A New Approach on Helices in Pseudo-Riemannian Manifolds

Abstract

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

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-space 𝐸³ 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 Hacisalihoğlu defined harmonic curvature functions 𝐻_𝑖(1≤𝑖≤𝑛−2) of a curve 𝛼 and generalized helices in 𝐸³ to those in 𝑛-dimensional Euclidean space 𝐸^𝑛. Moreover, they gave a characterization for the inclined curves in 𝐸^𝑛: “Acurveisaninclinedcurveifandonlyif^𝑛_𝑖=1𝐻²_𝑖=constant.” (1)

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 3-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 𝐸^𝑛, 𝑛>3, called 𝑉_𝑛-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 𝑉_𝑛-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 tensor 𝑔 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(𝑋_𝑝,𝑌_𝑝)=0 for all 𝑌_𝑝∈𝑇_𝑝(𝑀); then 𝑋_𝑝=0 (nondegenerate), where 𝑇_𝑝(𝑀) is the tangent space of 𝑀 at the point 𝑝 and 𝑔∶𝑇_𝑝(𝑀)×𝑇_𝑝(𝑀)→ℝ [17].

Definition 2. A pseudo-Riemannian manifold (or semi-Riemannian manifold) is a smooth manifold 𝑀 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 order 𝑛 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: ∇_𝑉1𝑉₁=𝑘₁𝑉₂,∇_𝑉1𝑉_𝑖=−𝜀_𝑖−2𝜀_𝑖−1𝑘_𝑖−1𝑉_𝑖−1+𝑘_𝑖𝑉_𝑖+1∇,1<𝑖<𝑛,_𝑉1𝑉_𝑛=−𝜀_𝑛−2𝜀_𝑛−1𝑘_𝑛−1𝑉_𝑛−1, (2) where 𝑘₁=∇_𝑉1𝑉₁𝑘>0,_𝑖=∇_𝑉1𝑉_𝑖+𝜀_𝑖−2𝜀_𝑖−1𝑘_𝑖−1𝑉_𝑖−1𝜀>0,2≤𝑖≤𝑛−1,_𝑗−1=𝑔𝑉_𝑗,𝑉_𝑗(=±1),1≤𝑗≤𝑛,on𝐼, (3) and ∇ is Levi-Civita connection of 𝑀.
One calls such a curve a proper curve of order 𝑛, 𝑘_𝑖(1≤𝑖≤𝑛−1) its 𝑖th curvature, and 𝑉₁,…,𝑉_𝑛 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. 𝑉_𝑛-Slant Helices and Their Harmonic Curvature Functions

In this section, we give definition of a 𝑉_𝑛-slant helix curve in a 𝑛-dimensional pseudo-Riemannian manifold. Furthermore, we give characterizations by using harmonic curvatures for 𝑉_𝑛-slant helices.

Definition 4. Let 𝑀 be a 𝑛-dimensional pseudo-Riemannian manifold and let 𝛼(𝑠) be a proper curve of order 𝑛 (nonnull) with the curvatures   𝑘_𝑖(𝑖=1,…,𝑛−1) in 𝑀. Then, harmonic curvature functions of 𝛼 are defined by 𝐻^∗_𝑖∶𝐼⊂ℝ⟶ℝ (4) along 𝛼 in 𝑀, where 𝐻^∗₀𝐻=0,_∗1=𝜀_𝑛−3𝜀_𝑛−2𝑘_𝑛−1𝑘_𝑛−2,𝐻^∗_𝑖=𝑘_𝑛−𝑖𝐻^∗_𝑖−2−∇_𝑉1𝐻^∗_𝑖−1𝜀_{𝑛−(𝑖+2)}𝜀_{𝑛−(𝑖+1)}𝑘_{𝑛−(𝑖+1)},2≤𝑖≤𝑛−2. (5) Note that ∇_𝑉1𝐻^∗_𝑖−1=𝑉₁(𝐻^∗_𝑖−1)=𝐻^∗′_𝑖−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 𝑉_𝑛-slant helix in 𝑀 if the function 𝑔𝑉_𝑛,𝑋 (6) is a nonzero constant along 𝛼 and 𝑋 is a parallel vector field along 𝛼 in 𝑀; that is, ∇_𝑉1𝑋=0. Here, 𝑉_𝑛 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 𝐻^∗_𝑛−2≠0 for 𝑖=𝑛−2. Then, 𝜀_𝑛−3𝐻₁^∗2+𝜀_𝑛−4𝐻₂^∗2+⋯+𝜀₀𝐻^∗2_𝑛−2 is nonzero constant if and only if 𝑉₁(𝐻^∗_𝑛−2)=𝐻^∗′_𝑛−2=𝑘₁𝐻^∗_𝑛−3, where 𝑉₁ and {𝐻^∗₁,…,𝐻^∗_𝑛−2} are the unit tangent vector field and the harmonic curvatures of 𝛼, respectively.

Proof. First, we assume that 𝜀_𝑛−3𝐻₁^∗2+𝜀_𝑛−4𝐻₂^∗2+⋯+𝜀₀𝐻^∗2_𝑛−2 is nonzero constant. Consider the following functions given in Definition 4: 𝐻^∗_𝑖=𝑘_𝑛−𝑖𝐻^∗_𝑖−2−𝐻^∗′_𝑖−1𝜀_{𝑛−(𝑖+2)}𝜀_{𝑛−(𝑖+1)}𝑘_{𝑛−(𝑖+1)} (7) for 3≤𝑖≤𝑛−2. So, from the equality, we can write 𝑘_{𝑛−(𝑖+1)}𝐻^∗_𝑖=𝜀_{𝑛−(𝑖+2)}𝜀_{𝑛−(𝑖+1)}𝑘_𝑛−𝑖𝐻^∗_𝑖−2−𝐻^∗′_𝑖−1. (8) Hence, in (8), if we take 𝑖+1 instead of 𝑖, we get 𝜀_{𝑛−(𝑖+3)}𝜀_{𝑛−(𝑖+2)}𝐻^∗′_𝑖=𝜀_{𝑛−(𝑖+3)}𝜀_{𝑛−(𝑖+2)}𝑘_{𝑛−(𝑖+1)}𝐻^∗_𝑖−1−𝑘_{𝑛−(𝑖+2)}𝐻^∗_𝑖+1,2≤𝑖≤𝑛−3 (9) together with 𝐻^∗′₁1=−𝜀_𝑛−4𝜀_𝑛−3𝑘_𝑛−3𝐻^∗₂ (10) or 𝐻^∗′₁=−𝜀_𝑛−4𝜀_𝑛−3𝑘_𝑛−3𝐻^∗₂. (11) On the other hand, since 𝜀_𝑛−3𝐻₁^∗2+𝜀_𝑛−4𝐻₂^∗2+⋯+𝜀₀𝐻^∗2_𝑛−2 is constant, we have 𝜀_𝑛−3𝐻^∗₁𝐻^∗′₁+𝜀_𝑛−4𝐻^∗₂𝐻^∗′₂+⋯+𝜀₀𝐻^∗_𝑛−2𝐻^∗′_𝑛−2=0 (12) and so 𝜀₀𝐻^∗_𝑛−2𝐻^∗′_𝑛−2=−𝜀_𝑛−3𝐻^∗₁𝐻^∗′₁−𝜀_𝑛−4𝐻^∗₂𝐻^∗′₂−⋯−𝜀₁𝐻^∗_𝑛−3𝐻^∗′_𝑛−3. (13) By using (9) and (11), we obtain 𝐻^∗₁𝐻^∗′₁=−𝜀_𝑛−4𝜀_𝑛−3𝑘_𝑛−3𝐻^∗₁𝐻^∗₂,𝜀_{𝑛−(𝑖+3)}𝜀_{𝑛−(𝑖+2)}𝐻^∗_𝑖𝐻^∗′_𝑖=𝜀_{𝑛−(𝑖+3)}𝜀_{𝑛−(𝑖+2)}𝑘_{𝑛−(𝑖+1)}𝐻^∗_𝑖−1𝐻^∗_𝑖−𝑘_{𝑛−(𝑖+2)}𝐻^∗_𝑖𝐻^∗_𝑖+1,2≤𝑖≤𝑛−3. (14)(15) Therefore, by using (13), (14), and (15), an algebraic calculus shows that 𝜀₀𝐻^∗_𝑛−2𝐻^∗′_𝑛−2=𝜀₀𝑘₁𝐻^∗_𝑛−3𝐻^∗_𝑛−2 (16) or 𝐻^∗_𝑛−2𝐻^∗′_𝑛−2=𝑘₁𝐻^∗_𝑛−3𝐻^∗_𝑛−2. (17) Since 𝐻^∗_𝑛−2≠0, we get the relation 𝐻^∗′_𝑛−2=𝑘₁𝐻^∗_𝑛−3. (18)
Conversely, we assume that 𝐻^∗′_𝑛−2=𝑘₁𝐻^∗_𝑛−3. (19) By using (19) and 𝐻^∗_𝑛−2≠0, we can write 𝐻^∗_𝑛−2𝐻^∗′_𝑛−2=𝑘₁𝐻^∗_𝑛−2𝐻^∗_𝑛−3. (20) From (15), we have the following equation system: for𝑖=𝑛−3,𝜀₁𝐻^∗_𝑛−3𝐻^∗′_𝑛−3=𝜀₁𝑘₂𝐻^∗_𝑛−4𝐻^∗_𝑛−3−𝜀₀𝑘₁𝐻^∗_𝑛−3𝐻^∗_𝑛−2,for𝑖=𝑛−4,𝜀₂𝐻^∗_𝑛−4𝐻^∗′_𝑛−4=𝜀₂𝑘₃𝐻^∗_𝑛−5𝐻^∗_𝑛−4−𝜀₁𝑘₂𝐻^∗_𝑛−4𝐻^∗_𝑛−3,for𝑖=𝑛−5,𝜀₃𝐻^∗_𝑛−5𝐻^∗′_𝑛−5=𝜀₃𝑘₄𝐻^∗_𝑛−6𝐻^∗_𝑛−5−𝜀₂𝑘₃𝐻^∗_𝑛−5𝐻^∗_𝑛−4,⋮for𝑖=2,𝜀_𝑛−4𝐻^∗₂𝐻^∗′₂=𝜀_𝑛−4𝑘_𝑛−3𝐻^∗₁𝐻^∗₂−𝜀_𝑛−5𝑘_𝑛−4𝐻^∗₂𝐻^∗₃. (21) Moreover, from (14) and (20), we obtain 𝜀_𝑛−3𝐻^∗₁𝐻^∗′₁=−𝜀_𝑛−4𝑘_𝑛−3𝐻^∗₁𝐻^∗₂,𝜀₀𝐻^∗_𝑛−2𝐻^∗′_𝑛−2=𝜀₀𝑘₁𝐻^∗_𝑛−2𝐻^∗_𝑛−3. (22) So, by using the above equation system and considering (22), an algebraic calculus shows that 𝜀_𝑛−3𝐻^∗₁𝐻^∗′₁+𝜀_𝑛−4𝐻^∗₂𝐻^∗′₂+⋯+𝜀₀𝐻^∗_𝑛−2𝐻^∗′_𝑛−2=0. (23) And, by integrating (23), we can easily say that 𝜀_𝑛−3𝐻₁^∗2+𝜀_𝑛−4𝐻₂^∗2+⋯+𝜀₀𝐻^∗2_𝑛−2 (24) 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 𝑉_𝑛-slant helix in 𝑀, then we have 𝑔𝑉_{𝑛−(𝑖+1)},𝑋=𝐻^∗_𝑖𝑔𝑉_𝑛,𝑋,𝑖=0,1,…,𝑛−2, (25) where 𝑋 is the axis of 𝛼. Here, {𝑉₁,𝑉₂,…,𝑉_𝑛} denote the Frenet frame of 𝛼 and {𝐻^∗₁,𝐻^∗₂,…,𝐻^∗_𝑛−2} denote the harmonic curvature functions of the curve 𝛼.

Proof. We will use the induction method. Let 𝑖=1. Since 𝑋 is the axis of the 𝑉_𝑛-slant helix 𝛼, we get 𝑋=𝜆₁𝑉₁+⋯+𝜆_𝑛𝑉_𝑛. (26) From the definition of 𝑉_𝑛-slant helix, we have 𝑔𝑉_𝑛,𝑋=𝜆_𝑛𝜀_𝑛−1=constant. (27) A differentiation in (27) and the Frenet formulas gives us 𝑔𝑉_𝑛−1,𝑋=0. (28) Again, differentiation in (28) and the Frenet formulas give 𝑔∇_𝑉1𝑉_𝑛−1,𝑋=0,−𝜀_𝑛−3𝜀_𝑛−2𝑘_𝑛−2𝑔𝑉_𝑛−2,𝑋+𝑘_𝑛−1𝑔𝑉_𝑛𝑔𝑉,𝑋=0,_𝑛−2,𝑋=𝜀_𝑛−3𝜀_𝑛−2𝑘_𝑛−1𝑘_𝑛−2𝑔𝑉_𝑛,𝑔𝑉,𝑋_𝑛−2,𝑋=𝐻^∗₁𝑔𝑉_𝑛,𝑋, (29) 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 𝑔𝑉_𝑛−𝑖,𝑋=𝐻^∗_𝑖−1𝑔𝑉_𝑛,𝑋. (30) A differentiation in (30) and the Frenet formulas give us that −𝜀_{𝑛−𝑖−2}𝜀_{𝑛−𝑖−1}𝑘_{𝑛−𝑖−1}𝑔𝑉_{𝑛−𝑖−1},𝑋+𝑘_𝑛−𝑖𝑔𝑉_{𝑛−𝑖+1},𝑋=∇_𝑉1𝐻^∗_𝑖−1𝑔𝑉_𝑛,𝑋. (31) Since we have the induction hypothesis, 𝑔(𝑉_{𝑛−𝑖+1},𝑋)=𝐻^∗_𝑖−2𝑔(𝑉_𝑛,𝑋), we get 𝑘_𝑛−𝑖𝐻^∗_𝑖−2−∇_𝑉1𝐻^∗_𝑖−1𝜀_{𝑛−(𝑖+2)}𝜀_{𝑛−(𝑖+1)}𝑘_{𝑛−(𝑖+1)}𝑔𝑉_𝑛𝑉,𝑋=𝑔_{𝑛−(𝑖+1)},𝑋, (32) which gives 𝑔𝑉_{𝑛−(𝑖+1)},𝑋=𝐻^∗_𝑖𝑔𝑉_𝑛,𝑋. (33)

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 𝑉_𝑛-slant helix in 𝑀 if and only if it satisfies that ^𝑛−2_𝑖=1𝜀_{𝑛−(𝑖+2)}𝐻_𝑖^∗2 (34) is equal to nonzero constant and 𝐻^∗_𝑛−2≠0.

Proof. Suppose 𝛼 to be a 𝑉_𝑛-slant helix. According to Definition 5 and the proof of Proposition 7, 𝑔𝑉_𝑛,𝑋=𝜆_𝑛𝜀_𝑛−1=constant, (35) where 𝑋 is the axis of 𝛼. From Proposition 7, we have 𝑔𝑉_{𝑛−(𝑖+1)},𝑋=𝐻^∗_𝑖𝑔𝑉_𝑛,𝑋 (36) for 1≤𝑖≤𝑛−2. Moreover, from (35) and Frenet formulas, we can write −𝜀_𝑛−2𝜀_𝑛−1𝑘_𝑛−1𝑔𝑉_𝑛−1,𝑋=0. (37) Since −𝜀_𝑛−2𝜀_𝑛−1𝑘_𝑛−1 is different from zero, 𝑔(𝑉_𝑛−1,𝑋)=0. It is known that the system {𝑉₁,…,𝑉_𝑛} is a basis of 𝜘(𝑀) (tangent bundle) along 𝛼. Hence, 𝑋 can be expressed in the form 𝑋=^𝑛_𝑖=1𝜆_𝑖𝑉_𝑖. (38) Moreover, from (38), we get the system 𝜀₀𝜆₁=𝑔𝑋,𝑉₁,𝜀₁𝜆₂=𝑔𝑋,𝑉₂,⋮𝜀_𝑛−3𝜆_𝑛−2=𝑔𝑋,𝑉_𝑛−2,𝜀_𝑛−2𝜆_𝑛−1=𝑔𝑋,𝑉_𝑛−1𝜀=0,_𝑛−1𝜆_𝑛=𝑔𝑋,𝑉_𝑛 (39) by using the metric 𝑔. Therefore, from Proposition 7 and the above system, we can see that the following system is true: 𝜆₁=𝑔𝑋,𝑉₁=𝜀₀𝐻^∗_𝑛−2𝑔𝑋,𝑉_𝑛,𝜆₂=𝑔𝑋,𝑉₂=𝜀₁𝐻^∗_𝑛−3𝑔𝑋,𝑉_𝑛,⋮𝜆_𝑛−2=𝑔𝑋,𝑉_𝑛−2=𝜀_𝑛−3𝐻^∗₁𝑔𝑋,𝑉_𝑛,𝜆_𝑛−1=𝑔𝑋,𝑉_𝑛−1𝜆=0,_𝑛=𝜀_𝑛−1𝑔𝑋,𝑉_𝑛. (40) Thus, the axis of the curve 𝛼 can be easily obtained as 𝑋=𝑔𝑋,𝑉_𝑛^𝑛−2_𝑖=1𝐻^∗_𝑖𝑉_{𝑛−(𝑖+1)}𝜀_{𝑛−(𝑖+2)}+𝜀_𝑛−1𝑉_𝑛 (41) by making use of equality (38) and the last system.
Therefore, from (41), we can write 𝑔(𝑋,𝑋)=𝑔𝑋,𝑉_𝑛²𝜀³₀𝐻^∗2_𝑛−2+⋯+𝜀³_𝑛−3𝐻₁^∗2+𝜀³_𝑛−1. (42) Moreover, by the definition of metric tensor, we have 𝑔(𝑋,𝑋)=‖𝑋‖². (43) Since 𝛼 is a 𝑉_𝑛-slant helix, ‖𝑋‖= constant and 𝑔(𝑋,𝑉_𝑛) is nonzero constant along 𝛼. Hence, from (42), we obtain that 𝜀³₀𝐻^∗2_𝑛−2+⋯+𝜀³_𝑛−3𝐻₁^∗2+𝜀³_𝑛−1 (44) is constant. In other words, 𝜀₀𝐻^∗2_𝑛−2+⋯+𝜀_𝑛−3𝐻₁^∗2=^𝑛−2_𝑖=1𝜀_{𝑛−(𝑖+2)}𝐻_𝑖^∗2 (45) is constant.
Now, we will show that 𝐻^∗_𝑛−2≠0. We assume that 𝐻^∗_𝑛−2=0. Then, for 𝑖=𝑛−2 in (36), we have 𝑔𝑉₁,𝑋=𝐻^∗_𝑛−2𝑔𝑋,𝑉_𝑛=0. (46) If we take derivative in each part of (46) in the direction 𝑉₁ on 𝑀, then we have 𝑔∇_𝑉1𝑉₁,𝑋+𝑔𝑉₁,∇_𝑉1𝑋=0. (47) On the other hand, ∇_𝑉1𝑋=0 since 𝛼 is a 𝑉_𝑛-slant helix. Then, from (47), we have 𝑔∇_𝑉1𝑉₁,𝑋=𝑘₁𝑔𝑉₂,𝑋=0 (48) by using the Frenet formulas. Since 𝑘₁ is positive, it must be 𝑔(𝑉₂,𝑋)=0. Now, for 𝑖=𝑛−3 in (36), 𝑔𝑉₂,𝑋=𝐻^∗_𝑛−3𝑔𝑉_𝑛,𝑋. (49) Since 𝑔(𝑉₂,𝑋)=0 and 𝑔(𝑉_𝑛,𝑋)≠0, it must be 𝐻^∗_𝑛−3=0. Continuing this process, we get 𝐻^∗₁=0. Let us recall that 𝐻^∗₁=𝜀_𝑛−3𝜀_𝑛−2(𝑘_𝑛−1/𝑘_𝑛−2); thus we have a contradiction because all the curvatures are nowhere zero. Consequently, 𝐻^∗_𝑛−2≠0.
Conversely, we assume that ∑^𝑛−2_𝑖=1𝜀_{𝑛−(𝑖+2)}𝐻_𝑖^∗2 = constant and 𝐻^∗_𝑛−2≠0. We take the vector field 𝑋=𝜆_𝑛𝑉_𝑛+^𝑛−2_𝑖=1𝜆_𝑛𝜀_𝑛−1𝜀_{𝑛−(𝑖+2)}𝐻^∗_𝑖𝑉_{𝑛−(𝑖+1)} (50) or 𝑋=𝜆_𝑛𝑉_𝑛+𝜆_𝑛𝜀^𝑛_{𝑛−1𝑖=3}𝜀_𝑛−𝑖𝐻^∗_𝑖−2𝑉_{𝑛−(𝑖−1)}, (51) where 𝜆_𝑛 is constant. We will show that it is parallel along 𝛼; that is, ∇_𝑉1𝑋=0. By direct calculation, we have ∇_𝑉1𝑋=∇_𝑉1𝜆_𝑛𝑉_𝑛+𝜆_𝑛𝜀^𝑛_{𝑛−1𝑖=3}𝜀_𝑛−𝑖∇_𝑉1𝐻^∗_𝑖−2𝑉_{𝑛−(𝑖−1)}=𝜆_𝑛∇_𝑉1𝑉_𝑛+𝜆_𝑛𝜀^𝑛_{𝑛−1𝑖=3}𝜀_𝑛−𝑖𝐻^∗′_𝑖−2𝑉_{𝑛−(𝑖−1)}+𝐻^∗_𝑖−2∇_𝑉1𝑉_{𝑛−(𝑖−1)}=𝜆_𝑛𝜀_𝑛−1−𝜀_𝑛−2𝑘_𝑛−1𝑉_𝑛−1+^𝑛−1_𝑖=3𝜀_𝑛−𝑖𝐻^∗′_𝑖−2𝑉_{𝑛−(𝑖−1)}−𝜀_{𝑛−(𝑖+1)}𝑘_𝑛−𝑖𝑉_𝑛−𝑖𝐻^∗_𝑖−2+𝑘_{𝑛−(𝑖−1)}𝑉_{𝑛−(𝑖−2)}𝐻^∗_𝑖−2𝜀_𝑛−𝑖+𝜀₀𝐻^∗′_𝑛−2𝑉₁+𝜀₀𝑘₁𝐻^∗_𝑛−2𝑉₂. (52) Here, in the case 𝑛=3, we omit the term of sum.
On the other hand, by using (9), we can write 𝜀_{𝑛−(𝑖+1)}𝜀_𝑛−𝑖𝐻^∗′_𝑖−2=𝜀_{𝑛−(𝑖+1)}𝜀_𝑛−𝑖𝑘_{𝑛−(𝑖−1)}𝐻^∗_𝑖−3−𝑘_𝑛−𝑖𝐻^∗_𝑖−1 (53) for 4≤𝑖≤𝑛−1 together with (11). Moreover, from Lemma 6, we know that 𝐻^∗′_𝑛−2=𝑘₁𝐻^∗_𝑛−3. (54) Therefore, by using (11), (53), and (54) and by the definition of 𝐻^∗₁, algebraic calculus shows that ∇_𝑉1𝑋=0. Besides, 𝑔(𝑉_𝑛,𝑋)=𝜆_𝑛𝜀_𝑛−1 is constant. Consequently, 𝛼 is a 𝑉_𝑛-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 𝑉_𝑛-slant helix in 𝑀 if and only if 𝐻^∗′_𝑛−2=𝑘₁𝐻^∗_𝑛−3 and 𝐻^∗_𝑛−2≠0, where {𝐻^∗₁,𝐻^∗₂,…,𝐻^∗_𝑛−2} denote the harmonic curvature functions of the curve 𝛼.

Proof. It is obvious by using Lemma 6. and Theorem 8.

4. Conclusion

In this work, 𝑉_𝑛-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.

Acknowledgment

The authors would like to express their sincere gratitude to the referees for valuable suggestions to improve the paper.