No Null-Helix Mannheim Curves in the Minkowski Space
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 .
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 .
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 . 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 . Another example is null Mannheim curves from ztekin and Ergt . 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 .
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 (). 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 :
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 .
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
some function . By taking
the derivative of (3.1) with
and applying the Frenet formulae, we have
coincides with ,
which means that
is a nonzero constant. Thus, we have
is null and from (3.4), we
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
is a null Mannheim curve with time-like Mannheim partner curve
the signs of
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.
Remarks. (a) Theorems hold for null dual Mannheim curve with space-like dual Mannheim
(b) Some results in  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  are related with a null-helix partner curve.
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