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.
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 . 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 , 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 . 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 .
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 .
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, , which we call de Sitter space. The other type is pseudosphere with a negative radius squared, , 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 . 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 . Grbović et al. discussed the null Mannheim curves in 3-dimensional Minkowski space . 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 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 .
Now we introduce some basic notions in semi-Euclidean space and curves.
Let denote the -dimensional pseudo-Euclidean space of index ; let be an canonical basis of . We choose two vectors , and the standard metric of is given by where stand for the coordinate components of and with respect to in , respectively.
Letting , the vector is said to be Define the norm of a nonnull vector by , where . We call the unit vector if .
We define the signature of a vector as follows:
Let denote the nonflat 3-dimensional space forms of , and constant curvature . Meanwhile, , if , and , if . Moreover, we will denote by the pseudo-Euclidean hypersphere or the pseudo-Euclidean hyperbolic space according to or , respectively, where is denoted by and the pseudo-Euclidean hyperbolic space of index and curvature is given by :
Many features of inner product spaces have analogues in the pseudo-Euclidean case. In , the Schwarz inequality permits the definition of the Euclidean angle between vectors and as the unique number , such that . For two nonnull vectors , which are not spacelike vectors in the Lorentzian space , 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 such that they span a plane . In this plane, we can choose an orthogonal basis , with and . Then we can write vectors , in this basis as and .
Definition 1 (see ). Let one consider two nonnull vectors .(a)Let one assume that and are spacelike vectors; then(i)if they span a spacelike plane, there is a unique number such that ;(ii)if they span a timelike plane, there is a unique number such that , where or according to or , respectively.(b)Let one assume that and are timelike vectors; then there is a unique number such that , where or according to and have different time orientation or the same time orientation, respectively.(c)Let one assume that is spacelike vector and is timelike vector; then there is a unique number such that , where or according to or , respectively.
Given two nonnull vectors , the corresponding number given above will be called simply the angle between and .
Definition 2. Let be a curve in and let be the velocity vector of , where is an open interval of . For any , the curve is called timelike curve, spacelike curve, or lightlike (null) curve if, for each , , or and , respectively. We call a nonnull curve if is a timelike curve or a spacelike curve.
The Frenet frame of a nonnull curve in is as follows.
Let , be a nonnull curve immersed in the 3-dimensional space , where is an open interval. If 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 , there exists the Frenet frame along and smooth functions , in such that where and are called the curvature and torsion of , respectively. Considering , , and , we denote by the casual characters of . When are spacelike, then , and otherwise, , where . It is well known that curvature and torsion are invariant under the isometries of . 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 along is said to be parallel along if , 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 by and review the exponential map at which is defined by , where is the constant speed geodesic starting from with the initial velocity . For any point in the curve , the principal normal geodesic in starting at is defined as the geodesic curve , , where the functions and are given by
In the following, we will recall the Frenet frame of a null curve in .
Let be a null curve in , where is an open interval of . Then there exists the Frenet frame along and smooth functions , in such thatwhere the first curvature if is geodesic; otherwise . In addition, the following conditions are satisfied: A null curve which is not a null geodesic is parameterized by pseudoarc length parameter if . In the circumstances, . For any point in the null curve , we define the principal normal geodesic in starting at as the geodesic curve: where , and the functions , .
3. Mannheim Curves in
In this section, we will discuss Mannheim curves in nonflat 3-dimensional space forms , 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 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 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 where denotes the Frenet frame along and is the point in corresponding to . We introduce the distance function as the distance in 3-dimensional nonflat space between and its corresponding point .
Proposition 4. Let and be a pair of nonnull Mannheim curves in ; 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 . Since the principal normal geodesics of curve coincide with the binormal geodesics of curve at corresponding points, we obtain and because of and , thenwhere with causal characters given by is the Frenet frame along curve . On the other hand, by the definition we get Then the tangent vector of is given by By taking the scalar product of (14) with (16), hence Thus (17) is reduced to , which implies is a constant.
(2) A straightforward computation shows that According to (14), and, by taking the derivative of this equation, Then apply (20) to (18): and finish the proof.
(3) By a direct computation, On the other hand, by taking the derivative of with respect to , We apply (16), (23), and Proposition 4(1) to (22), obtaining that if then
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 curve in with curvature and torsion is a Mannheim curve if and only if it satisfies where are the casual characters of , and .
Proof. Let be a unit speed curve in , 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 .
Let ; by differentiating with respect to parameter , where .
Let , , so we write 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 , its curvature and torsion satisfy (35); that is, there exists a constant that satisfies . Then we can choose some nonzero constant , such that Thus We define a curve and denote .
Let ; from (37), it is easy to know that (32) and (33) are satisfied. Moreover, By taking the derivative of (38) and applying (34), (28), The binormal geodesic of curve at is denoted by Then we put (39) in (40): 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 ; then is the Mannheim partner curve of the Mannheim curve in if and only if the curvature and torsion of curve satisfy the following equation:where 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 , Liu and Wang claimed that the Mannheim partner curve of a helix in is a straight line. Inspired by this result, we give the following example.
Example 7. Let be a 3-dimensional nonflat space form, which is equivalent to or , respectively. is a helix in parameterized by the arc length with curvature and the torsion .
According to (35), we have It means that is a Mannheim curve in . Thus, the Mannheim partner curve of in 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 . By taking the derivative of , we get Then
In the following, we take a concrete example.
Let be a curve in with equation
By simple calculation, we get By using the Schmidt orthogonalization method, we have
We can easily see that , , and ; that is, is a timelike vector, is a spacelike vector, and is a timelike vector. By using the Frenet frame, we obtain Moreover, by the formulae of the curvature and the torsion for a general parameter, we can calculate that Then and . Therefore, the curve is a helix in . Moreover, we know the helix in is Mannheim curve and satisfies Theorem 5.
The projected image of Mannheim curve in is obtained as Figure 1.
In general, we can take curve with where , , , and are constants such that and ; it is easy to know that curve is a unit speed curve in . In similar way to Example 7, we get that curve is Mannheim curve in .
4. Null Mannheim Curves in ,
In this section, we will apply ourself to discussing the null Mannheim curves in de Sitter space and anti-de Sitter space . Due to the fact that the conclusion about the null Mannheim curve in is similar to the case in , then we just give the proof of null Mannheim curve in .
Theorem 8. There is no null Mannheim curve in de Sitter space .
Proof. Let be a null curve in , and a Mannheim partner curve of in , where is the arc length parameter of . Suppose that and with casual characters are the frames of curves and , respectively. Then there exists a differential function such that , where and . Since the principal normal geodesics of curve coincide with the binormal geodesics of curve at corresponding points, we obtain and because of and , then so that . It means that is a spacelike vector. Thus is a nonnull curve whose Frenet equations are similar to Frenet equations (6) in . The tangent vector of is given by By taking the scalar product of (56) with (57), we get , and it means that is constant. For the sake of simplicity, we denote ; let ; by differentiating with respect to parameter , Next, we may distinguish two cases: (A) and (B) .
(A) If holds, This implies that null vector is collinear with nonnull vector , which is impossible.
(B) If holds, By using Frenet equations of curve , it follows that Then Relation (62) implies , and In addition, we assume that , and otherwise . We distinguish two subcases according to whether is geodesic or not: (B.1) and (B.2) .
(B.1) If , then is constant. By taking the derivative of (60) with respect to , By taking the scalar product of (56) and (64), Moreover, which is a contradiction to the fact that is nonzero constant.
(B.2) If , by applying (63) to (60), it follows thatBy differentiating (66) with respect to ,ThenBy taking the derivative of with respect to ,By taking the scalar product of (67) with and using (68), we obtain . Since is constant, (57) is reduced toBy applying to the Frenet frame of , we getOn the other hand,ThenTherefore,By taking the scalar product of (69) with itself,Then we apply (68) to the last relationThen or .
If holds, thuswhich is a contradiction to (74). It is impossible that .
Therefore, there is no null Mannheim curve in de Sitter space.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is partially supported by NSFC Grant no. 11271063.
M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ, USA, 1976.View at: MathSciNet