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Thai Journal of Mathematics

Volume 9 (2011) Number 1 : 127–137

www.math.science.cmu.ac.th/thaijournal

Online ISSN 1686-0209

Null Biminimal General Helices

in the Lorentzian Heisenberg Group

Essin Turhan and Talat K¨ orpinar

Department of Mathematics, Faculty of Science,

Fırat University, Elazı˘ g 23119, Turkey

e-mail : essin.turhan@gmail.com,

talatkorpinar@gmail.com

Abstract : In this paper, we study null biminimal curves in the Lorentzian

Heisenberg group Heis3. We characterize non-geodesic null biminimal general helix

in terms of its curvatures and torsions in the Lorentzian Heisenberg group Heis3.

Keywords : Heisenberg group; Biminimal curve; General helix.

2010 Mathematics Subject Classification : 58E20.

1 Introduction

Let f : (M,g) → (N,h) be a smooth function between two Lorentzian mani-

folds. f is harmonic over compact domain Ω ⊂ M if it is a critical point of the

energy

?

where dvg is the volume form of M. From the first variation formula it follows

that is harmonic if and only if its first tension field τ (f) = traceg∇df vanishes.

Harmonic maps between Riemannian manifolds were first introduced and es-

tablished by Eells and Sampson [1] in 1964. Afterwards, there were two reports

on harmonic maps by Eells and Lemaire [2, 3] in 1978 and 1988.

The bienergy E2(f) of f over compact domain Ω ⊂ M is defined by

?

E (f) =

Ω

h(df,df)dvg,

E2(f) =

Ω

h(τ (f),τ (f))dvg, (1.1)

Copyright

All rights reserved.

c ?

2011 by the Mathematical Association of Thailand.

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Thai J. Math. 9 (2011)/ E. Turhan and T. K¨ orpinar

where τ (f) = traceg∇df is the tension field of f. Using the first variational formula

one sees that f is a biharmonic function if and only if its bitension field vanishes

identically, i.e.

? τ(f) := −△f(τ(f)) − tracegRN(df,τ(f))df = 0,

△f= −traceg(∇f)2= −traceg

(1.2)

where

?

∇f∇f− ∇f

∇M

?

(1.3)

is the Laplacian on sections of the pull-back bundle f−1(TN) and RNis the

curvature operator of (N,h) defined by

R(X,Y )Z = −[∇X,∇Y]Z + ∇[X,Y ]Z.

Biharmonic maps, which generalized harmonic maps, were first studied by

Jiang [4] in 1986.

Recently, there has been a growing interest in the theory of biharmonic maps

which can be divided in two main research directions. On the one side, con-

structing the examples and classification results have become important from the

differential geometric aspect. The other side is the analytic aspect from the point

of view of partial differential equations [5–9], because biharmonic maps are solu-

tions of a fourth order strongly elliptic semilinear PDE. In differential geometry,

harmonic maps, candidate minimisers of the Dirichlet energy, can be described as

constraining a rubber sheet to fit on a marble manifold in a position of elastica

equilibrium, i.e. without tension [2]. However, when this scheme falls through,

and it can, as corroborated by the case of the two-torus and the two-sphere [10], a

best map will minimise this failure, measured by the total tension, called bienergy.

In the more geometrically meaningful context of immersions, the fact that the

tension field is normal to the image submanifold, suggests that the most effective

deformations must be sought in the normal direction [11–22].

An isometric immersion f : (M,g) −→ (N,h) is called a λ−biminimal immer-

sion if it is a critical point of the functional:

E2,λ(f) = E2(f) + λE(f) , λ ∈ R.

The Euler-Lagrange equation for λ−biminimal immersions is

? τ(f)⊥= λτ(f). (1.4)

Particularly, f is called a biminimal immersion if it is a critical point of the

bienergy functional E2with respect to all normal variation with compact support.

Here, a normal variation means a variation {ft} through f = f0 such that the

variational vector field V = dft/dt|t=0is normal to M.

The Euler-Lagrange equation of this variational problem is ? τ(f)⊥= 0. Here

In this paper, we study biminimal curves in Heisenberg group Heis3. Then we

prove that the non-geodesic null biminimal general helices are circular helices. We

characterize non-geodesic null biminimal general helix in terms of its curvatures

and torsions in the Lorentzian Heisenberg group Heis3.

? τ(f)⊥is the normal component of ? τ(f).

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Null Biminimal General Helices in the Lorentzian Heisenberg Group

129

2

Preliminaries

The Lorentzian Heisenberg group Heis3can be seen as the space R3endowed

with the following multiplication:

(x,y,z)(x,y,z) = (x + x,y + y,z + z − xy + xy).

Heis3is a three-dimensional, connected, simply connected and 2-step nilpotent

Lie group.

The Lorentz metric g is given by:

g = −dx2+ dy2+ (xdy + dz)2,

where

ω1= dz + xdy,ω2= dy,ω3= dx

is the left-invariant orthonormal coframe associated with the orthonormal left-

invariant frame,

∂

∂z, e2=

e1=

∂

∂y− x∂

∂z, e3=

∂

∂x

(2.1)

for which we have the Lie products

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

with

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

Proposition 2.1. For the covariant derivatives of the Levi-Civita connection of

the left-invariant metric g, defined above the following is true:

e2

−e1

∇ =

0

e3

e3

0

e2

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 curvature

operator:

R(X,Y )Z = −∇X∇YZ + ∇Y∇XZ + ∇[X,Y ]Z.

The Riemannian curvature tensor is given by

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

Moreover, we put

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

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Thai J. Math. 9 (2011)/ E. Turhan and T. K¨ orpinar

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)

3 Null Biminimal Curves in Lorentzian Heisen-

berg Group

Let γ : I −→ Heis3be a null curve on the Lorentzian Heisenberg group Heis3

parametrized by arc length. Let {T,N,B} be the Frenet frame fields tangent to

Lorentzian Heisenberg group Heis3along γ defined as follows: T is the unit vector

field γ′tangent to γ, N is the unit vector field in the direction of ∇TT (normal to

γ), and B is chosen so that {T,N,B} is a positively oriented orthonormal basis.

Then, we have the following Frenet formulas:

∇TT

∇TN

∇TB

=κ1N,

κ2T − κ1B,

−κ2N,

=

=

(3.1)

where

g(T,T)

g(T,N)

=

=

g(B,B) = 0, g(N,N) = 1,

g(N,B) = 0, g(T,B) = 1,

(3.2)

and κ1= |τ(γ)| = |∇TT| is the curvature of γ and κ2is its torsion.

With respect to the orthonormal basis {e1,e2,e3}, we can write

T

N

=

=

T1e1+ T2e2+ T3e3,

N1e1+ N2e2+ N3e3,

T × N = B1e1+ B2e2+ B3e3.

(3.3)

B=

Theorem 3.1. Let γ : I −→ Heis3be a non-geodesic null curve parametrized by

arc length. γ is a non-geodesic null biminimal curve if and only if

κ′′

1κ2+ κ′

1+ 2κ2

1κ2

2κ1

=

=

4κ1B2

−4κ1N1B1.

1, (3.4)

2κ′

Proof. Using Eq. (1.4) and Eq. (3.1), we have

τ2(γ)=∇3

(2κ′

0.

TT − κ1R(T,N)T

2κ1+ κ′

=

=

1κ2)T + (κ′′

1+ 2κ2

1κ2)N + (3κ1κ′

1)B + κ1R(T,N)T

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Null Biminimal General Helices in the Lorentzian Heisenberg Group

131

By Eq. (3.2), we see that γ is a biharmonic curve if and only if

κ′′

1+ 2κ2

2κ2κ′

1κ2

=

=

κ1R(T,N,T,N),

κ1R(T,N,T,B).

(3.5)

1+ κ1κ′

2

A direct computation using Eq. (2.4) yields

R(T,N,T,N) = −4B2

1,

and

R(T,N,T,B) = 4N1B1.

These, together with Eq. (3.5), complete the proof of the theorem.

4 Null Biminimal General Helix in the Lorentzian

Heisenberg Group

Definition 4.1. Let γ : I −→ Heis3be a curve and {T,N,B} be a Frenet frame

on Heis3along γ. If κ1 and κ2 are positive constant along γ, then γ is called

circular helix with respect to Frenet frame [11].

Definition 4.2. Let γ : I −→ Heis3be a curve and {T,N,B} be a Frenet frame

on Heis3along γ. A curve γ such that

κ1

κ2

= constant (4.1)

is called a general helix with respect to Frenet frame [11].

Theorem 4.3. Let γ : I −→ Heis3be a non-geodesic null biminimal general helix

parametrized by arc length. If N1B1= constant, then γ is circular helix.

Proof. We can use Eq. (3.3) to compute the covariant derivatives of the vector

fields T,N and B as:

∇TT

∇TN

=

=

T′

(N′

+(N′

(B′

+(B′

1e1+ (T′

1+ T2N3− T3N2)e1+ (N′

3+ T2N1− T1N2)e3,

1+ T2B3− T3B2)e1+ (B′

3+ T2B1− T1B2)e3.

2+ 2T1T3)e2+ (T′

3+ 2T1T2)e3,

2+ T1N3− T3N1)e2

(4.2)

∇TB=

2+ T1B3− T3B1)e2

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

< ∇TT,e1>= T′

< ∇TN,e1>= N′

< ∇TB,e1>= B′

1,

1+ T2N3− T3N2,

1+ T2B3− T3B2.

(4.3)