Mathematical Problems in Engineering

VolumeΒ 2011, Article IDΒ 539378, 19 pages

http://dx.doi.org/10.1155/2011/539378

## Generalized Timelike Mannheim Curves in Minkowski Space-Time

Department of Mathematics, Faculty of Arts and Sciences, Sakarya University, 54187 Sakarya, Turkey

Received 19 January 2011; Accepted 4 March 2011

Academic Editor: CarloΒ Cattani

Copyright Β© 2011 M. Akyig~it et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We give the definition of generalized timelike Mannheim curve in Minkowski space-time . The necessary and sufficient conditions for the generalized timelike Mannheim curve are obtained. We show some characterizations of generalized Mannheim curve.

#### 1. Introduction

The geometry of curves has long captivated the interests of mathematicians, from the ancient Greeks through to the era of Isaac Newton (1643β1727) and the invention of the calculus. It is a branch of geometry that deals with smooth curves in the plane and in the space by methods of differential and integral calculus. The theory of curves is the simpler and narrower in scope because a regular curve in a Euclidean space has no intrinsic geometry. One of the most important tools used to analyze curve is the Frenet frame, a moving frame that provides a coordinate system at each point of curve that is "best adopted" to the curve near that point. Every person of classical differential geometry meets early in his course the subject of Bertrand curves, discovered in 1850 by J. Bertrand. A Bertrand curve is a curve such that its principal normals are the principal normals of a second curve. There are many works related with Bertrand curves in the Euclidean space and Minkowski space, [1β3].

Another kind of associated curve is called Mannheim curve and Mannheim partner curve. The notion of Mannheim curves was discovered by A. 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 for some constant . The articles concerning Mannheim curves are rather few. In [4], a remarkable class of Mannheim curves is studied. General Mannheim curves in the Euclidean 3-space are obtained in [5β7]. Recently, Mannheim curves are generalized and some characterizations and examples of generalized Mannheim curves are given in Euclidean 4-space by [8].

In this paper, we study the generalized timelike Mannheim partner curves in 4-dimensional Minkowski space-time. We will give the necessary and sufficient conditions for the generalized timelike Mannheim partner curves.

#### 2. Preliminaries

To meet the requirements in the next sections, the basic elements of the theory of curves in Minkowski space-time are briefly presented in this section. A more complete elementary treatment can be found in [9].

Minkowski space-time is a usual vector space provided with the standard flat metric given by where is a rectangular coordinate system in .

Since is an indefinite metric, recall that a can have one of the three causal characters; it can be spacelike if or , timelike if , and null(ligthlike) if and . Similarly, an arbitrary curve in can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors are, respectively, spacelike, timelike, or null. The norm of is given by . If for all , then is a regular curve in . A timelike (spacelike) regular curve is parameterized by arc-length parameter which is given by , then the tangent vector along has unit length, that is, , for all .

Hereafter, curves considered are timelike and regular curves in . Let for all ; then the vector field is timelike and it is called timelike unit tangent vector field on .

The timelike curve is called special timelike Frenet curve if there exist three smooth functions , , on and smooth nonnull frame field along the curve . Also, the functions , , and are called the first, the second, and the third curvature function on , respectively. For the special timelike Frenet curve , the following Frenet formula issee [9].

Here, due to characters of Frenet vectors of the timelike curve, , , , and are mutually orthogonal vector fields satisfying equations For , the nonnull frame field and curvature functions , , and are determined as follows: where is determined by the fact that orthonormal frame field is of positive orientation. The function is determined by So the function never vanishes.

In order to make sure that the curve is a special timelike Frenet curve, above steps must be checked, from 1st step to 4th step, for .

Let be the moving Frenet frame along a unit speed timelike curve in , consisting of the tangent, the principal normal, the first binormal, and the second binormal vector field, respectively. Since is a timelike curve, its Frenet frame contains only nonnull vector fields.

#### 3. Generalized Timelike Mannheim Curves in

Mannheim curves are generalized by Matsuda and Yorozu in [8]. In this paper, we have investigated the generalization of timelike Mannheim curves in Minkowski space .

*Definition 3.1. *A special timelike curve in is a generalized timelike Mannheim curve if there exists a special timelike Frenet curve in such that the first normal line at each point of is included in the plane generated by the second normal line and the third normal line of at the corresponding point under . Here, is a bijection from to . The curve is called the generalized timelike Mannheim mate curve of .

By the definition, a generalized Mannheim mate curve is given by the map such that Here is a smooth function on . Generally, the parameter is not an arc-length of . Let be the arc-length of defined by If a smooth function is given by , then for all , we have The representation of timelike curve with arc-length parameter is For a bijection defined by , the reparameterization of is where is a smooth function on . Thus, we have

Theorem 3.2. *If a special timelike Frenet curve in is a generalized timelike Mannheim curve, then the following relation between the first curvature function and the second curvature function holds:
**
where is a constant number.*

* Proof. *Let be a generalized timelike Mannheim curve and the generalized timelike Mannheim mate curve of , as the following diagram:
(3.8)
A smooth function is defined by and is the arc-length parameter of . Also, is a bijection defined by . Thus, the timelike curve is reparametrized as follows
where is a smooth function. By differentiating both sides of (3.9) with respect to , we have
On the other hand, since the first normal line at each point of is lying in the plane generated by the second normal line and the third normal line of at the corresponding points under bijection , the vector field is given by
where and are some smooth functions on . If we take into consideration
and (3.10), then we have . So we rewrite (3.10) as
that is,
where
By taking the differentiations both sides of (3.13) with respect to , we get
Since
the coefficient of in (3.16) vanishes, that is,
Thus, this completes the proof.

Theorem 3.3. *In , let be a special timelike Frenet curve such that its nonconstant first and second curvature functions satisfy the equality for all . If the timelike curve given by
**
is a special timelike Frenet curve, then is a generalized timelike Mannheim mate curve of .*

* Proof. * The arc-length parameter of is given by
Under the assumption of
we obtain , .

Differentiating the equation with respect to , we reach
Thus, it is seen that
The differentiation of the last equation with respect to is
From our assumption, we have
Thus, the coefficient of in (3.24) is zero. It is seen from (3.23) that is a linear combination of and . Additionally, from (3.24), is given by linear combination of , , and . On the other hand, is a special timelike Frenet curve that the vector is given by linear combination of and .

Therefore, the first normal line of lies in the plane generated by the second normal line and third normal line of at the corresponding points under a bijection defined by .

This completes the proof.

*Remark 3.4. *In 4-dimensional Minkowski space , a special timelike Frenet curve with curvature functions and satisfying , it is not clear that a smooth timelike curve given by (3.1) is a special Frenet curve. Thus, it is unknown whether the reverse of Theorem 3.2 is true or not.

Theorem 3.5. *Let be a special timelike curve in with nonzero third curvature function . There exists a timelike special Frenet curve in such that the first normal line of is linearly dependent with the third normal line of at the corresponding points and , respectively, under a bijection , if and only if the curvatures and of are constant functions.*

* Proof. * Let be a timelike Frenet curve in with the Frenet frame field and curvature functions , , and . Also, we assume that is a timelike special Frenet curve in with the Frenet frame field and curvature functions , , and . Let the first normal line of be linearly dependent with the third normal line of at the corresponding points and , respectively. Then the parameterization of is
If the arc-length parameter of is given , then
Moreover, is a bijection given by .

Differentiating (3.26) with respect to and using the Frenet formulas, we get
Since , then
that is,
From the last equation, it is easily seen that is a constant number. Hereafter, we can denote , for all .

From (3.27), we have
Thus, we rewrite (3.29) as follows:
The differentiation of the last equation with respect to is
Since and for all ,
is satisfied. Then
is a nonzero constant number. Thus, from (3.34), we reach
where for all . Differentiating the last equation with respect to , then we have
for all . Considering
then we get
Arranging the last equation, we find
Moreover, the differentiation of (3.36) with respect to is
From the above equation, it is seen that
Substituting (3.36) and (3.43) into (3.41), we obtain
This means that the first curvature function is constant (that is, positive constant). Additionally, from (3.43) it is seen that the second curvature function is positive constant, too.

Conversely, suppose that is a timelike Frenet curve in with the Frenet frame field and curvature functions , , and . The first curvature function and the second curvature function of are of positive constant. Thus, is a positive constant number, say .

The representation of timelike curve with arc-length parameter is
Let denote the arc-length parameter of ; we have
Then, we obtain and
that is,
By differentiating the both sides of the above equality with respect to , we find
Hence, since does not vanish, we get
where denotes the sign of function . That is, is β1 or .

We can put
Then, we get
Differentiating of the last equation with respect to , we reach
and we have
Since is positive for , we have
Thus, we can put
By differentiation of the above equation with respect to , we get
Since and , we have
Thus, we obtain for , where . We must determine whether is β1 or under the condition that the frame field is of positive orientation.

We have, by for ,
and for any . Therefore, we get . Thus, we get
By the above facts, is a special Frenet curve in and the first normal line at each point of is the third normal line of at corresponding each point under the bijection .

Thus, the proof is completed.

The following theorem gives a parametric representation of a generalized timelike Mannheim curves .

Theorem 3.6. *Let be a timelike special curve defined by
**
Here, is a nonzero constant number, and are any smooth functions, and the positive-valued smooth function is given by
**
for . Then, the curvature functions and of satisfy
**
at each point of .*

* Proof. * Let be a timelike special curve defined by
where is a nonzero constant number and are any smooth functions. is a positive-valued smooth function. Thus, we obtain
where the subscript prime denotes the differentiation with respect to .

The arc-length parameter of is given by
where .

If denotes the inverse function of , then and we get
where the prime denotes the differentiation with respect to .

The unit tangent vector of the curve at the each point is given by
Some simplifying assumptions are made for the sake of brevity as follows:
Thus, we get
So, we rewrite (3.68) as
Differentiating the last equation with respect to , we find
that is,
From the last equation, we find
By the fact that , we get
In order to get second curvature function , we need to calculate . After a long process of calculations and using abbreviations, we obtain
where
If we simplify , then we have
Therefore, we rewrite (3.76) and (3.77) as
where
Consequently, from (3.79) and (3.80), we have
Substituting the abbreviations into the last equation, we have
After substituting (3.80) into the last equation and simplifying it, we get
Moreover, from (3.74) it is seen that
The last two equations show us that
By the fact , we obtain
According to our assumption
we obtain
Substituting the above equation into (3.86) and (3.87), we obtain
The proof is completed.

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