Mannheim Curves in Nonflat 3-Dimensional Space Forms

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.


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][2][3][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  and torsion  satisfy the relation  = ( 2 +  2 ), where  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][9][10][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, ⟨, ⟩ =  2 , which we call de Sitter space.The other type is pseudosphere with a negative radius squared, ⟨, ⟩ = − 2 , 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 2 Advances in Mathematical Physics 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 S 3  1 .Grbović et al. discussed the null Mannheim curves in 3dimensional 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 Section 2, we review some basic notions about the space R +1 V 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].

Preliminaries
Now we introduce some basic notions in semi-Euclidean space and curves.
We define the signature of a vector  as follows: V denote the nonflat 3-dimensional space forms of  = 0, 1, and constant curvature  ̸ = 0.Meanwhile, V = , if  = 1, and V =  + 1, if  = −1.Moreover, we will denote M 3  () by the pseudo-Euclidean hypersphere S 3  (1) or the pseudo-Euclidean hyperbolic space H 3  (−1) according to  = 1 or  = −1, respectively, where S 3  (1) is denoted by and the pseudo-Euclidean hyperbolic space of index  ≥ 0 and curvature  = −1 is given by H 3  (−1): Many features of inner product spaces have analogues in the pseudo-Euclidean case.In R  , the Schwarz inequality permits the definition of the Euclidean angle  between vectors  and  as the unique number 0 ≤  ≤ , such that ⟨, ⟩ = |||| cos .For two nonnull vectors ,  which are not spacelike vectors in the Lorentzian space R  1 , the definition of the angle  between  and  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 ,  ∈ R  1 such that they span a plane R 2 1 .In this plane, we can choose an orthogonal basis { 1 ,  2 }, with ⟨ 1 ,  1 ⟩ = −1 and ⟨ 2 ,  2 ⟩ = 1.Then we can write vectors ,  in this basis as  = ( 1 ,  2 ) and  = ( 1 ,  2 ).
(b) Let one assume that  and  are timelike vectors; then there is a unique number  ≥ 0 such that ⟨, ⟩ = |||| cosh , where  = 1 or  = −1 according to  and  have different time orientation or the same time orientation, respectively.
Given two nonnull vectors ,  ∈ R  1 , the corresponding number  given above will be called simply the angle between  and .
The Frenet frame of a nonnull curve in M 3  () is as follows.Let  = () :  → M 3  (),  = 0, 1 be a nonnull curve immersed in the 3-dimensional space M 3  (), where  is an open interval.If ‖  ()‖ = 1 for some  ∈ , the curve  is called a unit speed curve.Then in this paper we assume without loss of generality that  is parameterized by the arc length parameter .Letting ∇ be the Levi-Civita connection of R 4 V , there exists the Frenet frame {, , } along  and smooth functions ,  in M 3  () such that where  and  are called the curvature and torsion of , respectively.Considering ⟨, ⟩ =  1 , ⟨, ⟩ =  2 , and ⟨, ⟩ =  3 , we denote by { 1 ,  2 ,  3 } the casual characters of {, , }.When {, , } are spacelike, then   = 1, and otherwise,   = −1, where  ∈ {1, 2, 3}.It is well known that curvature and torsion are invariant under the isometries of M 3  ().Three vector fields , ,  consisting of the Frenet frame of  are called the tangent vector field, principal normal vector field, and binormal vector field, respectively.
A vector field  on M 3  () along  is said to be parallel along  if ∇   = 0, where ∇  denotes the covariant derivative along .A vector  () at () is called parallel displacement of vector  () at () along .If its tangent vector field   of curve  is parallel along , then the curve is called geodesic.
We can denote the exponential map at  ∈ M 3  () by exp  and review the exponential map exp  : , where   : [0, ∞) → M 3  () is the constant speed geodesic starting from  with the initial velocity    (0) = .For any point () in the curve , the principal normal geodesic in M 3  () starting at  is defined as the geodesic curve    () = exp () (()) = ()() + ()(),  ∈ R, where the functions  and  are given by In the following, we will recall the Frenet frame of a null curve in S 3  1 .
Let  :  → S 3 1 be a null curve in S 3 1 , where  is an open interval of R. Then there exists the Frenet frame { =   , , } along  and smooth functions  1 ,  2 in S 3  1 such that where the first curvature  1 = 0 if  is geodesic; otherwise  1 = 1.In addition, the following conditions are satisfied: A null curve  which is not a null geodesic is parameterized by pseudoarc length parameter if ⟨∇  , ∇  ⟩ = 1.In the circumstances,  1 = 1.For any point () in the null curve , we define the principal normal geodesic in S 3 1 starting at  as the geodesic curve: where  ∈ R, and the functions () = cos , () = sin .

Mannheim Curves in M 3 𝑞 (𝑐)
In this section, we will discuss Mannheim curves in nonflat 3-dimensional space forms M 3  (), and then we give the definition of Mannheim curves as follows.
Definition 3. A curve  with nonzero curvature is said to be a Mannheim curve if there exists another immersed curve  = () :  → M 3  () and a one-to-one corresponding between  and ,  ̸ = ±, such that the principal normal geodesics of curve  coincide with the binormal geodesics of curve  at corresponding points.One says that  is a Mannheim curve mate (or Mannheim partner curve) of curve .The curves  and  are called a pair of Mannheim curves.
Let  = () be a unit speed curve in M 3  () and let  = () be the Mannheim partner curve of , where  is the arc-length parameter of .From the definition, we know that there exists a differentiable function () such that
Proposition 4. Let () and () be a pair of nonnull Mannheim curves in M 3  (); then the following properties hold: (1) the function () is constant; (2) the angle between binormal vectors of Mannheim curves  and  at corresponding points is constant; (3) the angle between tangent vectors of Mannheim curves  and  at corresponding points is constant if and only if the curvature of  is nonzero constant: Proof.
(1) Suppose that () and () are a pair of nonnull Mannheim curves in M 3  ().Since the principal normal geodesics of curve  coincide with the binormal geodesics of curve  at corresponding points, we obtain By taking the scalar product of ( 14) with ( 16), hence Thus ( 17) is reduced to   () = 0, which implies () is a constant.
(   ( We apply ( 16), (23), and Proposition 4(1) to ( 22), obtaining that if In the following, we consider the characterizations in terms of the curvature and the torsion of Mannheim curves in nonflat 3-dimensional space forms.
Proof.Let  = () be a unit speed curve in M 3  (), and let  = () be a Mannheim partner curve of , where  is the arc length parameter of .We know that () is a nonzero constant from Proposition 4(1); we denote () =  0 .
Let (()) = ( 0 )() + ( 0 )  (); by differentiating (()) with respect to parameter , where By differentiating (28) with respect to , Since   is orthogonal to   , Therefore, we apply these equations to a computation: Then We change it as follows: Moreover, we have the following conclusion: Conversely, for some curve  in M 3  (), its curvature and torsion satisfy (35); that is, there exists a constant  that satisfies  3  2  +  1  2  =   +  1  2 .Then we can choose some nonzero constant  0 ∈ R, such that Thus We define a curve  and denote () =  0 .Let (()) = ( 0 )() + ( 0 )  (); from (37), it is easy to know that (32) and ( 33 Therefore,  is a Mannheim partner curve of ; it means that the curve  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 Theorem 5, we obtain the following theorem.Theorem 6.Let  be a curve with the arc length parameter  in M 3  (); then  is the Mannheim partner curve of the Mannheim curve  in M 3  () if and only if the curvature and torsion of curve  satisfy the following equation: where { 1 ,  2 ,  3 } are the causal characters of {  ,   ,   } and   and   are the curvature and torsion of curve , respectively.
As is known, a helix in a 3-dimensional manifold  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 R 3 is a straight line.Inspired by this result, we give the following example.
According to (35), we have It means that  is a Mannheim curve in M 3  ().Thus, the Mannheim partner curve  of  in M 3  () is given by where   is the principal normal vector field of  and  is the arc length parameter of .
In the following, we consider the curve  in S 3 1 .By taking the derivative of (), we get Then According to we obtain that On the other hand, by using relation ( 27), (28), and (32), we have ∇    = 0.Moreover,  is a geodesic in S 3 1 .In the following, we take a concrete example.Let  be a curve in S 3  1 with equation By simple calculation, we get By using the Schmidt orthogonalization method, we have We can easily see that ⟨, ⟩ = −1, ⟨, ⟩ = 1, and ⟨, ⟩ = −1; that is,  is a timelike vector,  is a spacelike vector, and  is a timelike vector.By using the Frenet frame, we obtain  (53) In this section, we will apply ourself to discussing the null Mannheim curves in de Sitter space S 3 1 and anti-de Sitter space H 3  1 .Due to the fact that the conclusion about the null Mannheim curve in H 3  1 is similar to the case in S 3 1 , then we just give the proof of null Mannheim curve in S 3  1 .
Theorem 8.There is no null Mannheim curve in de Sitter space S 3 1 .