# Biharmonic Curves in a Strict Walker 3-Manifold

Authors: Mamadou Gningue; Ameth Ndiaye; Rénovat Nkunzimana
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In this paper, we study the geometry of biharmonic curves in a strict Walker 3-manifold and we obtain explicit parametric equations for biharmonic curves and time-like biharmonic curves, respectively. We discuss the conditions for a speed curve to be a slant helix in a Walker manifold. We give an example of biharmonic curve for illustrating the main result.


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## Body

## 1. Introduction

The study of submanifolds of a given ambiant space is a natural interesting problem which enriches our knowledge and understanding of the geometry of the space itself. Here, the ambient space we will consider is a Lorentzian three-manifold admitting a parallel null vector field called strict Walker manifold. It is known that Walker metrics have served as a powerful tool of constructing interesting indefinite metrics which exhibit various aspects of geometric properties not given by any positive definite metrics. For details, see [1, 2].As a generalization of Legendre curve, the notion of slant curves was introduced in [3]. A curve in a Walker 3-manifold is said to be slant if its tangent vector field has a constant angle with a position vector field.Eells and Sampson [4] defined harmonic and biharmonic map between Riemannian manifolds. Jiang [5, 6] derived the first variation formula of the bienergy from the Euler–Lagrange equation. Harmonic maps are clearly biharmonic.Nonharmonic biharmonic maps are called proper biharmonic maps. Chen and Ishikawa [7] showed nonexistence of proper biharmonic curves in Euclidean 3-space E3. Moreover, they classified all proper biharmonic curves in Minkowski 3-space E13 (see [8]).Recently, Sasahara [9] introduced biharmonic maps between pseudo-Riemannian manifolds and studied proper biharmonic submanifolds in Lorentzian 3-space forms.Lee [10] studies biharmonic curves in 3-dimensional Lorentzian Heisenberg space ℍ3;g, and he shows that proper biharmonic space-like curve γ in Lorentzian Heisenberg space ℍ3;g is pseudohelix with some properties about their curvatures. Moreover, γ has the space-like normal vector field and is a slant curve. Finally, he found the parametric equations of them.In [11], the authors study the nongeodesic nonnull biharmonic curves in 3-dimensional hyperbolic Heisenberg group with a semi-Riemannian metric of index 2. They prove that all of the nongeodesic nonnull biharmonic curves in such a 3-dimensional hyperbolic Heisenberg group are helices. Moreover, they obtain explicit parametric equations for nongeodesic nonnull biharmonic curves and nongeodesic space-like horizontal biharmonic curves, respectively.And very recently, the author, in [12], studies proper biharmonic Frenet curves in 3-dimensional Lorentzian Sasakian space forms of constant holomorphic sectional curvature H. He gives a necessary and sufficient condition for a Frenet curve to be proper biharmonic in the Lorentzian Sasakian space forms. A classification of proper biharmonic Frenet curves in 3-dimensional Lorentzian Heisenberg space is also given.Motivated by the above works, in this paper, we study biharmonic curves in a strict Walker 3-manifold. We give a necessary and sufficient condition for curve in a Walker 3-manifold to be a slant helix.The paper is organised as follows. In Section2, we give some preliminaries tolls about biharmonic maps and Walker 3-manifold. In Section 3, we study biharmonic curves in a strict Walker 3-manifolds, and Section 4 talks about slant curves in this ambient space.


## 2. Preliminaries

### 2.1. Biharmonic Maps

In this section, we give some basic notions about biharmonic maps; for more details, see [11].LetM,g and N,h be Riemannian manifolds and ψ:M⟶N be a smooth map. The tension field of ψ (see [6]) is given by τψ=trace∇dψ, where ∇dψ is the second fundamental form of ψ defined by ∇dψX,Y=∇XψdψY−dψ∇XMY, X,Y∈ΓTM. For any compact domain Ω⊂M, the bienergy is defined by(1)E2ψ=12∫Ωτψ2νg.Then, a smooth mapψ is called biharmonic map if it is a critical point of the bienergy functional for any compact domain Ω⊂M. We have for the bienergy the following first variation formula:(2)ddtE2ψt,Ω|t=0=∫Ωτ2ψ,ωνg,where νg is the volume element, ω is the variational vector field associated to the variation ψt of ψ, and(3)τ2ψ=−Jτ2ψ=−Δψτψ−traceRNdψ,τψdψ.τ2ψ is called bitension field of ψ. Here, Δψ is the rough Laplacian on the sections of the pull-back bundle ψ−1TN which is defined by(4)ΔψV=−∑i=1m∇eiψ∇eiψV−∇∇eiMeiψV,V∈Γψ−1TN,where ∇ψ is the pull-back connection on the pull-back bundle ψ−1TN and eii=1m is an orthonormal frame on M. When the target manifold is semi-Riemannian manifold, the bienergy and bitension field can be defined in the same way.LetM be a semi-Riemannian manifold and γ:I⟶M be a nonnull curve parametrized by arc length. By using the definition of the tension field, we have(5)τγ=∇∂/∂sγdγ∂∂s=∇TT,where T=γ′. In this case, biharmonic equation for the curve γ reduces to(6)τ2γ=∇T3T−RT,∇TTT=0.


### 2.2. Walker Manifold

A Walkern-manifold is a pseudo-Riemannian manifold, which admits a field of null parallel r-planes, with r≤n/2. The canonical forms of the metrics were investigated by A. G. Walker [1]. Walker has derived adapted coordinates to a parallel plan field. Hence, the metric of a three-dimensional Walker manifold M,gfε with coordinates x,y,z is expressed as(7)gfε=dx∘dz+εdy2+fx,y,zdz2.And its matrix form is(8)gfε=0010ε010fwith inversegfε−1=−f010ε0100.For some functionfx,y,z, ε=±1, and thus, D=Span∂x as the parallel degenerate line field. Notice that when ε=1 and ε=−1, the Walker manifold has signature 2,1 and 1,2, respectively, and, therefore, is Lorentzian in both cases.It follows after a straightforward calculation that the Levi-Civita connection of any metric (1) is given by(9)∇∂x∂z=12fx∂x,∇∂y∂z=12fy∂x,∇∂z∂z=12ffx+fz∂x+12fy∂y−12fx∂z,where ∂x, ∂y, and ∂z are the coordinate vector fields ∂/∂x, ∂/∂y, and ∂/∂z, respectively. Hence, if M,gfε is a strict Walker manifolds, i.e., fx,y,z=fy,z, then the associated Levi-Civita connection satisfies(10)∇∂y∂z=12fy∂x,∇∂z∂z=12fz∂x−ε2fy∂y.Note that the existence of a null parallel vector field (i.e.,f=fy,z) simplifies the nonzero components of the Christoffel symbols and the curvature tensor of the metric gfε as follows:(11)Γ231=Γ321=12fy,Γ331=12fz,Γ332=−ε2fy.Starting from local coordinatesx,y,z for which (7) holds, it is easy to check that(12)e1=∂y,e2=2−f22∂x+12∂z,e3=2+f22∂x−12∂z,are local pseudo-orthonormal frame fields on M,gfε, with gfεe1,e1=ε, gfεe2,e2=1, and gfεe3,e3=−1. Thus, the signature of the metric gfϵ is ε,1,−1.Proposition 1.
For the covariant derivatives of the Levi-Civita connection of the left-invariant metricg defined above, we have(13)∇eiej=014fye2+e3−14fye2+e314fye2+e3−ε4fye1ε4fye1−14fye2+e3ε4fye1−ε4fye1.
The curvature tensor field of∇ is given by(14)RX,YZ=∇X∇YZ−∇Y∇XZ−∇X,YZ,where X,Y,Z∈ΓM. If we denote by(15)Rijk=Rei,ejek,where the indices i,j,andk take the values 1,2,and3, then the nonzero components of the curvature tensor field are(16)R121=−R131=−14fyye2+e3,R122=−R123=−R132=R133=ε4fyye1.
The vector product ofu and v in M,gfε with respect to the metric gfε is the vector denoted by u×fv in M defined by(17)gfεu×fv,w=detu,v,w,for all vector w in M, where detu,v,w is the determinant function associated to the canonical basis of ℝ3. If u=u1,u2,u3 and v=v1,v2,v3, then, by using (17), we have(18)u×fv=u1v1u2v2−fu2v2u3v3i⟶−εu1v1u3v3j⟶+u2v2u3v3k⟶.Lemma 1.
The Walker cross product inM has the following properties:(1)
The Walker cross product is bilinear and antisymmetric(2)
X×fY is perpendicular to both X and Y(3)
The frame defined in (5) verifies the following: e1×fe2=−e3, e2×fe3=−e1, and e3×fe1=e2


## 2.1. Biharmonic Maps

In this section, we give some basic notions about biharmonic maps; for more details, see [11].LetM,g and N,h be Riemannian manifolds and ψ:M⟶N be a smooth map. The tension field of ψ (see [6]) is given by τψ=trace∇dψ, where ∇dψ is the second fundamental form of ψ defined by ∇dψX,Y=∇XψdψY−dψ∇XMY, X,Y∈ΓTM. For any compact domain Ω⊂M, the bienergy is defined by(1)E2ψ=12∫Ωτψ2νg.Then, a smooth mapψ is called biharmonic map if it is a critical point of the bienergy functional for any compact domain Ω⊂M. We have for the bienergy the following first variation formula:(2)ddtE2ψt,Ω|t=0=∫Ωτ2ψ,ωνg,where νg is the volume element, ω is the variational vector field associated to the variation ψt of ψ, and(3)τ2ψ=−Jτ2ψ=−Δψτψ−traceRNdψ,τψdψ.τ2ψ is called bitension field of ψ. Here, Δψ is the rough Laplacian on the sections of the pull-back bundle ψ−1TN which is defined by(4)ΔψV=−∑i=1m∇eiψ∇eiψV−∇∇eiMeiψV,V∈Γψ−1TN,where ∇ψ is the pull-back connection on the pull-back bundle ψ−1TN and eii=1m is an orthonormal frame on M. When the target manifold is semi-Riemannian manifold, the bienergy and bitension field can be defined in the same way.LetM be a semi-Riemannian manifold and γ:I⟶M be a nonnull curve parametrized by arc length. By using the definition of the tension field, we have(5)τγ=∇∂/∂sγdγ∂∂s=∇TT,where T=γ′. In this case, biharmonic equation for the curve γ reduces to(6)τ2γ=∇T3T−RT,∇TTT=0.


## 2.2. Walker Manifold

A Walkern-manifold is a pseudo-Riemannian manifold, which admits a field of null parallel r-planes, with r≤n/2. The canonical forms of the metrics were investigated by A. G. Walker [1]. Walker has derived adapted coordinates to a parallel plan field. Hence, the metric of a three-dimensional Walker manifold M,gfε with coordinates x,y,z is expressed as(7)gfε=dx∘dz+εdy2+fx,y,zdz2.And its matrix form is(8)gfε=0010ε010fwith inversegfε−1=−f010ε0100.For some functionfx,y,z, ε=±1, and thus, D=Span∂x as the parallel degenerate line field. Notice that when ε=1 and ε=−1, the Walker manifold has signature 2,1 and 1,2, respectively, and, therefore, is Lorentzian in both cases.It follows after a straightforward calculation that the Levi-Civita connection of any metric (1) is given by(9)∇∂x∂z=12fx∂x,∇∂y∂z=12fy∂x,∇∂z∂z=12ffx+fz∂x+12fy∂y−12fx∂z,where ∂x, ∂y, and ∂z are the coordinate vector fields ∂/∂x, ∂/∂y, and ∂/∂z, respectively. Hence, if M,gfε is a strict Walker manifolds, i.e., fx,y,z=fy,z, then the associated Levi-Civita connection satisfies(10)∇∂y∂z=12fy∂x,∇∂z∂z=12fz∂x−ε2fy∂y.Note that the existence of a null parallel vector field (i.e.,f=fy,z) simplifies the nonzero components of the Christoffel symbols and the curvature tensor of the metric gfε as follows:(11)Γ231=Γ321=12fy,Γ331=12fz,Γ332=−ε2fy.Starting from local coordinatesx,y,z for which (7) holds, it is easy to check that(12)e1=∂y,e2=2−f22∂x+12∂z,e3=2+f22∂x−12∂z,are local pseudo-orthonormal frame fields on M,gfε, with gfεe1,e1=ε, gfεe2,e2=1, and gfεe3,e3=−1. Thus, the signature of the metric gfϵ is ε,1,−1.Proposition 1.
For the covariant derivatives of the Levi-Civita connection of the left-invariant metricg defined above, we have(13)∇eiej=014fye2+e3−14fye2+e314fye2+e3−ε4fye1ε4fye1−14fye2+e3ε4fye1−ε4fye1.
The curvature tensor field of∇ is given by(14)RX,YZ=∇X∇YZ−∇Y∇XZ−∇X,YZ,where X,Y,Z∈ΓM. If we denote by(15)Rijk=Rei,ejek,where the indices i,j,andk take the values 1,2,and3, then the nonzero components of the curvature tensor field are(16)R121=−R131=−14fyye2+e3,R122=−R123=−R132=R133=ε4fyye1.
The vector product ofu and v in M,gfε with respect to the metric gfε is the vector denoted by u×fv in M defined by(17)gfεu×fv,w=detu,v,w,for all vector w in M, where detu,v,w is the determinant function associated to the canonical basis of ℝ3. If u=u1,u2,u3 and v=v1,v2,v3, then, by using (17), we have(18)u×fv=u1v1u2v2−fu2v2u3v3i⟶−εu1v1u3v3j⟶+u2v2u3v3k⟶.Lemma 1.
The Walker cross product inM has the following properties:(1)
The Walker cross product is bilinear and antisymmetric(2)
X×fY is perpendicular to both X and Y(3)
The frame defined in (5) verifies the following: e1×fe2=−e3, e2×fe3=−e1, and e3×fe1=e2


## 3. Biharmonic Curves in Walker 3-Manifold

Letα:I⊂ℝ⟶M,gfε be a curve parametrized by its arc length s.The Frenet frame ofα is the vectors T, N, and B along α, where T is the tangent, N is the principal normal, and B is the binormal vector. They satisfied the Frenet formulas:(19)∇TTs=ε2κsNs,∇TNs=−ε1κTs−ε3τBs,∇TBs=ε2τsNs,where κ and τ are, respectively, the curvature and the torsion of the curve α, with ε1=gfT;T,ε2=gfN;N, and ε3=gfB,B.From (10), we obtain(20)∇T3T=−3ε1ε2κ′κT+−ε1κ3+ε2κ′′−ε3κτ2N−ε2ε32κ′τ+κτ′B.Using (16) and Lemma 1, we obtain(21)RT,∇TTT=ε2κfyyB2−B34B2−B3N+N2−N3B,where T=T1e1+T2e2+T3e3, N=N1e1+N2e2+N3e3, and B=B1e1+B2e2+B3e3.Hence, we obtain(22)τ2γ=−3ε1ε2κ′κT+ε1κ3−ε2κ″+ε3κτ2−ε2κfyyB2−B324N+ε2κfyyB3−B2N2−N34+ε2ε32κ′τ+κτ′B.Theorem 1.
Letα:I⊂ℝ⟶M,gfε be a curve parametrized by its arc length s. Then, α is a nongeodesic biharmonic curve if and only if(23)κ=Constant≠0,ε1ε3κ2+τ2=ε2ε3fyyB2−B324,τ′=fyyN2−N3B2−B3,where B2≠B3.Proof 1.
From (23), it follows that it is biharmonic if and only if(24)κ′κ=0,ε1κ3−ε2κ′′+ε3κτ2−ε2κfyyB2−B324=0,ε2κfyyB2−B3N2−N34+ε2ε32κτ′+κτ′=0.
The first equation of (24) shows that κ=constant≠0. Then, the second equation (24) becomes(25)ε1κ3+ε3κτ2−ε2κfyyB2−B324=0.
Sinceκ≠0, we obtain(26)ε1ε3κ2+τ2=ε2ε3fyyB2−B324.
Sinceκ=constant, the third equation also becomes(27)ε2κfyyB3−B2N2−N34+ε2ε3κτ′=0.
That implies also(28)−ε2fyyB2−B32N2−N34+ε2ε3τ′B2−B3=0.
Then, we have(29)τ′=ε1ε3κ2+τ2N2−N34ε2ε3B2−B3.
Sincefyy=ε1ε3κ2+τ2/4ε2ε3B2−B32, we obtain(30)τ′=fyyN2−N3B2−B3.Corollary 1.
Letα:I⊂ℝ⟶M,gfε be a time-like curve parameterized by its arc length s. We suppose that fy,z=A1zy+A2z, where Ai,i=1,2, are functions of z. Then, α is a nongeodesic biharmonic curve if and only if(31)κ=Constant≠0,τ=Constant≠0,τ2=ε3κ2.Remark 1.
In the condition of the corollary, the binormal must be space-like.


## 4. Slant Helix in Walker 3-Manifold

LetM,gf be a three-dimensional strict Walker manifold.Definition 1.
A unit speed curveα is called a slant helix if there exists a nonzero constant vector field U in M such that the function gfNs,U) is constant, where N is the normal of α.
We remark that, in Walker manifold like in Minkowski ambient space, we cannot define the angle between two vectors (except that both vectors are of time-like type). For this reason, we avoid to say about the angle between the vector fieldsNs and U. We have the following result in the three-dimensional Walker manifold.Theorem 2.
Letα be a speed unit curve in M. Then, α is a slant helix if only if(32)κ2τ/κ′ε1τ2+ε3κ23/2=constant,with ε1τ2+ε3κ2≠0.Proof 2.
Letα be a slant helix. Let U be a vector field such that the function gfNs;U=constant=b. There exist smooth functions a1 and a3 such that(33)U=a1Ts+bNs+a3Bs,where a1 and a3 are functions of s and T,N,B is the Frenet frame of the curve α.
Differentiating (33) with respect to s, one obtains(34)∇TU=a1′Ts+a1∇TT+b∇TN+a3′B+a3∇TB=a1′Ts+a1ε2kN−bε1kT−bε3τB+a3′B+a3ε2τN.
Then,∇TU=0 implies that(35)a1′−bε1k=0,a1ε2k+a3ε2τ=0,a3′−bε3τ=0.
From the second equation of (35), we obtain a1=−τ/ka3.
Moreover,(36)gfU;U=ε1a12+ε2b2+ε3a32=m,where m is a constant. According to a1=−τ/ka3, we obtain(37)gfU;U=ε1a32τk2+ε2b2+ε3a32=constant,and then, we get a32ε1τ/k2+ε3=constant−ε2b2.
We denote(38)3a32ε1τk2+ε3=εm2;m>0,ε∈−1,0,1.
Ifε=0, then we have a3=0 and a1=0. Therefore, b=1/ε1ka1′. Then, U=0, which is a contradiction. So, we have ε=1 or ε=−1.
The third equation of (35), a3′−bϵ3τ=0, implies that d/dsa3=bε3τ. And using the fact that a3=±m/ε1τ/k2+ε3, we obtain(39)dds±mε1τ/k2+ε3=bε3τ.
That is,(40)dds1ε1τ/k2+ε3=±bε3τm.
Finally,(41)k2τ/k′ε1τ2+ε3k23/2=±bm.
Conversely, assume that(42)k2τ/k′ε1τ2+ε3k23/2,is a constant, which we denote by b.
Now, we define the vectorU by(43)U=−ε1/ε3τε3k2+ε1τ2T+bN+ε3/ε1kε3k2+ε1τ2B.
A differentiation ofU together with the Frenet equations gives ∇TU=0, that is, U is a constant vector. On the contrary, gfNs;U=1, and this means that α is a slant helix. This concludes the proof of the theorem.Example 1.
Letγ:I⊂ℝ⟶M,gfε; the speed curve is given by(44)γs=9s3s+6,3s22,3s32.
An easy computation gives that the curvature and the torsion of the curveγ are κ2=τ2=9. Then, by Theorem 1, the curve γ is a helix.
By equations (21)–(23), we have τ2=0, and then, the curve γ is biharmonic [13].



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