MB·

by  Talat Korpinar

International J.Math. Combin. Vol.4(2012), 33-39
b Smarandache m m Curves of Biharmontic New Type b Slant
1 2
− −
Helices According to Bishop Frame in the Sol Space Sol3
Talat KO¨RPINAR and Essin TURHAN
(DepartmentofMathematicsofFıratUniversity,23119,Elazıg˘,TURKEY)
E-mail: [email protected],[email protected]
Abstract: Inthispaper,westudyb−Smarandachem1m2 curvesofbiharmonic newtype
b−slant helix in the Sol3. We characterize the b−Smarandache m1m2 curves in terms
of their Bishop curvatures. Finally, we find out their explicit parametric equations in the
Sol3.
Key Words: New typeb−slant helix, Sol space, curvatures.
AMS(2010): 53A04, 53A10
§1. Introduction
A smooth map φ : N M is said to be biharmonic if it is a critical point of the bienergy
−→
functional:
1
E (φ)= (φ)2dv ,
2 h
2|T |
ZN
where (φ):=tr φdφ is the tension field of φ.
T ∇
TheEuler–Lagrangeequationofthebienergyisgivenby (φ)=0. Herethesection (φ)
2 2
T T
is defined by
(φ)= ∆ (φ)+trR( (φ),dφ)dφ, (1.1)
2 φ
T − T T
andcalledthebitensionfieldofφ. Non-harmonicbiharmonicmapsarecalledproperbiharmonic
maps.
This study is organized as follows: Firstly, we study b−Smarandache m1m2 curves of
biharmonicnewtypeb slanthelix inthe Sol3. Secondly,we characterizethe b Smarandache
− −
m1m2 curves in terms of their Bishop curvatures. Finally, we find explicit equations of
b−Smarandache m1m2 curves in the Sol3.
§2. Riemannian Structure of Sol Space Sol3
Solspace,oneofThurston’seight3-dimensionalgeometries,canbeviewedasR3 providedwith
Riemannian metric
gSol3 =e2zdx2+e−2zdy2+dz2,
1ReceivedAugust28,2012. Accepted December2,2012.
34 TalatKO¨RPINARandEssinTURHAN
where (x,y,z) are the standard coordinates in R3 [11,12].
Note that the Sol metric can also be written as:
3
gSol3 = ωi ωi,
⊗
i=1
X
where
ω1 =ezdx, ω2 =e−zdy, ω3 =dz,
and the orthonormal basis dual to the 1-forms is
∂ ∂ ∂
e =e−z , e =ez , e = . (2.1)
1 2 3
∂x ∂y ∂z
Proposition2.1 ForthecovariantderivativesoftheLevi-Civitaconnectionoftheleft-invariant
metric gSol3, defined above the following is true:
e 0 e
3 1
−
= 0 e e , (2.2)
∇ 3 − 2
 0 0 0 
 
 
where the (i,j)-element in the table above equals ∇eiej for our basis
e ,k =1,2,3 = e ,e ,e .
{ k } { 1 2 3}
Lie brackets can be easily computed as:
[e ,e ]=0, [e ,e ]= e , [e ,e ]=e .
1 2 2 3 − 2 1 3 1
The isometry groupof Sol3 has dimension 3. The connected component of the identity is
generated by the following three families of isometries:
(x,y,z) (x+c,y,z),
→
(x,y,z) (x,y+c,z),
→
(x,y,z) e−cx,ecy,z+c .
→
(cid:0) (cid:1)
§3. Biharmonic New Type b Slant Helices in Sol Space Sol3
−
Assume that t,n,b be the Frenet frame field along γ. Then, the Frenet frame satisfies the
{ }
following Frenet–Serret equations:
tt = κn,
∇
tn = κt+τb, (3.1)
∇ −
tb = τn,
∇ −
b−Smarandachem1m2CurvesofBiharmonticNewTypeb−SlantHelices 35
where κ is the curvature of γ and τ its torsion [14,15] and
gSol3(t,t) = 1, gSol3(n,n)=1, gSol3(b,b)=1, (3.2)
gSol3(t,n) = gSol3(t,b)=gSol3(n,b)=0.
The Bishop frame or parallel transport frame is an alternative approach to defining a
movingframethatiswelldefinedevenwhenthecurvehasvanishingsecondderivative,[1]. The
Bishop frame is expressed as
tt = k1m1+k2m2,
∇
tm1 = k1t, (3.3)
∇ −
tm2 = k2t,
∇ −
where
gSol3(t,t) = 1, gSol3(m1,m1)=1, gSol3(m2,m2)=1, (3.4)
gSol3(t,m1) = gSol3(t,m2)=gSol3(m1,m2)=0.
Here, we shall call the set t,m ,m as Bishop trihedra, k and k as Bishop curvatures
{ 1 2} 1 2
and δ(s)=arctank2, τ(s)=δ′(s) and κ(s)= k2+k2.
k1 1 2
Bishop curvatures are defined by
p
k = κ(s)cosδ(s),
1
k = κ(s)sinδ(s).
2
The relation matrix may be expressed as
t=t,
n=cosδ(s)m +sinδ(s)m ,
1 2
b= sinδ(s)m +cosδ(s)m .
1 2
−
On the other hand, using above equation we have
t=t,
m =cosδ(s)n sinδ(s)b
1
−
m =sinδ(s)n+cosδ(s)b.
2
With respect to the orthonormal basis e ,e ,e we can write
{ 1 2 3}
t = t1e +t2e +t3e ,
1 2 3
m = m1e +m2e +m3e , (3.5)
1 1 1 1 2 1 3
m = m1e +m2e +m3e .
2 2 1 2 2 2 3
Theorem 3.1 γ :I Sol3 is a biharmonic curve according to Bishop frame if and only if
−→
k2+k2 = constant=0,
1 2 6
k′′ k2+k2 k = k 2m3 1 2k m3m3, (3.6)
1 − 1 2 1 − 1 2− − 2 1 2
k2′′−(cid:2)k12+k22(cid:3)k2 = 2k1m(cid:2)31m32−k2(cid:3) 2m31−1 .
(cid:2) (cid:3) (cid:2) (cid:3)
36 TalatKO¨RPINARandEssinTURHAN
Theorem 3.2 Let γ :I Sol3 be a unit speed non-geodesic biharmonic new type b slant
−→ −
helix with constant slope. Then, the position vector of γ is
cos
γ(s) = [ M [ cos[ s+ ]+sin sin[ s+ ]]+ e−sinMs+S3]e
2+sin2 −S1 S1 S2 M S1 S2 S4 1
S1 M
cos
+[ M [ sin cos[ s+ ]+ sin[ s+ ]]+ esinMs−S3]e
2+sin2 − M S1 S2 S1 S1 S2 S5 2
S1 M
+[ sin s+ ]e , (3.7)
3 3
− M S
where , , , , are constants of integration, [8].
1 2 3 4 5
S S S S S
We can use Mathematica in Theorem 3.4, yields
Fig.1
§4. b−Smarandache m1m2 Curves ofBiharmonicNewType b−SlantHelicesinSol3
To separate a Smarandache m1m2 curve according to Bishop frame from that of Frenet-
Serret frame, in the rest of the paper, we shall use notation for the curve defined above as
b−Smarandache m1m2 curve.
Definition 4.1 Let γ :I Sol3 be a unit speed non-geodesic biharmonic new type b slant
−→ −
helix and {t,m1,m2} be its moving Bishop frame. b−Smarandache m1m2 curves are defined
by
1
γm1m2 = k2+k2 (m1+m2). (4.1)
1 2
p
Theorem 4.2 Let γ :I Sol3 be a unit speed non-geodesic biharmonic new type b slant
−→ −
helix. Then, the equation of b−Smarandache m1m2 curves of biharmonic new type b−slant
b−Smarandachem1m2CurvesofBiharmonticNewTypeb−SlantHelices 37
helix is given by
1
γm1m2(s) = k2+k2[sinMsin[S1s+S2]+cos[S1s+S2]]e1
1 2
1
+p [sin cos[ s+ ] sin[ s+ ]]e (4.2)
1 2 1 2 2
k2+k2 M S S − S S
1 2
1
+p [cos ]e ,
3
k2+k2 M
1 2
p
where , are constants of integration.
1 2
C C
Proof Assume that γ is a non geodesic biharmonic new type b slant helix according to
−
Bishop frame.
From Theorem 3.2, we obtain
m =sin sin[ s+ ]e +sin cos[ s+ ]e +cos e , (4.3)
2 1 2 1 1 2 2 3
M S S M S S M
where , R.
1 2
S S ∈
Using Bishop frame, we have
m =cos[ s+ ]e sin[ s+ ]e . (4.4)
1 1 2 1 1 2 2
S S − S S
Substituting (4.3) and (4.4) in (4.1) we have (4.2), which completes the proof. (cid:3)
In terms of Eqs. (2.1) and (4.2), we may give:
Corollary 4.3 Let γ :I Sol3 be a unit speed non-geodesic biharmonic new type b slant
−→ −
helix. Then, the parametric equations of b−Smarandache tm1m2 curves of biharmonic new
type b slant helix are given by
−
− 1 [cosM]
e √k12+k22
xtm1m2(s) = k2+k2 [sinMsin[S1s+S2]+cos[S1s+S2]],
1 2
1 [cosM]
e√pk12+k22
ytm1m2(s) = k2+k2 [sinMcos[S1s+S2]−sin[S1s+S2]], (4.5)
1 2
1
ztm1m2(s) = kp2+k2[cosM],
1 2
p
where , are constants of integration.
1 2
S S
Proof Substituting (2.1) to (4.2), we have (4.5) as desired. (cid:3)
We may use Mathematica in Corollary 4.3, yields
38 TalatKO¨RPINARandEssinTURHAN
Fig.2
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