International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2011, Article IDΒ 580537, 7 pages

http://dx.doi.org/10.1155/2011/580537

## No Null-Helix Mannheim Curves in the Minkowski Space

Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan

Received 29 December 2010; Accepted 23 May 2011

Academic Editor: ManfredΒ Moller

Copyright Β© 2011 Jae Won Lee. 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 study a null Mannheim curve with time-like or space-like Mannheim partner curve in the Minkowski 3-space . We get the characterization of a null Mannheim curve. Then, we investigate there is no null-helix Mannheim curve in .

#### 1. Introduction

In the study of the fundamental theory and the characterizations of space curves, the related curves for which there exist corresponding relations between the curves are very interesting and important problems. The most fascinating examples of such curve are associated curves, the curves for which at the corresponding points of them one of the Frenet vectors of a curve coincides with the one of Frenet vectors of the other curve. The well-known associated curve is Bertrand curve which is characterized as a kind of corresponding relation between the two curves. The relation is that the principal normal of a curve is the principal normal of the other curve, that is, the Bertrand curve is a curve which shares the normal line with the other curve [1].

Furthermore, Bertrand curves are not only the example of associated curves. Recently, a new definition of the associated curves was given by Liu and Wang [2]. They called these new curves as Mannheim partner curves. They showed that the curve is the Mannheim partner of the other curve if and only if the curvature and of satisfy the following equation: for some nonzero constant . They also study the Mannheim curves in Minkowski 3-space. Some different characterizations of Mannheim partner curves are given by Orbay and Kasap [3]. Another example is null Mannheim curves from ztekin and Ergt [4]. Since a null vector and a nonnull vector are linearly independent in the Minkowski space , they have noticed that the Mannheim partner curve of a null curve cannot be a null curve. They defined the null Mannheim curves whose Mannheim partner curves are either time-like or space-like.

In this paper, we get the necessary and sufficient conditions for the null Mannheim curves. Then, we investigate there exists no null-helix Mannheim curve in the Minkowski 3-space .

#### 2. Preliminaries

Let be a 3-dimensional Lorentzian space and a smooth null curve in , given by Then, the tangent vector field in satisfies Denote by the tangent bundle of and the perpendicular. Clearly, is a vector bundle over of rank 2. Since is null, the tangent bundle of is a subbundle of of rank 1. This implies that is not complementary of in . Thus, we must find a complementary vector bundle to of in which will play the role of the normal bundle consistent with the classical non-degenerate theory.

Suppose denotes the
complementary vector subbundle to in
;
that is, we have where
means the orthogonal direct sum. It follows that is a
nondegenerate vector subbundle of ,
of rank of 1. We call * a screen
vector bundle of *,
which being non-degenerate, and we have
where
is a complementary orthogonal vector subbundle to in
of rank 2.

We denote by the algebra of smooth functions on and by the module of smooth sections of a vector bundle over . We use the same notation for any other vector bundle.

Theorem 2.1 (see [5, 6]). * Let be
a null curve of a Lorentzian space
and a screen
vector bundle of .
Then, there exists a unique vector bundle over of rank
1 such that there is a unique section satisfying *

We call the vector bundle the null transversal bundle of with respect to . Next consider the vector bundle which from (2.5) is complementary but not orthogonal to in .

More precisely, we have One calls the transversal vector bundle of with respect to . The vector field in Theorem 2.1 is called the null transversal vector field of with respect to . As is a null basis of satisfying (2.5), any screen vector bundle of is Euclidean.

Note that for any arbitrary parameter on and a screen vector bundle one finds a distinguished parameter given by where is the null transversal vector field with respect to and .

Let be a smooth null curve, parametrized by the distinguished parameter instead of such that ([6]). Using (2.5) and (2.7) and taking into account that the screen vector bundle is Euclidean of rank 1, one obtains the following Frenet equations [1]:

*Definition 2.2. *Let
be a curve in the Minkowski 3-space
and
a velocity of vector of .
The curve
is called * time-like (or space-like)* if
(or if ).

Let be the tangent, the principal normal, and the binormal of , respectively. Then, there are two cases for the Frenet formulae.

*Case 1. * and
are space-like vectors, and
is a time-like vector

*Case 2. * is
a time-like vector, and
and are
space-like vectors
where
and
are called the dual curvature and dual torsion of ,
respectively [5].

#### 3. Null Mannheim Curves in

*Definition 3.1. *Let be a Cartan
framed null curve and a time-like
or space-like curve in the Minkowski space .
If there exists a corresponding relationship between the space curves
and
such that the principal normal lines of
coincides with the binormal lines of
at the corresponding points of the curves, then called
a *null Mannheim curve* and
is called a * time-like or space-like Mannheim partner curve of
*.
The pair of is said to
be a * null Mannheim pair* [2, 4].

Theorem 3.2. *Let be a null
Mannheim curve with time-like Mannheim partner curve , and let
be the Cartan frame field along
and the Frenet frame field along
.
Then,
is the time-like Mannheim partner curve of
if and only if its torsion
is constant such that , where
is nonzero constant. *

*Proof. *Assume that
is a null Mannheim curve with time-like Mannheim partner curve
.
Then, by Definition 3.1, we can
write for
some function . By taking
the derivative of (3.1) with
respect to
and applying the Frenet formulae, we have
Since
coincides with ,
we get
which means that
is a nonzero constant. Thus, we have
Since
is null and from (3.4), we
obtain
which means that
is a time-like curve with constant torsion.

Conversely, let the torsion of
the time-like curve
be a constant with for some nonzero constant
.
By considering a null curve defined by
we prove that
is a null Mannheim and
is the time-like Mannheim partner curve of .
By differentiating (3.6) with
respect to ,
we get
If we use
in (3.7), we obtain
which
means that
lies in the plane which is spanned by
and ,
hence .
The proof is complete.

Theorem 3.3. *A Cartan framed null curve
in
is a null Mannheim curve with time-like Mannheim partner curve
if and only if the torsion
of
is nonzero constant.*

*Proof. *Let be a null
Mannheim curve in .
Suppose that is a
time-like curve whose binormal direction coincides with the principal normal of
.
Then, . Therefore,
we can write for
some function .
Differentiating (3.9) with
respect to ,
we obtain
Since the binormal direction of
coincides with the principal normal of ,
we get .
Therefore, we have
and
is constant. By taking the derivative of (3.10), we get
Since
is in the binormal direction of ,
we have
and hence
Conversely, similar to the proof of Theorem 3.2, we easily get a null Mannheim curve with time-like
Mannheim partner curve.

Proposition 3.4. *If be a
generalized null-helix in ,
then, the curve can not be a Mannheim curve.*

*Proof. *Suppose that is a
Mannheim curve in .
Then, there exists the Mannheim partner curve of
in
.
From Theorems 3.2 and 3.3, the torsions of the Mannheim
pair ,
and ,
are nonzero-constant. Since be a
generalized null-helix,
is constant, and thus
is constant. Using (3.10),
(3.13), and the fact that
is time-like, we have
From (3.11), we have
and thus, we get .
This shows that is a
straight line with nonzero torsion in ,
which is impossible. Therefore, cannot be a
dual Mannheim curve in .

Corollary 3.5. * (1) If a Cartan framed null curve
in
is a null Mannheim curve with time-like Mannheim partner curve
,
the signs of
and
are the same. ** (2) If a Cartan framed null curve
in
is a null Mannheim curve with space-like Mannheim partner curve
,
the signs of
and
are opposite.*

*Proof. *From (3.10) and (3.13),
The proof is complete.

*Remarks. *(a) Theorems hold for null dual Mannheim curve with space-like dual Mannheim
partner curve.

(b) Some results in [4] unfortunately are
not correct. For example, Theorem 3.3 gave necessary and sufficient conditions for null Mannheim
curve, which implies that the null Mannheim curve should be a null-helix from
Proposition 3.4. Moreover,
Propositions in [4] are related with a
null-helix partner curve.

#### References

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