In this paper, we firstly derived the equations for the curves of a Lorentz–Minkowski space to be -biharmonic. Then, using these equations, we classify such unit speed curves in .

1. Introduction

Biharmonic isometric immersions are critical points of the bienergy functional (proposed by Eells and Lemaire in [1])for isometric immersions from an -dimensional pseudo-Riemannian manifold into an -dimensional pseudo-Riemannian manifold , where (cf. [2, 3]), with be the mean curvature vector field of , is the tension field of vanishing of which means that is harmonic or is minimal. The first variation formula for the bienergy which is derived by Jiang in [4] shows that the Euler–Lagrange equation for iswhere , , and are the curvature tensor of , the induced connection by on the bundle , and the connection of , respectively (cf. [4, 5] for , and [3] for , for details).

As a generalization of biharmonic isometric immersions, the -biharmonic isometric immersion was introduced by Lu in [6] (cf. [7] for -biharmonic maps), as a critical point of the -bienergy functional:where is a fixed function .

The Euler–Lagrange equation gives the -biharmonic isometric immersion (derived by Lu in [6])where is the laplace operator of .

A submanifold is called a -biharmonic submanifold if its isometric immersion is -biharmonic (cf. [8]). When is a constant, -biharmonic submanifolds are called biharmonic submanifolds (i.e., its bitension field vanishes identically) (cf. [5]) which are called submanifolds with harmonic mean curvature vector field by Chen in [9].

The study of biharmonic submanifolds is a vibrant research subject, which was originated in [4, 5] by Jiang for his study of Euler–Lagrange’s equation of the bienergy functional and also independently by Chen (cf. [10]) in his program of understanding the finite type submanifolds in Euclidean spaces, and there were numerous important developments in this domain over the past 40 plus years. For example, Dimitríc proved (cf. [11]) that any biharmonic curve in a Euclidean space is a geodesic (Chen and Ishikawa in [12] obtained the same result independently). Then, Caddeo, Montaldo, and Piu in [13] considered biharmonic curves on a surface and giave some examples of nongeodesic biharmonic curves. Later, Caddeo, Montaldo, and Oniciuc (cf. [2]) showed nonexistence of nongeodesic biharmonic curves in a 3-dimensional hyperbolic space and proved that nongeodesic biharmonic curves in the unit 3-sphere are circles of geodesic curvature 1 or helices which are geodesics in the Clifford minimal torus. Also, Chen and Ishikawa (cf. [12, 14, 15]) classified completely unit speed biharmonic curves in pseudo-Euclidean spaces (when , is the Euclidean space ) and gave some examples of nonminimal biharmonic curves. More generally, Sasahara in [3] considered unit biharmonic curves in nonflat Lorentz 3-space forms and obtained full classification of such curves. For the study of biharmonic curves in other model spaces, we refer to [1619] with references therein. For some recent progress of biharmonic submanifolds (instead of biharmonic curves), we refer readers to [2, 5, 1214, 16, 17, 2024] and the references therein.

Naturally, the next step has been the study of -biharmonic curves. Ou in [8] derived equations for -biharmonic curves in a generic manifold and completely classified -biharmonic curves in 3-dimensional Euclidean space , where he proved that such curves in are planar curves or general helices and gave some examples of non-biharmonic -biharmonic curves in . After that, there are a few valuable results on -biharmonic curves in (generalized) Sasakian space forms, Sol spaces, Cartan–Vranceanu 3-dimensional spaces, or homogeneous contact 3-manifolds; we refer to [2527].

These facts motivate us to study -biharmonic curves in pseudo-Riemannian manifolds since it helps to bridge the gap between modern differential geometry and the mathematical physics of general relativity. In this paper, we will investigate unit speed -biharmonic curves with a positive function in Lorentz–Minkowski space and obtain the following classification theorems.

Theorem 1. A curve parametrized by arclength is an -biharmonic Frenet curve if and only if one of the following cases holds:(i) is a circular helix and is constant, and for timelike binormal, for timelike tangent(ii) is a planar curve and with and for spacelike binormal or for timelike binormal(iii) is a helix and with and for timelike binormal, or for timelike principal normal, or for timelike tangentwhere and are the curvature and torsion of .

Theorem 2. A curve parametrized by arclength is an -biharmonic unit speed curve with lightlike principal normal if and only if one of the following cases holds:(i) is a planar curve with and , and (ii) is a helix curve and with being nonzero constant, and (iii) is a helix curve and with being nonconstantwhere is the torsion of , , and are two constants.

2. Preliminaries

Let be a pseudo-Euclidean 3-space with metric given bywhere is the natural co-ordinate system of . The is also called 3-dimensional Lorentz–Minkowski space, denoted by . Since is an indefinite metric, an arbitrary vector can have one of three Lorentz causal characters: it can be spacelike if or , timelike if and lightlike or null if or .

Let be an arbitrary curve in and can have locally one of the following causal characters: is spacelike, lightlike (null), or timelike, if is bigger, equal or smaller than 0 on an interval .

When is non-lightlike, then is parametrized by arc length . Specifying that if a spacelike (resp. timelike) curve is parametrized by arc length, then the velocity vector satisfies respectively , (resp. ).

A curve is said to be a unit speed curve if the velocity vector field of satisfies

Differentiating (6), we have

In general, the causal character of may change in the interval , but the continuity assures that has the same spacelike, timelike, or lightlike in an interval around ; we refer the readers to (examples, pp: 15-16) [28]. Thus, we will assume that the causal character of or is the same in . Also, we have from (7) that is perpendicular to or the curvature of is zero identically (i.e., is a straight-line). In the following, we will give the Frenet formulas of with nonzero curvature depending on .

When is a unit speed curve with , is called a Frenet curve in . Every Frenet curve in admits a Frenet frame field along . Here, a Frenet frame field is an orthonormal frame field along such that , with being parallel to and being perpendicular to the plane . We call , , and the tangent vector field, principal normal vector field, and binormal vector field of , respectively, and satisfies the following Frenet formula.

Lemma 1. Let be a Frenet curve with arclength parameter in , then the Frenet formulas of are, in matrixflotation:where the functions and are called the curvature and torsion of , respectively, and

Proof. We setwhere is the curvature of . Note thatThen, differentiating the above equation, combining with (10), we getwhich mean that is parallel to , i.e.,where is the torsion of .
On the other hand, it is easy to see that the vector , then there exist three functions , , and , such thatTaking the scalar product with , , and , respectively, we obtainAlso, differentiating both of the following equations:using (10) and (13), we haveTogether with (15) leads toSubstituting into (14), and completing the proof of Lemma 1.

Remark 1. The Frenet formula 8 has appeared in [2931] in different forms, but the detail proof of (8) is not given in those papers. Thus, we give a brief proof of (8) for completeness and simplicity of our main results.
When is a unit speed curve with , then we choose a suitable pseudo-orthonormal frame field along in with , , and being tangent vector field, principal normal vector field, and binormal vector field of , respectively, such that (cf. [32])where the functions and are called the curvature and torsion, respectively, andIn this case, the curvature can take only two values: when is a straight line; in all other cases. Thus, we have from (19) that satisfies the following Frenet formula in matrixflotation:Next, we derived the equations for the unit speed curve to be -biharmonic.

Lemma 2. A curve parametrized by arclength is an -biharmonic unit speed curve if and only if

Proof. Let be parametrized by arclength . Then is an orthonormal frame on andThen the tension field of is given byFor a function , we haveA straightforward computation givesChoose a normal coordinates at a point in ; it follows from (24) thatPutting (26)–(28) into (4), we obtain that equation (22) holds and Lemma 2 follows.
Finally, we will give several definitions of curves in .
A helix is a curve parametrized by arclength such that there exists a vector with the property that the function is constant. Then the curve is a helix if and only if the ratio of the curvature and torsion of is a constant. If both the curvature (nonzero) and the torsion of are constant, then the curve is called a circular helix (cf. [28]).
A curve is called a planar curve if the torsion identically vanishes (cf. [32]).
It is obvious to see that a straight-line (i.e., the curvature identically vanishes) and a planar curve are helices.

3. Main Theorems and Their Proofs

Proof of Theorem 1. We have from (8) thatNow, taking into account the first and second equation of (29), we obtainwhich together with (29) shows thatPutting the first equation of (29)–(31) into (22), we get that is -biharmonic if and only ifIn the following, we will investigate the characteristics of curves according to different values of and .Case 1: When is a nonzero constant, then it follows from the first equation of (32) thatIt is obvious to see that is a constant, which implies that is a biharmonic curve. Together with the third equation of (32), we obtain that is a constant. Hence the second equation of (32) can be simplified towhich means that is a circular helix (cf. [28, 32]). We note that evaluates among the following two possible cases:(i)When and , i.e., is spacelike tangent and is timelike binormal, then we conclude from Theorem 3.3 in [32] that(ii)When and , i.e., is timelike tangent and is spacelike binormal, then we conclude from Theorem 3.6 in [32] thatConversely, according to the proof of Case 4 in Theorem 3.3 for (36) and Case 6 in Theorem 3.6 for (37), respectively, we know that their curvature and torsion are two constants which satisfy . Combining with being a constant, we prove that equation (32) holds for the corresponding curves; that is, is -biharmonic.Case 2: When is zero, then equation (32) is equivalent toFurthermore, we haveA direct derivative calculation for givesthen it follows from the first equation of (38) that with being a constant, which shows thatAlso, differentiating both sides (39) yieldsPutting (39) and (43) into (40), we deduce thatSince , we have from (44) that, for spacelike binormal vector field,and for timelike binormal vector field,Solving the ODEs (45) and (46), respectively, we have that, for (45),and for (46),Note that ; then is a planar curve. Thus, the curve is a planar curve with spacelike binormal and , or timelike binormal and and .
Conversely, because , it is not difficult to check that the first equation in (38) holds. Also, combining with , a long calculation for for spacelike binormal, or for timelike binormal, respectively, proves that the second equation in (38) holds, that is, is -biharmonic.Case 3: When is a nonzero constant, then equation (32) is equivalent toCombining with (i) and (iii) in (49), it is straightforward to prove that and are two constants. Substituting those facts into (ii) in (49), we obtain thatwhich implies that is also a nonzero constant. Since , following the similar process as in Case 1, we complete the case.Case 4: When constant and constant, then equation (32) is equivalent towhere and are two constants. On one hand, by the first equation in (51), it is obvious to see thatwhere . Moreover, it follows from (i) and (iii) in (51) that with being constant, then we conclude from [28] that is a helix.
On the other hand, combining with (43) and (i) and (ii) in (51), with much more tedious computations givesNote that evaluates among the following three possible cases:(i)When , equation (53) becomesSolving ODE (55), we have .(ii)When , equation (53) becomesSolving ODE (56), we have .(iii)When , equation (53) becomesSolving ODE (57), we have .Conversely, making similar discussions as in Case 2, we prove that is -biharmonic.

Proof of Theorem 2. According to (21), we haveUsing Lemma 2, we obtain that is -biharmonic if and only if(i)When is zero, then it follows from (59) that , which means that , where and are two constants. Meanwhile, we conclude from Theorem 3.5 in [32] thatwhich is a planar curve.(ii)When is a nonzero constant, then (59) is equivalent toSolving the above ODE, we obtain that , where and are two constants, and know from Theorem 3.5 in [32] thatwhich is a helix curve (cf. [28]).(iii)When constant, then by solving ODE (59), we get that , where and are two constants. Also, we know from [28] that is a helix.The converse is clear from the similar proof of Theorem 3.5 in [32], together with the corresponding function .
As a consequence of Theorems 1 and 2, we obtain the following.

Corollary 1. Any -biharmonic unit speed curve with arclength parameter in Lorentz-Minkowski 3-space is a helix.
Using (28), we obtain that is biharmonic if and only ifAccording to (22), it is easy to find that when is a constant, -biharmonic curves must be biharmonic ones. It is a natural and interesting problem: whethermust be a constant when-biharmonic curves are biharmonic. Unfortunately, this problem is not true.
For the -biharmonic curve in Theorem 2,where and are two constant. Using (63), it easily check that is biharmonic, but is a function.

Remark 2. As we all know, the theorem of existence and uniqueness for curves in asserts that given two functions and , there exists a unique (up a rigid motion) curve in with the curvature and the torsion . In Lorentz-Minkowski space , the result of existence for curves is the same as that of in (cf. [28], Thereoms 2.6-2.7). In general, the uniqueness for curves is not true by the causal character of the curve (cf. [28] for details). However, it holds if the causal character of the Frenet frame field agree for both curves and (i.e., the tangent, principal normal, and binormal of have the same causal character compared to the corresponding tangent, principal normal, and binormal of , respectively) [33]. This, together with our classification theorems, implies that there are many examples of proper -biharmonic curves in .
Finally, we give an example of a nonbiharmonic -biharmonic curve in . For the -biharmonic curve in Theorem 2,and with be a nonzero constant.
Using (63), a short computation proves that is a nonbiharmonic curve.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


This work was supported by the Natural Science Foundation of Chongqing (No. cstc2019jcyj-msxmX0172); the Scientific and Technological Research Program of Chongqing Municipal Education Commission (No. KJQN201901128); the Scientific Research Starting Foundation of Chongqing University of Technology (No. 2017ZD52).