Classification of f-Biharmonic Curves in Lorentz–Minkowski Space

for isometric immersions φ: Mnr⟶ N n+p q from an n-dimensional pseudo-Riemannian manifold Mnr into an (n + p)-dimensional pseudo-Riemannian manifold N q , where τ(φ) � tr∇dφ � nH → (cf. [2, 3]), with H → be the mean curvature vector field of Mnr , is the tension field of φ vanishing of which means that φ is harmonic or Mnr is minimal. (e first variation formula for the bienergy E2(φ) which is derived by Jiang in [4] shows that the Euler–Lagrange equation for E2(φ) is


Introduction
Biharmonic isometric immersions are critical points of the bienergy functional (proposed by Eells and Lemaire in [1]) for isometric immersions ϕ: M n r ⟶ N n+p q from an n-dimensional pseudo-Riemannian manifold M n r into an (n + p)-dimensional pseudo-Riemannian manifold N n+p q , where τ(ϕ) � tr∇dϕ � nH → (cf. [2,3]), with H → be the mean curvature vector field of M n r , is the tension field of ϕ vanishing of which means that ϕ is harmonic or M n r is minimal. e first variation formula for the bienergy E 2 (ϕ) which is derived by Jiang in [4] shows that the Euler-Lagrange equation for E 2 (ϕ) is where R, ∇ ϕ , and ∇ are the curvature tensor of N n+p q , the induced connection by ϕ on the bundle ϕ −1 TN n+p q , and the connection of M n r , respectively (cf. [4,5] for q � 0, and [3] for q > 0, for details).
As a generalization of biharmonic isometric immersions, the f-biharmonic isometric immersion ϕ was introduced by Lu in [6] (cf. [7] for f-biharmonic maps), as a critical point of the f-bienergy functional: where f is a fixed function M n r ⟶ (0, +∞). e Euler-Lagrange equation gives the f-biharmonic isometric immersion (derived by Lu in [6]) where Δ is the laplace operator of M n r . A submanifold is called a f-biharmonic submanifold if its isometric immersion ϕ is f-biharmonic (cf. [8]). When f is a constant, f-biharmonic submanifolds are called biharmonic submanifolds (i.e., its bitension field τ 2 (ϕ) vanishes identically) (cf. [5]) which are called submanifolds with harmonic mean curvature vector field by Chen in [9]. e 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). en, 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 E 3 q (q � 0, 1, 2, 3) (when q � 0, E 3 0 is the Euclidean space E 3 ) 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 [16][17][18][19] with references therein. For some recent progress of biharmonic submanifolds (instead of biharmonic curves), we refer readers to [2, 5, 12-14, 16, 17, 20-24] and the references therein.
Naturally, the next step has been the study of f-biharmonic curves. Ou in [8] derived equations for f-biharmonic curves in a generic manifold and completely classified f-biharmonic curves in 3-dimensional Euclidean space E 3 , where he proved that such curves in E 3 are planar curves or general helices and gave some examples of nonbiharmonic f-biharmonic curves in E 3 . After that, there are a few valuable results on f-biharmonic curves in (generalized) Sasakian space forms, Sol spaces, Cartan-Vranceanu 3-dimensional spaces, or homogeneous contact 3-manifolds; we refer to [25][26][27].
ese facts motivate us to study f-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 f-biharmonic curves with a positive function f in Lorentz-Minkowski space L 3 and obtain the following classification theorems.

Theorem 2. A curve c: (a, b) ⟶ L 3 parametrized by arclength s is an f-biharmonic unit speed curve with lightlike principal normal if and only if one of the following cases holds:
(i) c is a planar curve with τ � 0 and f � c 1 s + c 2 , and c � (s 2 /2, s, s 2 /2) (ii) c is a helix curve and f � (c 1 + c 2 s)e − τs with τ being nonzero constant, and where τ is the torsion of c, c 1 , and c 2 are two constants.

Preliminaries
Let E 3 1 be a pseudo-Euclidean 3-space with metric given by where Let c: (a, b) ⟶ L 3 be an arbitrary curve in L 3 and c can have locally one of the following causal characters: c is spacelike, lightlike (null), or timelike, if 〈c ′ , c ′ 〉 is bigger, equal or smaller than 0 on an interval (a, b).
A curve c: (a, b) ⟶ L 3 is said to be a unit speed curve if the velocity vector field of c ′ satisfies Differentiating (6), we have In general, the causal character of c ″ may change in the interval (a, b), but the continuity assures that c ″ has the same spacelike, timelike, or lightlike in an interval around p; we refer the readers to (examples, pp: 15-16) [28]. us, we will assume that the causal character of c ′ or c ″ is the same in  (a, b). Also, we have from (7) that c ″ is perpendicular to c ′ or the curvature of c is zero identically (i.e., c is a straightline). In the following, we will give the Frenet formulas of c with nonzero curvature depending on 〈c ″ , c ″ 〉.
When c is a unit speed curve with 〈c ″ , c ″ 〉 ≠ 0, c is called a Frenet curve in L 3 . Every Frenet curve c in L 3 admits a Frenet frame field along c. Here, a Frenet frame field P � (p 1 , p 2 , p 3 ) is an orthonormal frame field along c such that p 1 � c ′ , with p 2 being parallel to c ″ and p 3 being perpendicular to the plane p 1 , p 2 . We call p 1 , p 2 , and p 3 the tangent vector field, principal normal vector field, and binormal vector field of c, respectively, and p 1 , p 2 , p 3 satisfies the following Frenet formula.
where the functions κ(>0) and τ are called the curvature and torsion of c, respectively, and Proof. We set where κ(>0) is the curvature of c. Note that en, differentiating the above equation, combining with (10), we get which mean that ∇ L 3 c ′ p 3 is parallel to p 2 , i.e., where τ is the torsion of c.
On the other hand, it is easy to see that the vector ∇ L 3 c ′ p 2 ∈ L 3 , then there exist three functions f 1 , f 2 , and f 3 , such that Taking the scalar product with p 1 , p 2 , and p 3 , respectively, we obtain Also, differentiating both of the following equations: using (10) and (13), we have Together with (15) leads to Substituting into (14), and completing the proof of Lemma 1.

Remark 1.
e Frenet formula 8 has appeared in [29][30][31] in different forms, but the detail proof of (8) is not given in those papers. us, we give a brief proof of (8) for completeness and simplicity of our main results.
When c is a unit speed curve with 〈c ″ , c ″ 〉 � 0, then we choose a suitable pseudo-orthonormal frame field P � (p 1 , p 2 , p 3 ) along c in L 3 with p 1 , p 2 , and p 3 being tangent vector field, principal normal vector field, and binormal vector field of c, respectively, such that (cf. [32]) where the functions κ(≥0) and τ are called the curvature and torsion, respectively, and In this case, the curvature κ can take only two values: κ � 0 when c is a straight line; κ � 1 in all other cases. us, we have from (19) that p 1 , p 2 , p 3 satisfies the following Frenet formula in matrixflotation: Next, we derived the equations for the unit speed curve c to be f-biharmonic.

Lemma 2. A curve c: (a, b) ⟶ L 3 parametrized by arclength s is an f-biharmonic unit speed curve if and only if
dc e 1 � dc en the tension field of c is given by For a function f: A straightforward computation gives Choose a normal coordinates at a point in (a, b); it follows from (24) that Putting (26)- (28) into (4), we obtain that equation (22) holds and Lemma 2 follows.
Finally, we will give several definitions of curves in L 3 . A helix c: (a, b) ⟶ L 3 is a curve parametrized by arclength such that there exists a vector v ∈ L 3 with the property that the function 〈c ′ , v〉 is constant. en the curve c is a helix if and only if the ratio of the curvature and torsion of c is a constant. If both the curvature (nonzero) and the torsion of c are constant, then the c curve is called a circular helix (cf. [28]).
A curve c: (a, b) ⟶ L 3 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.

Main Theorems and Their Proofs
Proof of eorem 1. We have from (8) that Now, taking into account the first and second equation of (29), we obtain which together with (29) shows that Putting the first equation of (29)-(31) into (22), we get that c is f-biharmonic if and only if In the following, we will investigate the characteristics of curves according to different values of κ and τ.