An. Şt. Univ. Ovidius Constanţa Vol. 19(1), 2011, 285–296

Position vector of spacelike biharmonic curvesin the Lorentzian Heisenberg group Heis3

Essin TURHAN, Talat KÖRPINAR

Abstract

In this paper, we study spacelike biharmonic curves in the LorentzianHeisenberg group Heis3. We show that spacelike biharmonic curves aregeneral helices. We characterize position vector of spacelike biharmonicgeneral helices in terms of their curvature and torsion.

1 Introduction

Let (M, g) and (N, h) be Lorentzian manifolds and φ : M −→ N a smoothmap. Denote by ∇φ the connection of the vector bundle φ∗TN induced fromthe Levi-Civita connection ∇h of (N,h). The second fundamental form ∇dφis defined by

(∇dφ) (X, Y ) = ∇φXdφ (Y )− dφ (∇XY ) , X, Y ∈ Γ (TM) .Here ∇ is the Levi-Civita connection of (M, g). The tension field τ (φ) is

a section of φ∗TN defined by

τ (φ) = tr∇dφ. (1.1)A smooth map φ is said to be harmonic if its tension field vanishes. It is

well known that φ is harmonic if and only if φ is a critical point of the energy :

E (φ) =12

∫h (dφ, dφ) dvg

Key Words: Biharmonic curve, general helices, Heisenberg group.Mathematics Subject Classification: 31B30, 58E20Received: February, 2010Accepted: December, 2010

285

286 Essin TURHAN, Talat KÖRPINAR

over every compact region of M . Now let φ : M −→ N be a harmonic map.Then the Hessian H of E is given by

Hφ (V, W ) =∫

h (Jφ (V ) , W ) dvg, V, W ∈ Γ (φ∗TN) .

Here the Jacobi operator Jφ is defined by

Jφ (V ) := ∆φV − Rφ (V ) , V ∈ Γ (φ∗TN) , (1.2)

∆φ :=m∑

i=1

(∇φei∇φei −∇φ∇eiei

), Rφ (V ) =

m∑

i=1

RN (V, dφ (ei)) dφ (ei) , (1.3)

where RN and {ei} are the Riemannian curvature of N , and a local orthonor-mal frame field of M , respectively.

Let φ : (M, g) → (N, h) be a smooth map between two Lorentzian mani-folds. The bienergy E2(φ) of φ over compact domain Ω ⊂ M is defined by

E2 (φ) =∫

Ω

h (τ (φ) , τ (φ)) dvg.

A smooth map φ : (M, g) → (N,h) is said to be biharmonic if it is a criticalpoint of the E2(φ).

The section τ2(φ) is called the bitension field of φ and the Euler-Lagrangeequation of E2 is

τ2(φ) := −Jφ (τ(φ)) = 0. (1.4)Recently, there have been a growing interest in the theory of biharmonic

maps which can be divided into two main research directions. On the one side,the differential geometric aspect has driven attention to the construction ofexamples and classification results. The other side is the analytic aspect fromthe point of view of PDE: biharmonic maps are solutions of a fourth orderstrongly elliptic semilinear PDE.

Biharmonic functions are utilized in many physical situations, particu-larly in fluid dynamics and elasticity problems. Most important applicationsof the theory of functions of a complex variable were obtained in the planetheory of elasticity and in the approximate theory of plates subject to nor-mal loading. That is, in cases when the solutions are biharmonic functionsor functions associated with them. In linear elasticity, if the equations areformulated in terms of displacements for two-dimensional problems then theintroduction of a stress function leads to a fourth-order equation of biharmonictype. For instance, the stress function is proved to be biharmonic for an elasti-cally isotropic crystal undergoing phase transition, which follows spontaneous

Position vector of spacelike biharmonic curves ... 287

dilatation. Biharmonic functions also arise when dealing with transverse dis-placements of plates and shells. They can describe the deflection of a thinplate subjected to uniform loading over its surface with fixed edges. Bihar-monic functions arise in fluid dynamics, particularly in Stokes flow problems(i.e., low-Reynolds-number flows). There are many applications for Stokesflow such as in engineering and biological transport phenomena (for details,see [8, 13]). Fluid flow through a narrow pipe or channel, such as that usedin micro-fluidics, involves low Reynolds number. Seepage flow through cracksand pulmonary alveolar blood flow can also be approximated by Stokes flow.Stokes flow also arises in flow through porous media, which have been longapplied by civil engineers to groundwater movement. The industrial appli-cations include the fabrication of microelectronic components, the effect ofsurface roughness on lubrication, the design of polymer dies and the develop-ment of peristaltic pumps for sensitive viscous materials. In natural systems,creeping flows are important in biomedical applications and studies of animallocomotion.

Non-geodesic biharmonic curves are called proper biharmonic curves. Ob-viously geodesics are biharmonic. Caddeo, Montaldo and Piu showed thaton a surface with non-positive Gaussian curvature, any biharmonic curve isa geodesic of the surface [6]. So they gave a positive answer to generalizedChen’s conjecture. Caddeo et al. in [5] studied biharmonic curves in the unit3-sphere. More precisely, they showed that proper biharmonic curves in S3are circles of geodesic curvature 1 or helices which are geodesics in the Cliffordminimal torus.

In this paper, we first write down the conditions that any spacelike bihar-monic curve in the Lorentzian Heisenberg group Heis3 must satisfy. Then, weprove that the non-geodesic biharmonic curves in the Lorentzian Heisenberggroup Heis3 are general helices. Finally, we characterize position vector ofspacelike biharmonic general helices in terms of their curvature and torsion.

2 Preliminaries

The Heisenberg group Heis3 is a Lie group which is diffeomorphic to R3 andthe group operation is defined as

(x, y, z) ∗ (x, y, z) = (x + x, y + y, z + z − xy + xy).The identity of the group is (0, 0, 0) and the inverse of (x, y, z) is given by

(−x,−y,−z). The left-invariant Lorentz metric on Heis3 isg = −dx2 + dy2 + (xdy + dz)2.

288 Essin TURHAN, Talat KÖRPINAR

The following set of left-invariant vector fields forms an orthonormal basisfor the corresponding Lie algebra:

{e1 =

∂

∂z, e2 =

∂

∂y− x ∂

∂z, e3 =

∂

∂x

}. (2.1)

The characterising properties of this algebra are the following commutationrelations:

[e2, e3] = 2e1, [e3, e1] = 0, [e2, e1] = 0,

withg(e1, e1) = g(e2, e2) = 1, g(e3, e3) = −1. (2.2)

Proposition 2.1. For the covariant derivatives of the Levi-Civita connec-tion of the left-invariant metric g, defined above the following is true:

∇ =

0 e3 e2e3 0 e1e2 −e1 0

, (2.3)

where the (i, j)-element in the table above equals ∇eiej for our basis{ek, k = 1, 2, 3} = {e1, e2, e3}.

We adopt the following notation and sign convention for Riemannian cur-vature operator:

R(X, Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z.The Riemannian curvature tensor is given by

R(X, Y, Z, W ) = g(R(X, Y )W,Z).

Moreover, we put

Rabc = R(ea, eb)ec, Rabcd = R(ea, eb, ec, ed),

where the indices a, b, c and d take the values 1, 2 and 3.Then the non-zero components of the Riemannian curvature tensor field

and of the Riemannian curvature tensor are, respectively,

R121 = e2, R131 = e3, R232 = −3e3,and

R1212 = −1, R1313 = 1, R2323 = −3. (2.4)

Position vector of spacelike biharmonic curves ... 289

3 Position Vectors of Spacelike Biharmonic Curves InLorentzian Heisenberg Group Heis3

Let γ : I −→ Heis3 be a non geodesic curve on the Lorentzian Heisenberggroup Heis3 parametrized by arc length. A non geodesic curve γ is calledspacelike curve if g (γ′, γ′) > 0. Caddeo et al. [2], used the Frenet formulas inthe Riemannian case.

Let {T,N, B} be the Frenet frame fields tangent to the Lorentzian Heisen-berg group Heis3 along γ defined as follows:

T is the unit vector field γ′ tangent to γ, N is the unit vector field inthe direction of ∇T T (normal to γ) and B is chosen so that {T,N,B} is apositively oriented orthonormal basis. Then, we have the following Frenetformulas:

∇T T = κN∇T N = κT + τB∇T B = τN,

where κ (s) = |τ(γ)| = |∇T T | is the curvature of γ, τ (s) is its torsion andg (T, T ) = 1, g (N, N) = −1, g (B,B) = 1,g (T, N) = g (T,B) = g (N, B) = 0.

If we write this curve in the another parametric representation γ̃ = γ̃ (θ),where θ =

∫κ (s) ds. We have new Frenet equations as follows:

∇T̃ (θ)T̃ (θ) = Ñ (θ)∇T̃ (θ)Ñ (θ) = T̃ (θ) + f (θ) B̃ (θ) (3.1)∇T̃ (θ)B̃ (θ) = f (θ) Ñ (θ) ,

where f (θ) = τ(θ)κ(θ) .With respect to the orthonormal basis {e1, e2, e3}, we can write

T̃ (θ) = T̃1 (θ) e1 + T̃2 (θ) e2 + T̃3 (θ) e3,Ñ (θ) = Ñ1 (θ) e1 + Ñ2 (θ) e2 + Ñ3 (θ) e3, (3.2)B̃ (θ) = T̃ (θ)× Ñ (θ) = B̃1 (θ) e1 + B̃2 (θ) e2 + B̃3 (θ) e3.

Theorem 3.1. γ̃ = γ̃ (θ) is a non geodesic spacelike biharmonic curve inthe Lorentzian Heisenberg group Heis3 if and only if

f2 (θ) = −2 + 4B̃21 (θ) , (3.3)f ′ (θ) = 2Ñ1 (θ) B̃1 (θ) .

290 Essin TURHAN, Talat KÖRPINAR

Proof. Using (3.1), we have

τ2(γ̃) = ∇3T̃ (θ)T̃ (θ)−R(T̃ (θ) ,∇T̃ (θ)T̃ (θ))T̃ (θ)= (1 + f2 (θ))Ñ (θ) + f ′ (θ) B̃ (θ)−R(T̃ (θ) , Ñ (θ))T̃ (θ) .

Using (1.2), we see that γ̃ is a biharmonic curve if and only if

1 + f2 (θ) = −R(T̃ (θ) , Ñ (θ) , T̃ (θ) , Ñ (θ)), (3.4)f ′ (θ) = R(T̃ (θ) , Ñ (θ) , T̃ (θ) , B̃ (θ)).

A direct computation using (2.4), yields

R(T̃ (θ) , Ñ (θ) , T̃ (θ) , Ñ (θ)) = 1− 4B̃21 (θ) (3.5)R(T̃ (θ) , Ñ (θ) , T̃ (θ) , B̃ (θ)) = 2Ñ1 (θ) B̃1 (θ) ,

These, together with (3.4), complete the proof of the theorem.

Theorem 3.2. Let γ̃ = γ̃ (θ) is a non-geodesic spacelike biharmonic curvein the Lorentzian Heisenberg group Heis3. Then, γ̃ = γ̃ (θ) is a general helix.

Proof. Suppose that f ′ (θ) = 2Ñ1 (θ) B̃1 (θ) 6= 0. We shall derive a con-tradiction by showing that must be f (θ) =constant.

We can use (2.3) to compute the covariant derivatives of the vector fieldsT̃ (θ) , Ñ (θ) and B̃ (θ) as:

∇T̃ (θ)T̃ (θ) = T̃ ′1 (θ) e1 + (T̃ ′2 (θ) + 2T̃1 (θ) T̃3 (θ))e2+(T̃ ′3 (θ) + 2T̃1 (θ) T̃2 (θ))e3,

∇T̃ (θ)Ñ (θ) = (Ñ ′1 (θ) + T̃2 (θ) Ñ3 (θ)− T̃3 (θ) Ñ2 (θ))e1+(Ñ ′2 (θ) + T̃1 (θ) Ñ3 (θ)− T̃3 (θ) Ñ1 (θ))e2+(N ′3 (θ) + T̃2 (θ) Ñ1 (θ)− T̃1 (θ) Ñ2 (θ))e3, (3.6)

∇T̃ (θ)B̃ (θ) = (B̃′1 (θ) + T̃2 (θ) B̃3 (θ)− T̃3 (θ) B̃2 (θ))e1+(B′2 (θ) + T̃1 (θ) B̃3 (θ)− T̃3 (θ) B̃1 (θ))e2+(B̃′3 (θ) + T̃2 (θ) B̃1 (θ)− T̃1 (θ) B̃2 (θ))e3.

It follows that the first components of these vectors are given by

< ∇T̃ (θ)T̃ (θ) , e1 >= T̃ ′1 (θ) ,< ∇T̃ (θ)N (θ) , e1 >= Ñ ′1 (θ) + T̃2 (θ) Ñ3 (θ)− T̃3 (θ) Ñ2 (θ) , (3.7)< ∇T̃ (θ)B̃ (θ) , e1 >= B̃′1 (θ) + T̃2 (θ) B̃3 (θ)− T̃3 (θ) B̃2 (θ) .

Position vector of spacelike biharmonic curves ... 291

On the other hand, using Frenet formulas (3.1), we have,

< ∇T̃ (θ)T̃ (θ) , e1 >= Ñ1 (θ) ,< ∇T̃ (θ)Ñ (θ) , e1 >= T̃1 (θ) + f (θ) B̃1 (θ) , (3.8)< ∇T̃ (θ)B̃ (θ) , e1 >= f (θ) Ñ1 (θ) .

These, together with (3.7) and (3.8), give

T̃ ′1 (θ) = Ñ1 (θ) ,Ñ ′1 (θ) + B̃1 (θ) = T̃1 (θ) + f (θ) B̃1 (θ) , (3.9)

B̃′1 (θ) + T̃2 (θ) B̃3 (θ)− T̃3 (θ) B̃2 (θ) = f (θ) Ñ1 (θ) .Assume now that γ̃ is biharmonic.Differentiating (3.2) with respect to θ,

we obtainUsing f ′ (θ) = 2Ñ1 (θ) B̃1 (θ) 6= 0 and (3.3), we obtain

2f (θ) f ′ (θ) = 8B̃1 (θ) B̃′1 (θ) .

We substitute f ′ (θ) = 2Ñ1 (θ) B̃1 (θ) above equation, give

f (θ) Ñ1 (θ) B̃1 (θ) = 2B̃1 (θ) B̃′1 (θ) .

Then

f (θ) =2B̃′1 (θ)Ñ1 (θ)

. (3.10)

If we use T̃2 (θ) B̃3 (θ)− T̃3 (θ) B̃2 (θ) = −Ñ1 (θ) and (3.9), we getB̃′1 (θ) = (1 + f (θ))Ñ1 (θ) .

We substitute B̃′1 (θ) in equation (3.10):

f (θ) = −23

= constant.

Therefore also f (θ) is constant and we have a contradiction that is f ′ (θ) 6=0. Therefore γ̃ (θ) is not biharmonic.

Corollary 3.3. γ̃ = γ̃ (θ) is a spacelike biharmonic curve in the LorentzianHeisenberg group Heis3 if and only if

f (θ) = constant,Ñ1 (θ) B̃1 (θ) = 0, (3.11)

f2 (θ) = −2 + 4B̃21 (θ) .

292 Essin TURHAN, Talat KÖRPINAR

Theorem 3.4. The position vector of the spacelike biharmonic curve γ̃ =γ̃ (θ) in the Lorentzian Heisenberg group Heis3 is given by

γ̃ (θ) =((

1− 1µ

)θ − c1

µeµθ +

c2µ

e−µθ + c3

)T̃ (θ)

+(

c1eµθ + c2e−µθ +

1µ2

)Ñ (θ) (3.12)

+(

µ2 − 1µ

θ + c1µ2 − 1

µeµθ − c2 µ

2 − 1µ

e−µθ + c4

)B̃ (θ) ,

where µ =√

1 + f2 (θ) and c1, c2, c3, c4 are constants of integration.

Proof. If γ̃ (θ) is a non-geodesic biharmonic curve in the LorentzianHeisenberg group Heis3, then we can write its position vector as follows:

γ̃ (θ) = ξ (θ) T̃ (θ) + η (θ) Ñ (θ) + ρ (θ) B̃ (θ) (3.13)

for some differentiable functions ξ, η and ρ of θ ∈ I ⊂ R. These functions arecalled component functions (or simply components) of the position vector.

Differentiating (3.9) with respect to θ and by using the correspondingFrenet equation (3.1), we find

ξ′ (θ) + η (θ) = 1,η′ (θ) + ξ (θ) + f (θ) ρ (θ) = 0, (3.14)

ρ′ (θ) + f (θ) η (θ) = 0.

From (3.14), we get the following differential equation:

η′′ (θ)− (1 + f2 (θ))η (θ) + 1 = 0. (3.15)

The solution of (3.15) is

η (θ) = c1e√

1+f2(θ)θ + c2e−√

1+f2(θ)θ +1

1 + f2 (θ)(3.16)

= c1eµθ + c2e−µθ +1µ2

,

where c1, c2 ∈ R.The picture of η (θ) at c1 = c2 = f (θ) = 1 : From ξ′ (θ) = 1 − η (θ) and

Position vector of spacelike biharmonic curves ... 293

using (3.16), we find the solution of this equation as follows:

ξ (θ) =

(1− 1√

1 + f2 (θ)

)θ +

c1√1 + f2 (θ)

e√

1+f2(θ)θ

− c2√1 + f2 (θ)

e−√

1+f2(θ)θ + c3 (3.17)

=(

1− 1µ

)θ − c1

µeµθ +

c2µ

e−µθ + c3,

where c1, c2, c3 ∈ RThe picture of ξ (θ) at c1 = c2 = c3 = f (θ) = 1 : By using (3.16), we find

294 Essin TURHAN, Talat KÖRPINAR

the solution of ρ′ (θ) = η (θ) f (θ) as follows:

ρ (θ) = − f (θ)√1 + f2 (θ)

θ − f (θ) c1√1 + f2 (θ)

e√

1+f2(θ)θ

+f (θ) c2√1 + f2 (θ)

e−√

1+f2(θ)θ + c4 (3.18)

=µ2 − 1

µθ + c1

µ2 − 1µ

eµθ − c2 µ2 − 1µ

e−µθ + c4,

where c1, c2, c3, c4 ∈ RThe picture of ρ (θ) at c1 = c2 = c4 = f (θ) = 1 :

Substituting (3.16), (3.17) and (3.18) into the equation in (3.13), we have(3.12). This concludes the proof of Theorem 3.4.

The picture of γ̃ (θ) at c1 = c2 = c3 = c4 = f (θ) = 1 :

Acknowledgement. The author would like to thank the referee for sig-nificant corrections and kind comments.

Position vector of spacelike biharmonic curves ... 295

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Fırat University, Department of Mathematicse-mail: essin.turhan@gmail.com