# A New Approach on Helices in Pseudo-Riemannian Manifolds

Authors: Evren Zıplar; Yusuf Yaylı; İsmail Gök
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A proper curveα in the n-dimensional pseudo-Riemannian manifold (M,g) is called a Vn-slant helix if the function g(Vn,X) is a nonzero constant along α, where X is a
parallel vector field along α and Vn is nth Frenet frame. In this work, we study such curves
and give important characterizations about them.


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## Body

## 1. Introduction

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” [3].A curve of constant slope or general helix in Euclidean 3-spaceE3 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, κ/τ is constant along the curve, where κ and τ denote the first and second curvatures of the curve, respectively. In [6], Özdamar and Hacisalihog˘lu defined harmonic curvature functions Hi(1≤i≤n-2) of a curve α and generalized helices in E3 to those in n-dimensional Euclidean space En. Moreover, they gave a characterization for the inclined curves in En:
(1)“Acurveisaninclinedcurveifandonlyif∑i=1n‍Hi2=constant.”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 Euclidean3-space E3 [7]. In 2008, Önder et al. defined a new kind of slant helix in Euclidean 4-space E4 which is called B2-slant helix and they gave some characterizations of these slant helices in Euclidean 4-space E4 [8]. And then in 2009 Gök et al. generalized B2-slant helix in E4 to En, n>3, called Vn-slant helix in Euclidean and Minkowski n-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 n-space [11–15]. But, Şenol et al. [16] 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  n-space. Then, they consider the precharacterizations about inclined curves and slant helices and restructure them with the necessary and sufficient condition [16].Similar to the work in [16], in this work, we define Vn-slant helix and give characterizations about the helix with necessary and sufficient condition in n-dimensional pseudo-Riemannian manifolds for the first time.


## 2. Preliminaries

In this section, we give some basic definitions from differential geometry.Definition 1.
A metric tensorg on a smooth manifold M is a symmetric nondegenerate (0,2) tensor field on M.
In other words,g(X,Y)=g(Y,X) for all X,Y∈TM (tangent bundle) and at each point p of M if g(Xp,Yp)=0 for all Yp∈Tp(M); then Xp=0 (nondegenerate), where Tp(M) is the tangent space of M at the point p and g:Tp(M)×Tp(M)→ℝ [17].Definition 2.
A pseudo-Riemannian manifold (or semi-Riemannian manifold) is a smooth manifoldM furnished with a metric tensor g. That is, a pseudo-Riemannian manifold is an ordered pair (M,g) [17].Definition 3.
One will recall the notion of a proper curve of ordern in n -dimensional pseudo-Riemannian manifold M with the metric tensor g. Let α:I→M be a nonnull curve in M parametrized by the arc length s, where I is an open interval of the real line ℝ. One denotes the tangent vector field of α by V1. One assumes that α satisfies the following Frenet formula:
(2)∇V1V1=k1V2,∇V1Vi=-εi-2εi-1ki-1Vi-1+kiVi+1,1<i<n,∇V1Vn=-εn-2εn-1kn-1Vn-1,
where
(3)k1=∥∇V1V1∥>0,ki=∥∇V1Vi+εi-2εi-1ki-1Vi-1∥>0,2≤i≤n-1,εj-1=g(Vj,Vj)(=±1),1≤j≤n,on⁡I,
and ∇ is Levi-Civita connection of M.
One calls such a curve a proper curve of ordern, ki(1≤i≤n-1) its ith curvature, and V1,…,Vn its Frenet Frame field.
Moreover, let one recall that∥X∥=|g(X,X)| for X∈TM, where TM is the tangent bundle of M [18].


## 3.Vn-Slant Helices and Their Harmonic Curvature Functions

In this section, we give definition of aVn-slant helix curve in a n-dimensional pseudo-Riemannian manifold. Furthermore, we give characterizations by using harmonic curvatures for Vn-slant helices.Definition 4.
LetM be a n-dimensional pseudo-Riemannian manifold and let α(s) be a proper curve of order n (nonnull) with the curvatures   ki(i=1,…,n-1) in M. Then, harmonic curvature functions of α are defined by
(4)Hi*:I⊂ℝ⟶ℝ
along α in M, where
(5)H0*=0,H1*=εn-3εn-2kn-1kn-2,Hi*=(kn-iHi-2*-∇V1Hi-1*)εn-(i+2)εn-(i+1)kn-(i+1),2≤i≤n-2.
Note that ∇V1Hi-1*=V1(Hi-1*)=Hi-1*′.Definition 5.
Let(M,g) be a n-dimensional pseudo-Riemannian manifold and let α(s) be a proper curve of order n (nonnull). One calls α as a Vn-slant helix in M if the function
(6)g(Vn,X)
is a nonzero constant along α and X is a parallel vector field along α in M; that is, ∇V1X=0. Here, Vn is nth Frenet frame field and X∈TM. Also, X is called the axis of α.Lemma 6.
Let(M,g) be a n-dimensional pseudo-Riemannian manifold and let α(s) be a proper curve of order n (nonnull). Let one assume that Hn-2*≠0 for i=n-2. Then, εn-3H1*2+εn-4H2*2+⋯+ε0Hn-2*2 is nonzero constant if and only if V1(Hn-2*)=Hn-2*′=k1Hn-3*, where V1 and {H1*,…,Hn-2*} are the unit tangent vector field and the harmonic curvatures of α, respectively.Proof.
First, we assume thatεn-3H1*2+εn-4H2*2+⋯+ε0Hn-2*2 is nonzero constant. Consider the following functions given in Definition 4:
(7)Hi*=(kn-iHi-2*-Hi-1*′)εn-(i+2)εn-(i+1)kn-(i+1)
for 3≤i≤n-2. So, from the equality, we can write
(8)kn-(i+1)Hi*=εn-(i+2)εn-(i+1)(kn-iHi-2*-Hi-1*′).
Hence, in (8), if we take i+1 instead of i, we get
(9)εn-(i+3)εn-(i+2)Hi*′=εn-(i+3)εn-(i+2)kn-(i+1)Hi-1*-kn-(i+2)Hi+1*,2≤i≤n-3
together with
(10)H1*′=-1εn-4εn-3kn-3H2*
or
(11)H1*′=-εn-4εn-3kn-3H2*.
On the other hand, since εn-3H1*2+εn-4H2*2+⋯+ε0Hn-2*2 is constant, we have
(12)εn-3H1*H1*′+εn-4H2*H2*′+⋯+ε0Hn-2*Hn-2*′=0
and so
(13)ε0Hn-2*Hn-2*′=-εn-3H1*H1*′-εn-4H2*H2*′-⋯-ε1Hn-3*Hn-3*′.
By using (9) and (11), we obtain
(14)H1*H1*′=-εn-4εn-3kn-3H1*H2*,(15)εn-(i+3)εn-(i+2)Hi*Hi*′=εn-(i+3)εn-(i+2)kn-(i+1)Hi-1*Hi*-kn-(i+2)Hi*Hi+1*,2≤i≤n-3.
Therefore, by using (13), (14), and (15), an algebraic calculus shows that
(16)ε0Hn-2*Hn-2*′=ε0k1Hn-3*Hn-2*
or
(17)Hn-2*Hn-2*′=k1Hn-3*Hn-2*.
Since Hn-2*≠0, we get the relation
(18)Hn-2*′=k1Hn-3*.
Conversely, we assume that(19)Hn-2*′=k1Hn-3*.
By using (19) and Hn-2*≠0, we can write
(20)Hn-2*Hn-2*′=k1Hn-2*Hn-3*.
From (15), we have the following equation system:
(21)fori=n-3,ε1Hn-3*Hn-3*′=ε1k2Hn-4*Hn-3*0000000000000000000000000-ε0k1Hn-3*Hn-2*,fori=n-4,ε2Hn-4*Hn-4*′=ε2k3Hn-5*Hn-4*0000000000000000000000000-ε1k2Hn-4*Hn-3*,fori=n-5,ε3Hn-5*Hn-5*′=ε3k4Hn-6*Hn-5*0000000000000000000000000-ε2k3Hn-5*Hn-4*,⋮fori=2,εn-4H2*H2*′=εn-4kn-3H1*H2*-εn-5kn-4H2*H3*.
Moreover, from (14) and (20), we obtain
(22)εn-3H1*H1*′=-εn-4kn-3H1*H2*,ε0Hn-2*Hn-2*′=ε0k1Hn-2*Hn-3*.
So, by using the above equation system and considering (22), an algebraic calculus shows that
(23)εn-3H1*H1*′+εn-4H2*H2*′+⋯+ε0Hn-2*Hn-2*′=0.
And, by integrating (23), we can easily say that
(24)εn-3H1*2+εn-4H2*2+⋯+ε0Hn-2*2
is a nonzero constant. This completes the proof.Proposition 7.
Let(M,g) be a n-dimensional pseudo-Riemannian manifold and let α(s) be a proper curve of order n (nonnull). If α is Vn-slant helix in M, then we have
(25)g(Vn-(i+1),X)=Hi*g(Vn,X),i=0,1,…,n-2,
where X is the axis of α. Here, {V1,V2,…,Vn} denote the Frenet frame of α and {H1*,H2*,…,Hn-2*} denote the harmonic curvature functions of the curve α.Proof.
We will use the induction method. Leti=1. Since X is the axis of the Vn-slant helix α, we get
(26)X=λ1V1+⋯+λnVn.
From the definition of Vn-slant helix, we have
(27)g(Vn,X)=λnεn-1=constant.
A differentiation in (27) and the Frenet formulas gives us
(28)g(Vn-1,X)=0.
Again, differentiation in (28) and the Frenet formulas give
(29)g(∇V1Vn-1,X)=0,-εn-3εn-2kn-2g(Vn-2,X)+kn-1g(Vn,X)=0,g(Vn-2,X)=εn-3εn-2kn-1kn-2g(Vn,X),g(Vn-2,X)=H1*g(Vn,X),
respectively. Hence, it is shown that (25) is true for i=1.
We now assume that (25) is true for the first i-1. Then, we have
(30)g(Vn-i,X)=Hi-1*g(Vn,X).
A differentiation in (30) and the Frenet formulas give us that
(31)-εn-i-2εn-i-1kn-i-1g(Vn-i-1,X)+kn-ig(Vn-i+1,X)=∇V1Hi-1*g(Vn,X).
Since we have the induction hypothesis, g(Vn-i+1,X)=Hi-2*g(Vn,X), we get
(32)(kn-iHi-2*-∇V1Hi-1*)εn-(i+2)εn-(i+1)kn-(i+1)g(Vn,X)=g(Vn-(i+1),X),
which gives
(33)g(Vn-(i+1),X)=Hi*g(Vn,X).Theorem 8.
Let(M,g) be a n-dimensional pseudo-Riemannian manifold and let α(s) be a proper curve of order n (nonnull). Then, α is a Vn-slant helix in M if and only if it satisfies that
(34)∑i=1n-2‍εn-(i+2)Hi*2
is equal to nonzero constant and Hn-2*≠0.Proof.
Supposeα to be a Vn-slant helix. According to Definition 5 and the proof of Proposition 7,
(35)g(Vn,X)=λnεn-1=constant,
where X is the axis of α. From Proposition 7, we have
(36)g(Vn-(i+1),X)=Hi*g(Vn,X)
for 1≤i≤n-2. Moreover, from (35) and Frenet formulas, we can write
(37)-εn-2εn-1kn-1g(Vn-1,X)=0.
Since -εn-2εn-1kn-1 is different from zero, g(Vn-1,X)=0. It is known that the system {V1,…,Vn} is a basis of ϰ(M) (tangent bundle) along α. Hence, X can be expressed in the form
(38)X=∑i=1n‍λiVi.
Moreover, from (38), we get the system
(39)ε0λ1=g(X,V1),ε1λ2=g(X,V2),⋮εn-3λn-2=g(X,Vn-2),εn-2λn-1=g(X,Vn-1)=0,εn-1λn=g(X,Vn)
by using the metric g. Therefore, from Proposition 7 and the above system, we can see that the following system is true:
(40)λ1=g(X,V1)=ε0Hn-2*g(X,Vn),λ2=g(X,V2)=ε1Hn-3*g(X,Vn),⋮λn-2=g(X,Vn-2)=εn-3H1*g(X,Vn),λn-1=g(X,Vn-1)=0,λn=εn-1g(X,Vn).
Thus, the axis of the curve α can be easily obtained as
(41)X=g(X,Vn){∑i=1n-2‍Hi*Vn-(i+1)εn-(i+2)+(εn-1Vn)}
by making use of equality (38) and the last system.
Therefore, from (41), we can write
(42)g(X,X)=[g(X,Vn)]2(ε03Hn-2*2+⋯+εn-33H1*2+εn-13).
Moreover, by the definition of metric tensor, we have
(43)|g(X,X)|=∥X∥2.
Since α is a Vn-slant helix, ∥X∥= constant and g(X,Vn) is nonzero constant along α. Hence, from (42), we obtain that
(44)ε03Hn-2*2+⋯+εn-33H1*2+εn-13
is constant. In other words,
(45)ε0Hn-2*2+⋯+εn-3H1*2=∑i=1n-2‍εn-(i+2)Hi*2
is constant.
Now, we will show thatHn-2*≠0. We assume that Hn-2*=0. Then, for i=n-2 in (36), we have
(46)g(V1,X)=Hn-2*g(X,Vn)=0.
If we take derivative in each part of (46) in the direction V1 on M, then we have
(47)g(∇V1V1,X)+g(V1,∇V1X)=0.
On the other hand, ∇V1X=0 since α is a Vn-slant helix. Then, from (47), we have
(48)g(∇V1V1,X)=k1g(V2,X)=0
by using the Frenet formulas. Since k1 is positive, it must be g(V2,X)=0. Now, for i=n-3 in (36),
(49)g(V2,X)=Hn-3*g(Vn,X).
Since g(V2,X)=0 and g(Vn,X)≠0, it must be Hn-3*=0. Continuing this process, we get H1*=0. Let us recall that H1*=εn-3εn-2(kn-1/kn-2); thus we have a contradiction because all the curvatures are nowhere zero. Consequently, Hn-2*≠0.
Conversely, we assume that∑i=1n-2‍εn-(i+2)Hi*2 = constant and Hn-2*≠0. We take the vector field
(50)X=λnVn+∑i=1n-2‍λnεn-1εn-(i+2)Hi*Vn-(i+1)
or
(51)X=λnVn+λnεn-1∑i=3n‍εn-iHi-2*Vn-(i-1),
where λn is constant. We will show that it is parallel along α; that is, ∇V1X=0. By direct calculation, we have
(52)∇V1X=∇V1(λnVn)+λnεn-1∑i=3n‍εn-i∇V1(Hi-2*Vn-(i-1))=λn∇V1Vn+λnεn-1∑i=3n‍εn-i[Hi-2*′Vn-(i-1)+Hi-2*∇V1(Vn-(i-1))]=λnεn-1[∑i=3n-1‍-εn-2kn-1Vn-1+(∑i=3n-1‍εn-iHi-2*′Vn-(i-1)-εn-(i+1)kn-iVn-iHi-2*+kn-(i-1)Vn-(i-2)Hi-2*εn-i∑i=3n-1‍)+ε0Hn-2*′V1+ε0k1Hn-2*V2∑i=3n-1‍].
Here, in the case n=3, we omit the term of sum.
On the other hand, by using (9), we can write
(53)εn-(i+1)εn-iHi-2*′=εn-(i+1)εn-ikn-(i-1)Hi-3*-kn-iHi-1*
for 4≤i≤n-1 together with (11). Moreover, from Lemma 6, we know that
(54)Hn-2*′=k1Hn-3*.
Therefore, by using (11), (53), and (54) and by the definition of H1*, algebraic calculus shows that ∇V1X=0. Besides, g(Vn,X)=λnεn-1 is constant. Consequently, α is a Vn-slant helix in M.Corollary 9.
Let(M,g) be a n-dimensional pseudo-Riemannian manifold and let α(s) be a proper curve of order n (non-null). Then, α is a Vn-slant helix in M if and only if Hn-2*′=k1Hn-3* and Hn-2*≠0, where {H1*,H2*,…,Hn-2*} denote the harmonic curvature functions of the curve α.Proof.
It is obvious by using Lemma6. and Theorem 8.


## 4. Conclusion

In this work,Vn-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.



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