# Mannheim Curves in Nonflat 3-Dimensional Space Forms

Authors: Wenjing Zhao; Donghe Pei; Xinyu Cao
Journal: Advances in Mathematical Physics
 (2015)

---

## Abstract

We consider the Mannheim curves in nonflat 3-dimensional space forms (Riemannian or Lorentzian) and we give the concept of Mannheim curves. In addition, we investigate the properties of nonnull Mannheim curves and their partner curves. We come to the conclusion that a necessary and sufficient condition is that a linear relationship with constant coefficients will exist between the curvature and the torsion of the given original curves. In the case of null curve, we reveal that there are no null Mannheim curves in the 3-dimensional de Sitter space.


---

## Body

## 1. Introduction

Semi-Riemannian geometry is important both in differential geometry and in physics, where it plays a central role in the theory of relativity. Curve which is the basic object of study has attracted much more attention by many mathematicians and physicists. In particular, there has been an increase in research on null curves in geometry and physics [1–4]. From physical significance point of view, there is a particle model entirely based on geometry of null curves [2, 3]. The other important physical reason is the application of null curves theory to general relativity. Many basic properties about curves could be seen in [5]. The renewed interest in the theory of curves has developed from the need to observe the properties of special curves such as Mannheim curves and Bertrand curves in different space. Mannheim curves are well studied classical curves and may be defined by their property that space curves whose principle normals are binormals of another curve at corresponding points. The notion of Mannheim curve was put forward by Mannheim in 1878. These curves in Euclidean 3-space are characterized in terms of the curvature and torsion as follows: a space curve is a Mannheim curve if and only if its curvature κ and torsion τ satisfy the relation κ=λ(κ2+τ2), where λ is a nonzero constant. On this basis, many mathematicians systematically investigated Mannheim curves in different space and they obtained a number of important results [6]. In addition, the properties of Mannheim curve in Minkowski space have been studied extensively by, among others, Liu and Wang who studied Mannheim partner curves in Euclidean 3-space and Minkowski 3-space in detail; in particular, they provided the necessary and sufficient condition for the judgment of the Mannheim partner curves [7].The other kind of curve which has been focusing a lot of researchers’ attention since the beginning is Bertrand curve [8–11]. In particular, Lucas and Ortega-Yagües have devoted their work to the research of properties of Bertrand curves which include nonnull Bertrand curves and null Bertrand curves and they obtained many perfect characterizations of Bertrand curves in nonflat 3-dimensional space forms [9].Inspired by their work, we use some fundamental results of differential geometry as basic tools in our research on the Mannheim curve in nonflat 3-dimensional space forms. It is well known that, in nonflat semi-Euclidean space, there are two types of pseudospheres: pseudosphere with a positive radius squared,〈x,x〉=r2, which we call de Sitter space. The other type is pseudosphere with a negative radius squared, 〈x,x〉=-r2, which we call anti-de Sitter space. When we choose one of these spheres as the ambient space of curves, the curves will show some special properties. The first goal of this paper is to define the Mannheim curve in 3-dimensional nonflat space forms which include two types of sphere as above. In addition, the definition of angle in Euclidean is well known by us. However, the concept of general angle in semi-Euclidean space has been drawing our attention [9]. On this basis, we show and proof some properties of the angle between tangent vectors or binormal vectors of Mannheim curves and their partner curves at corresponding points. Furthermore, we investigate the characterization of Mannheim curves and Mannheim partner curves in nonflat 3-dimensional space forms. We give a necessary and sufficient condition for a curve to be Mannheim curve or Mannheim partner curve and obtain some explicit equations. Meanwhile, as is known, the Mannheim curves in nonflat space forms have two cases: one case is the nonnull Mannheim curves; the other case is the null Mannheim curves which are mainly considered in 3-dimensional de Sitter space S13. Grbović et al. discussed the null Mannheim curves in 3-dimensional Minkowski space [12]. We know that the case of null curve that is immersed in a three-dimensional de Sitter space is more sophisticated and interesting than nonnull curve in de Sitter space. To the authors’ knowledge, there is no article dedicated to studying the existence of null Mannheim curves in de Sitter space. For this reason, we consider the null Mannheim curve and discover that there is no null Mannheim curve in 3-dimensional de Sitter space.The brief description of the organization of this paper is as follows. In Section2, we review some basic notions about the space Rvn+1 and curves which include nonnull curves and null curves. In addition, we give the definition of Mannheim curve in nonflat 3-dimensional space forms in Section 3. One of the main results in this paper is stated in Section 3, Theorem 5. Section 4 is devoted to claiming and showing that a null Mannheim curve that lies in 3-dimensional de Sitter space is nonexistent.Throughout the paper, all the symbols here could be found in [13].


## 2. Preliminaries

Now we introduce some basic notions in semi-Euclidean space and curves.LetRvn+1 denote the (n+1)-dimensionalpseudo-Euclidean space of index v≥0; let E={e1,…,en+1} be an canonical basis of Rvn+1. We choose two vectors x,y∈Rvn+1, and the standard metric of Rvn+1 is given by (1)x,y=-∑i=1v‍xiyi+∑j=v+1n+1‍xjyj,where xi,yi stand for the coordinate components of x and y with respect to E in Rvn+1, respectively.Lettingx∈Rvn+1, the vector x is said to be (2)spacelike,ifx,x>0orx=0,timelike,ifx,x<0,lightlike(null),ifx,x=0,x≠0.Define thenorm of a nonnull vector u by u=|〈u,u〉|1/2, where u∈Rvn+1. We call u theunit vector if u=1.We define thesignature of a vector x as follows: (3)sign(x)=1,xisspacelike,0,xisnull,-1,xistimelike.LetMq3(c)⊂Rv4 denote thenonflat 3-dimensional space forms of q=0,1, and constant curvature c≠0. Meanwhile, v=q, if c=1, and v=q+1, if c=-1. Moreover, we will denote Mq3(c) by the pseudo-Euclidean hypersphere Sq3(1) or the pseudo-Euclidean hyperbolic space Hq3(-1) according to c=1 or c=-1, respectively, where Sq3(1) is denoted by (4)Sq3(1)={x=(x1,…,x4)∈Rq4∣〈x,x〉=1},and the pseudo-Euclidean hyperbolic space of index q≥0 and curvature c=-1 is given by Hq3(-1): (5)Hq3(-1)={x=(x1,…,x4)∈Rq+14∣〈x,x〉=-1}.Many features of inner product spaces have analogues in the pseudo-Euclidean case. InRn, the Schwarz inequality permits the definition of the Euclidean angle θ between vectors X and Y as the unique number 0≤θ≤π, such that 〈X,Y〉=|X||Y|cos⁡θ. For two nonnull vectors X, Y which are not spacelike vectors in the Lorentzian space R1n, the definition of the angle θ between X and Y is of great interest and importance. Then definition of general angle which is similar to Euclidean angle is as Definition 1. We consider two nonnull vectors X,Y∈R1n such that they span a plane R12. In this plane, we can choose an orthogonal basis {e1,e2}, with 〈e1,e1〉=-1 and 〈e2,e2〉=1. Then we can write vectors X, Y in this basis as X=(X1,X2) and Y=(Y1,Y2).Definition 1 (see [9]).
Let one consider two nonnull vectorsX,Y∈R1n.(a)
Let one assume thatX and Y are spacelike vectors; then
(i)
if they span a spacelike plane, there is a unique number0≤θ≤π such that 〈X,Y〉=|X||Y|cos⁡θ;(ii)
if they span a timelike plane, there is a unique numberθ≥0 such that 〈X,Y〉=ε|X||Y|cosh⁡θ, where ε=1 or ε=-1 according to sgn(X2)=sgn(Y2) or sgn(X2)≠sgn(Y2), respectively.(b)
Let one assume thatX and Y are timelike vectors; then there is a unique number θ≥0 such that 〈X,Y〉=ε|X||Y|cosh⁡θ, where ε=1 or ε=-1 according to X and Y have different time orientation or the same time orientation, respectively.(c)
Let one assume thatX is spacelike vector and Y is timelike vector; then there is a unique number θ≥0 such that 〈X,Y〉=ε|X||Y|sinh⁡θ, where ε=1 or ε=-1 according to sgn(X2)=sgn(Y1) or sgn(X2)≠sgn(Y1), respectively.Given two nonnull vectorsX,Y∈R1n, the corresponding number θ given above will be called simply the angle between X and Y.Definition 2.
Letα:I→Rvn+1 be a curve in Rvn+1 and let α′ be the velocity vector of α, where I is an open interval of R. For any s∈I, the curve α is called timelike curve, spacelike curve, or lightlike (null) curve if, for each s,〈α′(s),α′(s)〉<0, 〈α′(s),α′(s)〉>0, or 〈α′(s),α′(s)〉=0 and α′(s)≠0, respectively. We call α a nonnull curve if α is a timelike curve or a spacelike curve.The Frenet frame of a nonnull curve inMq3(c) is as follows.Letα=α(s):I→Mq3(c), q=0,1 be a nonnull curve immersed in the 3-dimensional space Mq3(c), where I is an open interval. If α′(s)=1 for some s∈I, the curve α is called aunit speed curve. Then in this paper we assume without loss of generality that α is parameterized by the arc length parameter s. Letting ∇ be the Levi-Civita connection of Rv4, there exists the Frenet frame {T,N,B} along α and smooth functions κ, τ in Mq3(c) such that (6)∇TT=-ε1cα+ε2κN,∇TN=-ε1κT+ε3τB,∇TB=-ε2τN,where κ and τ are called thecurvature andtorsion of α, respectively. Considering 〈T,T〉=ε1, 〈N,N〉=ε2, and 〈B,B〉=ε3, we denote by {ε1,ε2,ε3} thecasual characters of {T,N,B}. When {T,N,B} are spacelike, then εi=1, and otherwise, εi=-1, where i∈{1,2,3}. It is well known that curvature and torsion are invariant under the isometries of Mq3(c). Three vector fields T, N, B consisting of the Frenet frame of α are called the tangent vector field, principal normal vector field, and binormal vector field, respectively.A vector fieldX on Mq3(c) along α is said to beparallel alongα if ∇sX=0, where ∇s denotes the covariant derivative along α. A vector Xαs at α(s) is called parallel displacement of vector Xαt at αt along α. If its tangent vector field α′ of curve α is parallel along α, then the curve is called geodesic.We can denote the exponential map atx∈Mq3(c) by exp⁡x and review theexponential mapexp⁡x:TxMq3(c)→Mq3(c) at x∈Mq3(c) which is defined by exp⁡x(υ)=γυ(1), where γυ:[0,∞)→Mq3(c) is the constant speed geodesic starting from x with the initial velocity γυ′(0)=υ. For any point α(s) in the curve α, theprincipal normal geodesic in Mq3(c) starting at α is defined as the geodesic curve γsα(t)=expα(s)(tN(s))=f(t)α(s)+g(t)N(s),  t∈R, where the functions f and g are given by (7)f(t)=cos⁡t,g(t)=sint,ifε2c=1,f(t)=cosh⁡t,g(t)=sinh⁡t,ifε2c=-1.In the following, we will recall the Frenet frame of a null curve inS13.Letα:I→S13 be a null curve in S13, where I is an open interval of R. Then there exists the Frenet frame {T=α′,N,B} along α and smooth functions κ1, κ2 in S13 such that(8)∇TT=κ1N,∇TN=-κ2T+κ1B,∇TB=-κ2N+α,where the first curvature κ1=0 if α is geodesic; otherwise κ1=1. In addition, the following conditions are satisfied: (9)〈T,T〉=〈B,B〉=〈T,N〉=〈B,N〉=0,〈N,N〉=1,〈T,B〉=1.A null curve α which is not a null geodesic is parameterized by pseudoarc length parameter if 〈∇TT,∇TT〉=1. In the circumstances, κ1=1. For any point α(s) in the null curve α, we define the principal normal geodesic in S13 starting at α as the geodesic curve: (10)γsα(t)=expα(s)(tN(s))=f(t)α(s)+g(t)N(s),where t∈R, and the functions f(t)=cos⁡t, g(t)=sint.


## 3. Mannheim Curves inMq3(c)

In this section, we will discuss Mannheim curves in nonflat 3-dimensional space formsMq3(c), and then we give the definition of Mannheim curves as follows.Definition 3.
A curveα with nonzero curvature is said to be a Mannheim curve if there exists another immersed curve β=β(σ):I→Mq3(c) and a one-to-one corresponding between α and β, β≠±α, such that the principal normal geodesics of curve α coincide with the binormal geodesics of curve β at corresponding points. One says that β is a Mannheim curve mate (or Mannheim partner curve) of curve α. The curves α and β are called a pair of Mannheim curves.Letα=α(s) be a unit speed curve in Mq3(c) and let β=β(σ) be the Mannheim partner curve of α, where σ is the arc-length parameter of β. From the definition, we know that there exists a differentiable function a(s) such that (11)βσs=fasα(s)+gasNα(s),where {Tα,Nα,Bα} denotes the Frenet frame along α and β(σ(s)) is the point in β corresponding to α(s). We introduce the distance function d(s) as the distance in 3-dimensional nonflat space between α(s) and its corresponding point β(σ(s)).Proposition 4.
Letα(s) and β(σ) be a pair of nonnull Mannheim curves in Mq3(c); then the following properties hold:(1)
the functiona(s) is constant;(2)
the angle between binormal vectors of Mannheim curvesα and β at corresponding points is constant;(3)
the angle between tangent vectors of Mannheim curvesα and β at corresponding points is constant if and only if the curvature of α is nonzero constant: (12)κα=-ε1ε2cg(a(s))f(a(s)).Proof.
(1) Suppose thatα(s) and β(σ) are a pair of nonnull Mannheim curves in Mq3(c). Since the principal normal geodesics of curve α coincide with the binormal geodesics of curve β at corresponding points, we obtain (13)ddtt=a(s)γsα(t)=εBβ(σ(s)),ε=±1,and because of f′=-ε2cg and g′=f, then(14)Bβ(σ(s))=-εε2cg(a(s))α(s)+εf(a(s))Nα(s),where {Tβ,Nβ,Bβ} with causal characters given by {δ1,δ2,δ3} is the Frenet frame along curve β. On the other hand, by the definition we get (15)βσs=fasαs+gasNαs.Then the tangent vector of β is given by (16)σ′(s)Tβ(σ(s))=ddsβ(σ(s))=-ε2cgasa′sαs+-ε1καgas+fasTα(s)+a′sfasNα(s)+ε3gasταBα(s).By taking the scalar product of (14) with (16), hence (17)0=Bβσs,ddsβσs=εa′scg2as+ε2f2as=εε2a′(s).Thus (17) is reduced to a′(s)=0, which implies a(s) is a constant.
(2) A straightforward computation shows that(18)ddsBαs,Bβσs=-ε2ταNαs,Bβσs+σ′sBαs,-δ2τβNβσs.According to (14), (19)Bβ(σ(s))=-εε2cg(a(s))α(s)+εf(a(s))Nα(s),and, by taking the derivative of this equation, (20)∇TβBβσs=-σ′sδ2τβσsNβσs=-εε2cgas+ε1fasκαTαs+εε3f(a(s))ταBα(s).
Then apply (20) to (18): (21)ddsBαs,Bβσs=-ετα(s)f(a(s))+εf(a(s))τα(s)=0and finish the proof.
(3) By a direct computation,(22)ddsTαs,Tβσs=-ε1cαs+ε2καNαs,Tβσs+σ′sTα,-δ1cβσs+δ2κβNβσs.On the other hand, by taking the derivative of Bβ(σ(s)) with respect to s, (23)∇TβBβσs=-δ2σ′sτβNβσs=-εε2cgasTαs+εfas-ε1καTαs+ε3ταBαs=-εε2cgas+ε1fasκαTαs+εε3f(a(s))ταBα(s).We apply (16), (23), and Proposition 4(1) to (22), obtaining that if (24)κα=-ε1ε2cg(a(s))f(a(s)),then (25)ddsTαs,Tβσs=0.In the following, we consider the characterizations in terms of the curvature and the torsion of Mannheim curves in nonflat 3-dimensional space forms.Theorem 5.
A curveα in Mq3(c) with curvature κ and torsion τ is a Mannheim curve if and only if it satisfies (26)ε3τα2+ε1κα2=f2(θ0)-cε2g2(θ0)f(θ0)g(θ0)κα+cε1ε2,where {ε1,ε2,ε3} are the casual characters of {Tα,Nα,Bα}, and c=±1.Proof.
Letα=α(s) be a unit speed curve in Mq3(c), and let β=β(σ) be a Mannheim partner curve of α, where σ is the arc length parameter of β. We know that a(s) is a nonzero constant from Proposition 4(1); we denote a(s)=θ0.
Letβ(σ(s))=f(θ0)α(s)+g(θ0)Nα(s); by differentiating β(σ(s)) with respect to parameter σ, (27)σ′sTβσs=(f(θ0)-ε1g(θ0)κα)Tα(s)+g(θ0)ε3ταBα(s),where σ′(s)=(f(θ0)-ε1g(θ0)κα)2+g2(θ0)τα2.
Leta1(s)=(1/σ′s)(f(θ0)-ε1g(θ0)κα), a2(s)=(1/σ′s)ε3ταg(θ0), so we write (28)Tβ(σ(s))=a1(s)Tα(s)+a2(s)Bα(s).By differentiating (28) with respect to s, (29)σ′s-δ1cβσs+δ2κβNβσs=a1′sTαs-ε1ca1sαs+(ε2a1(s)κα-ε2a2(s)τα)Nα+a2′(s)Bα(s).Since Bβ is orthogonal to Nβ, (30)βσs=fθ0αs+gθ0Nαs,Bβσs=-εε2cg(θ0)α(s)+εf(θ0)Nα(s).Therefore, we apply these equations to a computation: (31)δ2σ′(s)κβNβ(σ(s))=a1′(s)Tα(s)+δ1cσ′sfθ0-ε1ca1sα(s)+ε2a1(s)κα-ε2a2(s)ταδ1cσ′sgθ0+δ1cσ′sgθ0Nα(s)+a2′(s)Bα(s),(32)δ1cσ′sfθ0-ε1ca1(s)=0,ε2a1(s)κα-ε2a2(s)τα+δ1cσ′sgθ0=0.Then (33)fθ0=ε1δ1a1sσ′s,a2(s)τα=σ′sε1δ1fθ0καs+δ1ε2cgθ0.We change it as follows: (34)fθ0ε1δ1fθ0καs+δ1ε2cgθ0=ε1δ1τα·f(θ0)-ε1g(θ0)καε3ταg(θ0).Moreover, we have the following conclusion: (35)ε3τα2+ε1κα2=f2(θ0)-cε2g2(θ0)f(θ0)g(θ0)κα+cε1ε2.
Conversely, for some curveα in Mq3(c), its curvature and torsion satisfy (35); that is, there exists a constant λ that satisfies ε3τα2+ε1κα2=λκα+cε1ε2. Then we can choose some nonzero constant θ0∈R, such that (36)λ=f2(θ0)-cε2g2(θ0)f(θ0)g(θ0).Thus (37)ε1δ1(f(θ0)-ε1g(θ0)κα)=ε3f(θ0)g(θ0)τα2δ1ε1f(θ0)κα+δ1ε2cg(θ0).We define a curve β and denote a(s)=θ0.
Letβ(σ(s))=f(θ0)α(s)+g(θ0)Nα(s); from (37), it is easy to know that (32) and (33) are satisfied. Moreover, (38)δ2σ′sκβNβσs=a1′(s)Tα(s)+a2′(s)Bα(s).By taking the derivative of (38) and applying (34), (28), (39)Bβ(σ(s))=-εε2cg(a(s))α(s)+εf(a(s))Nα(s).The binormal geodesic of curve β at β(σ(s0)) is denoted by (40)γσ0βt=ftβσs0+gtBβσs0.Then we put (39) in (40): (41)γσ0β(t0)=f(t+a)α(s0)+g(t+a)Nα(s0)=γσ0α(s0).Therefore, β is a Mannheim partner curve of α; it means that the curve α is a Mannheim curve.After that, we investigate the relationship with respect to the curvature and the torsion of Mannheim partner curves. In similar way to Theorem5, we obtain the following theorem.Theorem 6.
Letβ be a curve with the arc length parameter σ in Mq3(c); then β is the Mannheim partner curve of the Mannheim curve α in Mq3(c) if and only if the curvature and torsion of curve β satisfy the following equation:(42)τβ′=δ1κβf2θ1+g2θ1τβ2δ2f(θ1)g(θ1),where {δ1,δ2,δ3} are the causal characters of {Tβ,Nβ,Bβ} and κβ and τβ are the curvature and torsion of curve β, respectively.As is known, a helix in a 3-dimensional manifoldM is defined by a curve whose curvature and torsion are constants. In [7], Liu and Wang claimed that the Mannheim partner curve of a helix in R3 is a straight line. Inspired by this result, we give the following example.Example 7.
LetMq3(c) be a 3-dimensional nonflat space form, which is equivalent to Mq3(c)=S13(1) or Mq3(c)=H03(-1), respectively. α=α(s) is a helix in Mq3(c) parameterized by the arc length with curvature κα and the torsion τα.According to (35), we have (43)λ=ε3τα2+ε1κα2-cε1ε2κα.It means that α is a Mannheim curve in Mq3(c). Thus, the Mannheim partner curve β of α in Mq3(c) is given by (44)β(σ)=f(θ)α(s)+g(θ)Nα(s),where Nα is the principal normal vector field of α and σ is the arc length parameter of β.In the following, we consider the curveα in S13. By taking the derivative of β(σ), we get (45)Tβsdσds=cos⁡θTαs+sinθ-ε1καTαs+ε3ταBαs=(cos⁡θ-ε1καsinθ)Tα(s)+ε3ταsinθBα(s).Then (46)σ=cos⁡θ-ε1καsinθ2+τα2sin2θs.According to(47)λ=f2(θ)-cε2g2(θ)f(θ)g(θ),we obtain that (48)θ=12arctan2κακα2+τα2-1.On the other hand, by using relation (27), (28), and (32), we have ∇σTβ=0. Moreover, β is a geodesic in S13.In the following, we take a concrete example.Letα be a curve in S13 with equation (49)α(s)=22sinh⁡2s,22cosh⁡2s,22sins,22cos⁡s.By simple calculation, we get(50)α′s=2cosh⁡2s,2sinh⁡2s,22cos⁡s,-22sins,α′′s=22sinh⁡2s,22cosh⁡2s,-22sins,-22cos⁡s,α′′′s=42cosh⁡2s,42sinh⁡2s,-22cos⁡s,22sins.By using the Schmidt orthogonalization method, we have (51)Tαs=233cosh⁡2s,233sinh⁡2s,33cos⁡s,-233sins,Nαs=417sinh⁡2s,417cosh⁡2s,-117sins,-117cos⁡s,Bαs=837cosh⁡2s,837sinh⁡2s,-137cos⁡s,137sins.We can easily see that〈T,T〉=-1, 〈N,N〉=1, and B,B=-1; that is, T is a timelike vector, N is a spacelike vector, and B is a timelike vector. By using the Frenet frame, we obtain (52)∇TB=1637sinh⁡2s,1637cosh⁡2s,137sins,137cos⁡s,∇TN=817cosh⁡2s,817sinh⁡2s,-117cos⁡s,117sins.Moreover, by the formulae of the curvature and the torsion for a general parameter, we can calculate that (53)∇TB,N=-τ=3717,∇TN,T=-κ=-173.Then τ=-37/17 and κ=17/3. Therefore, the curve α is a helix in S13. Moreover, we know the helix in S13 is Mannheim curve and satisfies Theorem 5.The projected image of Mannheim curveα in S13 is obtained as Figure 1.Figure 1
The projected image of Mannheim curveα in S13.In general, we can take curveα with (54)α(s)=asinh⁡(ps),acosh⁡(ps),bsin(qs),bcos⁡(qs),where a, b, p, and q are constants such that a2+b2=1 and -a2p2+b2q2=±1; it is easy to know that curve α is a unit speed curve in S13. In similar way to Example 7, we get that curve α is Mannheim curve in S13.


## 4. Null Mannheim Curves inS13, H13

In this section, we will apply ourself to discussing the null Mannheim curves in de Sitter spaceS13 and anti-de Sitter space H13. Due to the fact that the conclusion about the null Mannheim curve in H13 is similar to the case in S13, then we just give the proof of null Mannheim curve in S13.Theorem 8.
There is no null Mannheim curve in de Sitter spaceS13.Proof.
Letα=α(s) be a null curve in S13, and β=β(σ) a Mannheim partner curve of α in S13, where σ is the arc length parameter of β. Suppose that {Tα,Nα,Bα} and {Tβ,Nβ,Bβ} with casual characters {ε1,ε2,ε3} are the frames of curves α and β, respectively. Then there exists a differential function a(s) such that β(σ(s))=f(a(s))α(s)+g(a(s))Nα(s), where f(t)=cos⁡t and g(t)=sint. Since the principal normal geodesics of curve α coincide with the binormal geodesics of curve β at corresponding points, we obtain (55)ddtt=a(s)γsα(t)=εBβ(σ(s)),ε=±1,and because of f′(t)=-g(t) and g′(t)=f(t), then (56)Bβσs=εf′asαs+g′asNαs=-εg(a(s))α(s)+εf(a(s))Nα(s),so that 〈Bβ(σ(s)),Bβ(σ(s))〉=ε2(f2+g2)=1. It means that Bβ is a spacelike vector. Thus β is a nonnull curve whose Frenet equations are similar to Frenet equations (6) in S13. The tangent vector of β is given by (57)σ′sTβσs=ddsβσs=-gasa′sαs+-κ2gas+fasTαs+a′(s)f(a(s))Nα(s)+g(a(s))κ1Bα(s).By taking the scalar product of (56) with (57), we get a′(s)=0, and it means that a(s) is constant. For the sake of simplicity, we denote a(s)=θ0; let α(s)=f(θ0)β(σ(s))+g(θ0)Bβ(σ(s)); by differentiating α(s) with respect to parameter σ, (58)dsdσTαs=fθ0Tβσs-ε2gθ0τβNβσs.Next, we may distinguish two cases: (A) τβ(σ(s))=0 and (B) τβ(σ(s))≠0.
(A) Ifτβ(σ(s))=0 holds, (59)dsdσTαs=fθ0Tβσs.This implies that null vector Tα is collinear with nonnull vector Tβ, which is impossible.
(B) Ifτβ(σ(s))≠0 holds, (60)dsdσTαs=fθ0Tβσs-ε2gθ0τβNβσs.By using Frenet equations of curve β, it follows that (61)0=dsdσTαs,dsdσTαs=ε1f2θ0+ε2g2θ0τβ2.Then (62)τβ2=-ε1f2(θ0)ε2g2(θ0).Relation (62) implies ε1ε2=-1, and (63)τβ=±f(θ0)g(θ0).In addition, we assume that g(θ0)≠0, and otherwise β=±α. We distinguish two subcases according to whether α is geodesic or not: (B.1) κ1=0 and (B.2) κ1=1.
(B.1) Ifκ1=0, then Tα(s) is constant. By taking the derivative of (60) with respect to σ, (64)d2sdσ2Tαs=-ε1fθ0βσs-gθ0κβτβTβσs+ε2f(θ0)κβNβ(σ(s))-ε2g(θ0)τβ2Bβ(σ(s)).By taking the scalar product of (56) and (64), (65)0=d2sdσ2Tαs,Bβσs=-ε2g(θ0)τβ2.Moreover, τβ=0 which is a contradiction to the fact that τβ is nonzero constant.
(B.2) Ifκ1=1, by applying (63) to (60), it follows that(66)dsdσTα(s)=f(θ0)Tβ(σ(s))∓ε2f(θ0)Nβ(σ(s)).By differentiating (66) with respect to σ,(67)d2sdσ2Tα(s)+dsdσ2Nα(s)=-ε1fθ0βσs∓fθ0κβTβσs+ε2fθ0κβNβσs∓ε2fθ0τβBβσs.Then(68)dsdσ4=d2sdσ2Tα(s)+dsdσ2Nα(s),mmmd2sdσ2Tα(s)+dsdσ2Nα(s)=τβ2.By taking the derivative of Bβ(σ(s)) with respect to s,(69)ddsBβσs=-σ′sτβNβσs=-εgθ0+fθ0κ2Tα(s)+εfθ0Bα(s).By taking the scalar product of (67) with Nβ(σ(s)) and using (68), we obtain κβ=0. Since a(s) is constant, (57) is reduced to(70)Tβσs=dsdσ-κ2gθ0+fθ0Tαs+gθ0Bαs.By applying κβ=0 to the Frenet frame of β, we get(71)∇TβTβ=-ε1β.On the other hand,(72)∇TβTβ=dsdσ2fθ0-2κ2gθ0Nαs+gθ0αs.Then(73)βσs=-ε1dsdσ2fθ0-2κ2gθ0Nαs+gθ0αs=fθ0αs+gθ0Nαs,βσs,βσs=1=-ε1dsdσ22fθ0gθ0-2κ2g2θ0.Therefore,(74)κ2=dσ/ds2+2ε1fθ0gθ02ε1g2θ0=f(θ0)g(θ0)+ε12g2(θ0)dσds2.By taking the scalar product of (69) with itself,(75)-σ′sτβNβσs,-σ′sτβNβσs=-f(θ0)(g(θ0)+f(θ0)κ2)=ε2dσds2τβ2.Then we apply (68) to the last relation(76)τβ2τβ2-f2θ0gθ0+fθ0κ22=0.Then τβ=0 or τβ2-f2(θ0)(g(θ0)+f(θ0)κ2)2=0.
Ifτβ2-f2(θ0)(g(θ0)+f(θ0)κ2)2=0 holds, thus(77)κ2=f(θ0)g(θ0),which is a contradiction to (74). It is impossible that τβ=0.
Therefore, there is no null Mannheim curve in de Sitter space.



---

*Source: 319046-2015-04-20.xml*