A New Approach on Helices in Pseudo-Riemannian Manifolds
A proper curve in the -dimensional pseudo-Riemannian manifold is called a -slant helix if the function is a nonzero constant along , where is a parallel vector field along and is th Frenet frame. In this work, we study such curves and give important characterizations about them.
Curves theory is an essential structure in the differential geometry works. Helix is one of the most spectacular curves because of its helical structure in nature and science. Helices are used in the field of imitation of kinematic motion or the shape of DNA and carbon nanotubes. Moreover, the helical structure can be seen in fractal geometry, especially in hyperhelices [1, 2].
Furthermore, “helices share common origins in the geometries of the platonic solids, with inherent hierarchical potential that is typical of biological structures. The helices provide an energy-efficient solution to close-packing in molecular biology, a common motif in protein construction, and a readily observable pattern at many size levels throughout the body. The helices are described in a variety of anatomical structures, suggesting their importance to structural biology and manual therapy” .
A curve of constant slope or general helix in Euclidean 3-space is defined by the property that its tangent vector field makes a constant angle with a fixed straight line (the axis of general helix). A classical result stated by Lancret in 1802 and first proved by de Saint Venant in 1845 [4, 5] as follows: A necessary and sufficient condition for a curve to be a general helix is that the ratio of the first curvature to the second curvature should be constant; that is, is constant along the curve, where and denote the first and second curvatures of the curve, respectively. In , Özdamar and Hacisaliholu defined harmonic curvature functions of a curve and generalized helices in to those in -dimensional Euclidean space . Moreover, they gave a characterization for the inclined curves in :
Harmonic curvature functions have important role in characterizations of general helices in higher dimensions, because the notion of a general helix can be generalized to higher dimension in different ways. However, these ways are not easy to show which curves are general helices and finding the axis of a general helix is complicated in higher dimension. Thanks to harmonic curvature functions, we can easily obtain the axis of such curves. Moreover, this way is confirmed in 3-dimensional spaces.
Izumiya and Takeuchi defined a new kind of helix (slant helix) and they gave a characterization of slant helices in Euclidean -space . In 2008, Önder et al. defined a new kind of slant helix in Euclidean -space which is called -slant helix and they gave some characterizations of these slant helices in Euclidean -space . And then in 2009 Gök et al. generalized -slant helix in to , , called -slant helix in Euclidean and Minkowski -space [9, 10]. Lots of authors in their papers have investigated inclined curves and slant helices using the harmonic curvature functions in Euclidean and Minkowski -space [11–15]. But, Şenol et al.  see for the first time that the characterization of inclined curves and slant helices in (1) is true only for the case necessity but not true for the case sufficiency in Euclidean -space. Then, they consider the precharacterizations about inclined curves and slant helices and restructure them with the necessary and sufficient condition .
Similar to the work in , in this work, we define -slant helix and give characterizations about the helix with necessary and sufficient condition in -dimensional pseudo-Riemannian manifolds for the first time.
In this section, we give some basic definitions from differential geometry.
Definition 1. A metric tensor on a smooth manifold is a symmetric nondegenerate tensor field on .
In other words, for all (tangent bundle) and at each point of if for all ; then (nondegenerate), where is the tangent space of at the point and .
Definition 2. A pseudo-Riemannian manifold (or semi-Riemannian manifold) is a smooth manifold furnished with a metric tensor . That is, a pseudo-Riemannian manifold is an ordered pair .
Definition 3. One will recall the notion of a proper curve of order in -dimensional pseudo-Riemannian manifold with the metric tensor . Let be a nonnull curve in parametrized by the arc length , where is an open interval of the real line . One denotes the tangent vector field of by . One assumes that satisfies the following Frenet formula:
and is Levi-Civita connection of .
One calls such a curve a proper curve of order , its th curvature, and its Frenet Frame field.
Moreover, let one recall that for , where is the tangent bundle of .
3. -Slant Helices and Their Harmonic Curvature Functions
In this section, we give definition of a -slant helix curve in a -dimensional pseudo-Riemannian manifold. Furthermore, we give characterizations by using harmonic curvatures for -slant helices.
Definition 4. Let be a -dimensional pseudo-Riemannian manifold and let be a proper curve of order (nonnull) with the curvatures in . Then, harmonic curvature functions of are defined by along in , where Note that .
Definition 5. Let be a -dimensional pseudo-Riemannian manifold and let be a proper curve of order (nonnull). One calls as a -slant helix in if the function is a nonzero constant along and is a parallel vector field along in ; that is, . Here, is th Frenet frame field and . Also, is called the axis of .
Lemma 6. Let be a -dimensional pseudo-Riemannian manifold and let be a proper curve of order (nonnull). Let one assume that for . Then, is nonzero constant if and only if , where and are the unit tangent vector field and the harmonic curvatures of , respectively.
Proof. First, we assume that is nonzero constant. Consider the following functions given in Definition 4:
for . So, from the equality, we can write
Hence, in (8), if we take instead of , we get
On the other hand, since is constant, we have
By using (9) and (11), we obtain
Therefore, by using (13), (14), and (15), an algebraic calculus shows that
Since , we get the relation
Conversely, we assume that By using (19) and , we can write From (15), we have the following equation system: Moreover, from (14) and (20), we obtain So, by using the above equation system and considering (22), an algebraic calculus shows that And, by integrating (23), we can easily say that is a nonzero constant. This completes the proof.
Proposition 7. Let be a -dimensional pseudo-Riemannian manifold and let be a proper curve of order (nonnull). If is -slant helix in , then we have where is the axis of . Here, denote the Frenet frame of and denote the harmonic curvature functions of the curve .
Proof. We will use the induction method. Let . Since is the axis of the -slant helix , we get
From the definition of -slant helix, we have
A differentiation in (27) and the Frenet formulas gives us
Again, differentiation in (28) and the Frenet formulas give
respectively. Hence, it is shown that (25) is true for .
We now assume that (25) is true for the first . Then, we have A differentiation in (30) and the Frenet formulas give us that Since we have the induction hypothesis, , we get which gives
Theorem 8. Let be a -dimensional pseudo-Riemannian manifold and let be a proper curve of order (nonnull). Then, is a -slant helix in if and only if it satisfies that is equal to nonzero constant and .
Proof. Suppose to be a -slant helix. According to Definition 5 and the proof of Proposition 7,
where is the axis of . From Proposition 7, we have
for . Moreover, from (35) and Frenet formulas, we can write
Since is different from zero, . It is known that the system is a basis of (tangent bundle) along . Hence, can be expressed in the form
Moreover, from (38), we get the system
by using the metric . Therefore, from Proposition 7 and the above system, we can see that the following system is true:
Thus, the axis of the curve can be easily obtained as
by making use of equality (38) and the last system.
Therefore, from (41), we can write Moreover, by the definition of metric tensor, we have Since is a -slant helix, constant and is nonzero constant along . Hence, from (42), we obtain that is constant. In other words, is constant.
Now, we will show that . We assume that . Then, for in (36), we have If we take derivative in each part of (46) in the direction on , then we have On the other hand, since is a -slant helix. Then, from (47), we have by using the Frenet formulas. Since is positive, it must be . Now, for in (36), Since and , it must be . Continuing this process, we get . Let us recall that ; thus we have a contradiction because all the curvatures are nowhere zero. Consequently, .
Conversely, we assume that = constant and . We take the vector field or where is constant. We will show that it is parallel along ; that is, . By direct calculation, we have Here, in the case , we omit the term of sum.
On the other hand, by using (9), we can write for together with (11). Moreover, from Lemma 6, we know that Therefore, by using (11), (53), and (54) and by the definition of , algebraic calculus shows that . Besides, is constant. Consequently, is a -slant helix in .
Corollary 9. Let be a -dimensional pseudo-Riemannian manifold and let be a proper curve of order (non-null). Then, is a -slant helix in if and only if and , where denote the harmonic curvature functions of the curve .
In this work, -slant helix is defined and new characterizations are given about the helix by using the harmonic curvature functions in pseudo-Riemannian manifolds. Links with other disciplines of the curve can be examined since the curve is important for other branches of science such as molecular biology and mathematical physics. On the other hand, the curve can be analyzed in different spaces of mathematics.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to express their sincere gratitude to the referees for valuable suggestions to improve the paper.
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