# Classification of f-Biharmonic Curves in Lorentz–Minkowski Space

Authors: Li Du; Juan Zhang
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we firstly derived the equations for the curves of a Lorentz–Minkowski spaceL3 to be f-biharmonic. Then, using these equations, we classify such unit speed curves in L3.


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## Body

## 1. Introduction

Biharmonic isometric immersions are critical points of the bienergy functional (proposed by Eells and Lemaire in [1])(1)E2ϕ=12∫Mrnτϕ,τϕvg,for isometric immersions ϕ:Mrn⟶Nqn+p from an n-dimensional pseudo-Riemannian manifold Mrn into an n+p-dimensional pseudo-Riemannian manifold Nqn+p, where τϕ=tr∇dϕ=nH⟶ (cf. [2, 3]), with H⟶ be the mean curvature vector field of Mrn, is the tension field of ϕ vanishing of which means that ϕ is harmonic or Mrn is minimal. The first variation formula for the bienergy E2ϕ which is derived by Jiang in [4] shows that the Euler–Lagrange equation for E2ϕ is(2)τ2ϕ≔trace∇ϕ∇ϕ−∇∇ϕτϕ−traceR˜dϕ,τϕdϕ=0,where R˜, ∇ϕ, and ∇ are the curvature tensor of Nqn+p, the induced connection by ϕ on the bundle ϕ−1TNqn+p, and the connection of Mrn, respectively (cf. [4, 5] for q=0, and [3] for q>0, for details).As a generalization of biharmonic isometric immersions, thef-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:(3)E2,fϕ=12∫Mrnfτϕ,τϕvg,where f is a fixed function Mrn⟶0,+∞.The Euler–Lagrange equation gives thef-biharmonic isometric immersion (derived by Lu in [6])(4)τ2,fϕ≔fτ2ϕ+Δfτϕ+2∇gradfϕτϕ=0,where Δ is the laplace operator of Mrn.A submanifold is called af-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 τ2ϕ 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 Eq3q=0,1,2,3 (when q=0, E03 is the Euclidean space E3) 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 [16–19] 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 off-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 E3, where he proved that such curves in E3 are planar curves or general helices and gave some examples of non-biharmonic f-biharmonic curves in E3. 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 [25–27].These facts motivate us to studyf-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 L3 and obtain the following classification theorems.Theorem 1.
A curveγ:a,b⟶L3 parametrized by arclength s is an f-biharmonic Frenet curve if and only if one of the following cases holds:(i)
γ is a circular helix and f is constant, and γ=1/6κτs3,−κ2s3+6s,3κs2 for timelike binormal, γ=1/6κ2s3+6s,3κs2,κτs3 for timelike tangent(ii)
γ is a planar curve and f=c1κ−3/2 with τ=0 and κ=4c2/c22s+c32−16 for spacelike binormal or κ=4c2/c22s+c32+16 for timelike binormal(iii)
γ is a helix and f=c1κ−3/2 with τ=cκ and κ=4c2/c22s+c32+161−c2 for timelike binormal, or κ=4c2/c22s+c32−161+c2 for timelike principal normal, or κ=4c2/c22s+c32+16c2−1 for timelike tangentwhere κ>0 and τ are the curvature and torsion of γ.Theorem 2.
A curveγ:a,b⟶L3 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)
γ is a planar curve with τ˜=0 and f=c1s+c2, and γ=s2/2,s,s2/2(ii)
γ is a helix curve and f=c1+c2se−τ˜s with τ˜ being nonzero constant, and γ=1/τ˜2coshτ˜s+sinhτ˜s,τ˜2s,coshτ˜s+sinhτ˜s(iii)
γ is a helix curve and f=c1+c2se−∫τ˜ds with τ˜ being nonconstantwhere τ˜ is the torsion of γ, c1, and c2 are two constants.


## 2. Preliminaries

LetE13 be a pseudo-Euclidean 3-space with metric given by(5),=−dx12+dx22+dx32,where x1,x2,x3 is the natural co-ordinate system of E13. The E13 is also called 3-dimensional Lorentz–Minkowski space, denoted by L3. Since , is an indefinite metric, an arbitrary vector v∈L3 can have one of three Lorentz causal characters: it can be spacelike if v,v>0 or v=0, timelike if v,v<0 and lightlike or null if v,v=0 or v≠0.Letγ:a,b⟶L3 be an arbitrary curve in L3 and γ can have locally one of the following causal characters: γ is spacelike, lightlike (null), or timelike, if γ′,γ′ is bigger, equal or smaller than 0 on an interval a,b.Whenγ is non-lightlike, then γ is parametrized by arc length s. Specifying that if a spacelike (resp. timelike) curve is parametrized by arc length, then the velocity vector satisfies respectively γ′,γ′=1, (resp. γ′,γ′=−1).A curveγ:a,b⟶L3 is said to be a unit speed curve if the velocity vector field of γ′ satisfies(6)γ′,γ′=ε1=±1.Differentiating (6), we have(7)γ″,γ′=0.In general, the causal character ofγ″ may change in the interval a,b, but the continuity assures that γ″ has the same spacelike, timelike, or lightlike in an interval around p; we refer the readers to (examples, pp: 15-16) [28]. Thus, we will assume that the causal character of γ′ or γ″ is the same in a,b. Also, we have from (7) that γ″ is perpendicular to γ′ or the curvature of γ is zero identically (i.e., γ is a straight-line). In the following, we will give the Frenet formulas of γ with nonzero curvature depending on γ″,γ″.Whenγ is a unit speed curve with γ″,γ″≠0, γ is called a Frenet curve in L3. Every Frenet curve γ in L3 admits a Frenet frame field along γ. Here, a Frenet frame field P=p1,p2,p3 is an orthonormal frame field along γ such that p1=γ′, with p2 being parallel to γ″ and p3 being perpendicular to the plane p1,p2. We call p1, p2, and p3 the tangent vector field, principal normal vector field, and binormal vector field of γ, respectively, and p1,p2,p3 satisfies the following Frenet formula.Lemma 1.
Letγ:a,b⟶L3 be a Frenet curve with arclength parameter in L3, then the Frenet formulas of γ are, in matrixflotation:(8)∇γ′L3P=0κ0−ε1ε2κ0−ε2ε3τ0τ0p1p2p3,where the functions κ>0 and τ are called the curvature and torsion of γ, respectively, and(9)pi,pi=εi,i=1,2,3,pi,pj=0,i,j=1,2,3,i≠j.Proof.
We set(10)∇γ′L3p1=κp2,where κ>0 is the curvature of γ. Note that(11)p3,p1=0,p3,p3=ε3.
Then, differentiating the above equation, combining with (10), we get(12)∇γ′L3p3⊥p1,∇γ′L3p3⊥p3,which mean that ∇γ′L3p3 is parallel to p2, i.e.,(13)∇γ′L3p3=τp2,where τ is the torsion of γ.
On the other hand, it is easy to see that the vector∇γ′L3p2∈L3, then there exist three functions f1, f2, and f3, such that(14)∇γ′L3p2=f1p1+f2p2+f3p3.
Taking the scalar product withp1, p2, and p3, respectively, we obtain(15)∇γ′L3p2,p1=ε1f1,∇γ′L3p2,p2=ε2f2,∇γ′L3p2,p3=ε3f3.
Also, differentiating both of the following equations:(16)p2,p2=ε2,p2,p1=0,p2,p3=0,using (10) and (13), we have(17)∇γ′L3p2,p2=0,∇γ′L3p2,p1=−κε2,∇γ′L3p2,p3=−τε2.
Together with (15) leads to(18)f1=−ε1ε2κ,f2=0,f3=−ε2ε3τ.
Substituting into (14), and completing the proof of Lemma 1.Remark 1.
The Frenet formula8 has appeared in [29–31] 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.
Whenγ is a unit speed curve with γ″,γ″=0, then we choose a suitable pseudo-orthonormal frame field P˜=p˜1,p˜2,p˜3 along γ in L3 with p˜1, p˜2, and p˜3 being tangent vector field, principal normal vector field, and binormal vector field of γ, respectively, such that (cf. [32])(19)∇γ′L3P˜=0κ˜00τ˜0−κ˜0τ˜p˜1p˜2p˜3,where the functions κ˜≥0 and τ˜ are called the curvature and torsion, respectively, and(20)p˜1,p˜1=1,p˜2,p˜2=p˜3,p˜3=0,p˜2,p˜3=1,p˜1,p˜2=p˜1,p˜3=0.
In this case, the curvatureκ˜ can take only two values: κ˜=0 when γ is a straight line; κ˜=1 in all other cases. Thus, we have from (19) that p˜1,p˜2,p˜3 satisfies the following Frenet formula in matrixflotation:(21)∇γ′L3P˜=0100τ˜0−10τ˜p˜1p˜2p˜3.
Next, we derived the equations for the unit speed curveγ to be f-biharmonic.Lemma 2.
A curveγ:a,b⟶L3 parametrized by arclength s is an f-biharmonic unit speed curve if and only if(22)f∇γ′L3∇γ′L3∇γ′L3γ′+2f′∇γ′L3∇γ′L3γ′+f″∇γ′L3γ′=0.Proof.
Letγ=γs be parametrized by arclength s. Then e1=∂/∂s is an orthonormal frame on a,b,ds and(23)dγe1=dγ∂∂s=γ′.
Then the tension field ofγ is given by(24)τγ=ε1∇e1γdγe1=ε1∇γ′L3γ′.
For a functionf:a,b⟶0,∞, we have(25)gradf=ε1f′e1,Δf=ε1f″.
A straightforward computation gives(26)∇gradγτγ=ε1f′∇e1γτγ=ε1f′∇γ′L3ε1∇γ′L3γ′=f′∇γ′L3∇γ′L3γ′,(27)Δfτγ=f″∇γ′L3γ′.
Choose a normal coordinates at a point ina,b; it follows from (24) that(28)τ2γ=ε1∇e1γ∇e1γτγ=∇γ′L3∇γ′L3∇γ′L3γ′.
Putting (26)–(28) into (4), we obtain that equation (22) holds and Lemma 2 follows.
Finally, we will give several definitions of curves inL3.
A helixγ:a,b⟶L3 is a curve parametrized by arclength such that there exists a vector v∈L3 with the property that the function γ′,v is constant. Then the curve γ is a helix if and only if the ratio of the curvature and torsion of γ is a constant. If both the curvature (nonzero) and the torsion of γ are constant, then the γ curve is called a circular helix (cf. [28]).
A curveγ:a,b⟶L3 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) that(29)∇γ′L3p1=κp2,∇γ′L3p2=−ε1ε2κp1+−ε2ε3τp3,∇γ′L3p3=τp3.
Now, taking into account the first and second equation of (29), we obtain(30)∇γ′L3∇γ′L3γ′=κ′p2+κ∇γ′L3p2=−ε1ε2κ2p1+κ′p2+−ε2ε3κτp3,which together with (29) shows that(31)∇γ′L3∇γ′L3∇γ′L3γ′=∇γ′L3−ε1ε2κ2p1+κ′p2+−ε2ε3κτp3=2−ε1ε2κκ′p1+−ε1ε2κ2∇γ′L3p1+κ″p2+κ′∇γ′L3p2+−ε2ε3κ′τp3+−ε2ε3κτ′p3+−ε2ε3κτ∇γ′L3p3=3−ε1ε2κκ′p1+−ε1ε2κ3+κ″+−ε1ε2κτ2p2+2−ε2ε3κ′τ+−ε2ε3κτ′p3.
Putting the first equation of (29)–(31) into (22), we get that γ is f-biharmonic if and only if(32)2κ2f′+3κκ′f=0,κf″+2κ′f′+−ε1ε2κ3+κ″+−ε2ε3κτ2f=0,2κτf′+2κ′τf+τ′κf=0.
In the following, we will investigate the characteristics of curves according to different values ofκ and τ.Case 1: Whenκ is a nonzero constant, then it follows from the first equation of (32) that(33)f′=0.It is obvious to see thatf is a constant, which implies that γ is a biharmonic curve. Together with the third equation of (32), we obtain that τ is a constant. Hence the second equation of (32) can be simplified to(34)τ2=−ε1ε3κ2a nonzero constant,which means thatγ is a circular helix (cf. [28, 32]). We note that ε1,ε3 evaluates among the following two possible cases:(35)1,−1,−1,1.(i)
Whenε1=1 and ε3=−1, i.e., p1 is spacelike tangent and p3 is timelike binormal, then we conclude from Theorem 3.3 in [32] that(36)γs=16κτs3,−κ2s3+6s,3κs2.(ii)
Whenε1=−1 and ε3=1, i.e., p1 is timelike tangent and p3 is spacelike binormal, then we conclude from Theorem 3.6 in [32] that(37)γs=16κ2s3+6s,3κs2,κτs3.Conversely, 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 κ and torsion τ are two constants which satisfy κ=τ. Combining with f being a constant, we prove that equation (32) holds for the corresponding curves; that is, γ is f-biharmonic.Case 2: Whenτ is zero, then equation (32) is equivalent to(38)2κ2f′+3κκ′f=0,κf″+2κ′f′+−ε1ε2κ3f+κ″f=0.
Furthermore, we have(39)2f′f+3κ′κ=0,(40)κ″κ+2κ′κf′f+f″f+−ε1ε2κ2=0.
A direct derivative calculation forf2κ3 gives(41)f2κ3′=fκ2κ2f′+3κκ′f,then it follows from the first equation of (38) that f2κ3=c with c being a constant, which shows that(42)f=cκ−3/2.
Also, differentiating both sides (39) yields(43)f″f=−32κ″κ+154κ′2κ2.
Putting (39) and (43) into (40), we deduce that(44)3κ′2−2κ″κ−4ε1ε2κ4=0.
Sinceε1ε2=−ε3, we have from (44) that, for spacelike binormal vector field,(45)3κ′2−2κ″κ+4κ4=0,and for timelike binormal vector field,(46)3κ′2−2κ″κ−4κ4=0.
Solving the ODEs (45) and (46), respectively, we have that, for (45),(47)κ=4c2c22s2+2c22c3s+c2c32−16,and for (46),(48)κ=4c216+c22s2+2c22c3s+c2c32.
Note thatτ=0; then γ is a planar curve. Thus, the curve γ is a planar curve with spacelike binormal and κ=4c2/c22s2+2c22c3s+c2c32−16, or timelike binormal and κ=4c2/16+c22s2+2c22c3s+c2c32 and f=cκ−3/2.
Conversely, becausef=cκ−3/2, it is not difficult to check that the first equation in (38) holds. Also, combining with f=cκ−3/2, a long calculation for κ=4c2/c22s2+2c22c3s+c2c32−16 for spacelike binormal, or κ=4c2/16+c22s2+2c22c3s+c2c32 for timelike binormal, respectively, proves that the second equation in (38) holds, that is, γ is f-biharmonic.Case 3: Whenτ is a nonzero constant, then equation (32) is equivalent to(49)i2κ2f′+3κ′κf=0,iif″f+2κ′f′+−ε1ε2κ3+κ″+−ε2ε3κτ2f=0,iiiκf′+κ′f=0.Combining with (i) and (iii) in (49), it is straightforward to prove that f and κ are two constants. Substituting those facts into (ii) in (49), we obtain that(50)κ2=ε2τ2,which implies thatκ is also a nonzero constant. Since ε2=−ε1ε3, following the similar process as in Case 1, we complete the case.Case 4: Whenκ≠ constant and τ≠ constant, then equation (32) is equivalent to(51)if2κ3=c1,iif″f+2κ′f′+κ″f=ε1ε2κ2+ε2ε3τ2κf,iiiκ2τf2=c˜2,wherec1 and c˜2 are two constants. On one hand, by the first equation in (51), it is obvious to see that(52)f=c1κ−3/2,where c1>0. Moreover, it follows from (i) and (iii) in (51) that τ/κ=c with c being constant, then we conclude from [28] that γ is a helix.
On the other hand, combining with (43) and (i) and (ii) in (51), with much more tedious computations gives(53)3κ′2−2κ″κ−4ε1ε2+ε2ε3c2κ4=0.
Note thatε1,ε2,ε3 evaluates among the following three possible cases:(54)1,1,−1,1,−1,1,−1,1,1.(i)
Whenε1=1,ε2=1,ε3=−1, equation (53) becomes(55)3κ′2−2κ″κ−41−c2κ4=0.Solving ODE (55), we have κ=4c2/161−c2+c22s2+2c22c3s+c2c32.(ii)
Whenε1=1,ε2=−1,ε3=1, equation (53) becomes(56)3κ′2−2κ′′κ+41+c2κ4=0.Solving ODE (56), we have κ=4c2/c22s2+2c22c3s+c2c32−161+c2.(iii)
Whenε1=−1,ε2=1,ε3=1, equation (53) becomes(57)3κ′2−2κ″κ−4c2−1κ4=0.Solving ODE (57), we have κ=4c2/16c2−1+c22s2+2c22c3s+c2c32.
Conversely, making similar discussions as in Case 2, we prove thatγ is f-biharmonic.Proof of Theorem 2.
According to (21), we have(58)∇γ′L3p˜1=p˜2,∇γ′L3∇γ′L3γ′=τ˜p˜2,∇γ′L3∇γ′L3∇γ′L3γ′=τ˜′+τ˜2p˜2.
Using Lemma2, we obtain that γ is f-biharmonic if and only if(59)f″+2τ˜f′+τ˜′+τ˜2f=0.(i)
Whenτ˜ is zero, then it follows from (59) that f″=0, which means that f=c1s+c2, where c1 and c2 are two constants. Meanwhile, we conclude from Theorem 3.5 in [32] that(60)γs=s22,s,s22,which is a planar curve.(ii)
Whenτ˜ is a nonzero constant, then (59) is equivalent to(61)f″+2τ˜f′+τ˜2f=0.Solving the above ODE, we obtain thatf=c1+c2se−τ˜s, where c1 and c2 are two constants, and know from Theorem 3.5 in [32] that(62)γ=1τ˜2coshτ˜s+sinhτ˜s,τ˜2s,coshτ˜s+sinhτ˜s,which is a helix curve (cf. [28]).(iii)
Whenτ˜≠ constant, then by solving ODE (59), we get that f=c1+c2se−∫τ˜ds, where c1 and c2 are two constants. Also, we know from [28] that γ is a helix.
The converse is clear from the similar proof of Theorem 3.5 in [32], together with the corresponding function f.
As a consequence of Theorems1 and 2, we obtain the following.Corollary 1.
Anyf-biharmonic unit speed curve with arclength parameter in Lorentz-Minkowski 3-space is a helix.
Using (28), we obtain that γ is biharmonic if and only if(63)γiv=0.
According to (22), it is easy to find that when f is a constant, f-biharmonic curves must be biharmonic ones. It is a natural and interesting problem: whetherfmust be a constant whenf-biharmonic curves are biharmonic. Unfortunately, this problem is not true.
For thef-biharmonic curve γ in Theorem 2,(64)γ=s22,s,s22,f=c1s+c2,where c1 and c2 are two constant. Using (63), it easily check that γ is biharmonic, but f is a function.Remark 2.
As we all know, the theorem of existence and uniqueness for curves inℝ3 asserts that given two functions κ>0 and τ, there exists a unique (up a rigid motion) curve in ℝ3 with the curvature κ and the torsion τ. In Lorentz-Minkowski space L3, the result of existence for curves is the same as that of in ℝ3 (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 γ1 and γ2 (i.e., the tangent, principal normal, and binormal of γ1 have the same causal character compared to the corresponding tangent, principal normal, and binormal of γ2, respectively) [33]. This, together with our classification theorems, implies that there are many examples of proper f-biharmonic curves in L3.
Finally, we give an example of a nonbiharmonicf-biharmonic curve in L3. For the f-biharmonic curve in Theorem 2,(65)γ=1τ˜2coshτ˜s+sinhτ˜s,τ˜2s,coshτ˜s+sinhτ˜s,and f=c1+c2se−τ˜s with τ˜ be a nonzero constant.
Using (63), a short computation proves that γ is a nonbiharmonic curve.



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