Biharmonic Curves in a Strict Walker 3-Manifold

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.

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 . Moreover, they classified all proper biharmonic curves in Minkowski 3-space (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 , and he shows that proper biharmonic space-like curve in Lorentzian Heisenberg space is pseudohelix with some properties about their curvatures. Moreover, 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 Section 2, 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].

Let and be Riemannian manifolds and be a smooth map. The tension field of (see [6]) is given by , where is the second fundamental form of defined by , . For any compact domain , the bienergy is defined by

Then, a smooth map is called biharmonic map if it is a critical point of the bienergy functional for any compact domain . We have for the bienergy the following first variation formula:where is the volume element, is the variational vector field associated to the variation of , and

is called bitension field of . Here, is the rough Laplacian on the sections of the pull-back bundle which is defined bywhere is the pull-back connection on the pull-back bundle and is an orthonormal frame on . When the target manifold is semi-Riemannian manifold, the bienergy and bitension field can be defined in the same way.

Let be a semi-Riemannian manifold and be a nonnull curve parametrized by arc length. By using the definition of the tension field, we havewhere . In this case, biharmonic equation for the curve reduces to

2.2. Walker Manifold

A Walker -manifold is a pseudo-Riemannian manifold, which admits a field of null parallel -planes, with . 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 with coordinates is expressed as

And its matrix form is

For some function , , and thus, as the parallel degenerate line field. Notice that when and , the Walker manifold has signature and , 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 bywhere , , and are the coordinate vector fields , , and , respectively. Hence, if is a strict Walker manifolds, i.e., , then the associated Levi-Civita connection satisfies

Note that the existence of a null parallel vector field (i.e., ) simplifies the nonzero components of the Christoffel symbols and the curvature tensor of the metric as follows:

Starting from local coordinates for which (7) holds, it is easy to check thatare local pseudo-orthonormal frame fields on , with , , and . Thus, the signature of the metric is .

Proposition 1. For the covariant derivatives of the Levi-Civita connection of the left-invariant metric defined above, we haveThe curvature tensor field of is given bywhere . If we denote bywhere the indices take the values , then the nonzero components of the curvature tensor field areThe vector product of and in with respect to the metric is the vector denoted by in defined byfor all vector in , where is the determinant function associated to the canonical basis of . If and , then, by using (17), we have

Lemma 1. The Walker cross product in has the following properties:(1)The Walker cross product is bilinear and antisymmetric(2) is perpendicular to both and (3)The frame defined in (5) verifies the following: , , and

3. Biharmonic Curves in Walker 3-Manifold

Let be a curve parametrized by its arc length .

The Frenet frame of is the vectors , , and along , where is the tangent, is the principal normal, and is the binormal vector. They satisfied the Frenet formulas:where and are, respectively, the curvature and the torsion of the curve , with , and .

From (10), we obtain

Using (16) and Lemma 1, we obtainwhere , , and .

Hence, we obtain

Theorem 1. Let be a curve parametrized by its arc length . Then, is a nongeodesic biharmonic curve if and only ifwhere .

Proof 1. From (23), it follows that it is biharmonic if and only ifThe first equation of (24) shows that . Then, the second equation (24) becomesSince , we obtainSince , the third equation also becomesThat implies alsoThen, we haveSince , we obtain

Corollary 1. Let be a time-like curve parameterized by its arc length . We suppose that , where , are functions of . Then, is a nongeodesic biharmonic curve if and only if

Remark 1. In the condition of the corollary, the binormal must be space-like.

4. Slant Helix in Walker 3-Manifold

Let be a three-dimensional strict Walker manifold.

Definition 1. A unit speed curve is called a slant helix if there exists a nonzero constant vector field in such that the function ) is constant, where is the normal of .
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 fields and . We have the following result in the three-dimensional Walker manifold.

Theorem 2. Let be a speed unit curve in . Then, is a slant helix if only ifwith .

Proof 2. Let be a slant helix. Let be a vector field such that the function . There exist smooth functions and such thatwhere and are functions of and is the Frenet frame of the curve .
Differentiating (33) with respect to , one obtainsThen, implies thatFrom the second equation of (35), we obtain .
Moreover,where is a constant. According to , we obtainand then, we get .
We denoteIf , then we have and . Therefore, . Then, , which is a contradiction. So, we have or .
The third equation of (35), , implies that . And using the fact that , we obtainThat is,Finally,Conversely, assume thatis a constant, which we denote by .
Now, we define the vector byA differentiation of together with the Frenet equations gives , that is, is a constant vector. On the contrary, , and this means that is a slant helix. This concludes the proof of the theorem.

Example 1. Let ; the speed curve is given byAn easy computation gives that the curvature and the torsion of the curve are . Then, by Theorem 1, the curve is a helix.
By equations (21)–(23), we have , and then, the curve is biharmonic [13].

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.