On Slant Magnetic Curves in S-manifolds

Güvenç Şaban; Cihan Özgür

doi:10.1080/14029251.2019.1640463

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Volume 26, Issue 4, July 2019, Pages 536 - 554

On Slant Magnetic Curves in S-manifolds

Authors

Güvenç Şaban

Department of Mathematics, Balikesir University, Balikesir, 10145, Turkey,sguvenc@balikesir.edu.tr

Cihan Özgür

Department of Mathematics, Balikesir University, Balikesir, 10145, Turkey,cozgur@balikesir.edu.tr

Received 1 September 2018, Accepted 22 April 2019, Available Online 9 July 2019.

DOI: 10.1080/14029251.2019.1640463 How to use a DOI?
Keywords: Magnetic curve; slant curve; S-manifold
Abstract: We consider slant normal magnetic curves in (2n + 1)-dimensional S-manifolds. We prove that γ is a slant normal magnetic curve in an S-manifold (M^2m+s, φ, ξ_α, η^α, g) if and only if it belongs to a list of slant φ-curves satisfying some special curvature equations. This list consists of some specific geodesics, slant circles, Legendre and slant helices of order 3. We construct slant normal magnetic curves in ℝ^2n+s(−3s) and give the parametric equations of these curves.
Copyright: © 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

Let (M, g) be a Riemannian manifold, F a closed 2-form and let us denote the Lorentz force on M by Φ, which is a (1, 1)-type tensor field. If F is associated by the relation

g(ΦX,Y)=F(X,Y), ∀X,Y∈χ(M),(1.1)

then it is called a magnetic field ([1], [4] and [9]). Let ∇ be the Riemannian connection associated to the Riemannian metric g and γ : I → M a smooth curve. If γ satisfies the Lorentz equation

∇γ′(t)γ′(t)=Φ(γ′(t)),(1.2)

then it is called a magnetic curve or a trajectory for the magnetic field F. The Lorentz equation is a generalization of the equation for geodesics. A curve which satisfies the Lorentz equation is called magnetic trajectory. Magnetic trajectories have constant speed. If the speed of the magnetic curve γ is equal to 1, then it is called a normal magnetic curve [10]. For extensive information about almost contact metric manifolds and Sasakian manifolds, we refer to Blair’s book [2].

In [1], Adachi studied curvature bound and trajectories for magnetic fields on a Hadamard surface. He showed that every normal trajectory is unbounded in both directions in a 2-dimensional complete simply connected Riemannian manifold satisfying some special curvature conditions. In [5], Baikoussis and Blair considered Legendre curves in contact 3-manifolds and they proved that the torsion of a Legendre curve in a 3-dimensional Sasakian manifold is equal to 1. Moreover, in [8], Cho, Inoguchi and Lee proved that a non-geodesic curve in a Sasakian 3-manifold is a slant curve if and only if the ratio of (τ ± 1) and κ is constant, where τ is the geodesic torsion and κ is the geodesic curvature. Cabrerizo, Fernandez and Gomez gave a nice geometric construction of an almost contact metric structure compatible with an assigned metric on a 3-dimensional oriented Riemannian manifold in [6]. In the paper [10], Druţă-Romaniuc, Inoguchi, Munteanu and Nistor studied the magnetic trajectories of the contact magnetic field F_q = qΩ on a Sasakian (2n + 1)-manifold (M²ⁿ⁺¹, φ, ξ, η, g), where Ω is the fundamental 2-form. The main objective of [11] is the study of trajectories for particles moving under the inӾuence of a contact magnetic curve in a cosymplectic manifold. The paper [14] is concerned with closed magnetic trajectories on 3-dimensional Berger spheres. In [15], the authors studied magnetic trajectories in an almost contact metric manifold. They proved that normal magnetic curves are helices of maximum order 5. Moreover, in [16], Jleli and Munteanu worked in the context of a para-Kaehler manifold, showing that spacelike and timelike normal magnetic curves corresponding to the para-Kaehler 2-forms are circles. In [17], the authors gave a complete classification of Killing magnetic curves with unit speed. Furthermore, in [18], the same authors proved that a normal magnetic curve on the Sasakian sphere S²ⁿ⁺¹ lies on a totally geodesic sphere S³. They also considered two particular magnetic fields on three-dimensional torus obtained from two different contact forms on the Euclidean space E³ and studied their closed normal magnetic trajectories in their recent paper [19]. In [23], the authors investigated some special curves in 3-dimensional semi-Riemannian manifolds, such as T-magnetic curves, N-magnetic curves and B-magnetic curves, that are defined by means of their Frenet elements. Calvaruso, Munteanu and Perrone provided a complete classification of the magnetic trajectories of a Killing characteristic vector field on an arbitrary normal paracontact metric manifold of dimension 3 in [7]. The present authors considered biharmonic Legendre curves of S-space forms in [21]. The second author studied magnetic curves in the 3-dimensional Heisenberg group in [22]. In [20], Nakagawa introduced the notion of framed f-structures, which is a generalization of almost contact structures. On the other hand, Hasegawa, Okuyama and Abe defined a pth Sasakian manifold and gave some typical examples in [13].

Motivated by the above studies, in the present paper, we consider slant normal magnetic curves in (2n + s)-dimensional S-manifolds. S-manifolds are interesting generalizations of Sasakian manifolds, which have a tensor field of type (1, 1) called a φ-structure, satisfying φ³ + φ = 0. So, in Section 2, we give the definitions and brief information on S-manifolds and magnetic curves. In Section 3, we prove that γ is a slant normal magnetic curve in an S-manifold (M^2m+s, φ, ξ_α, η^α, g) if and only if it belongs to a list of slant φ-curves. This list consists of some specific geodesics, slant circles, Legendre and slant helices of order 3. Finally, in Section 4, after the definition of ℝ^2n+s(−3s) and its structures, we construct slant normal magnetic curves in ℝ^2n+s(−3s) and give the parametric equations of these curves in two cases.

2. Preliminaries

In this section, we give brief information on S-manifolds and magnetic curves. Let (M^2n+s, g) be a differentiable manifold, φ a (1, 1)-type tensor field, η^α 1-forms, ξ_α vector fields for α = 1,...,s, satisfying

ϕ2=−I+∑α=1sηα⊗ξα,(2.1)

ηα(ξβ)=δβα, ϕξα=0, ηα(ϕX)=0, ηα(X)=g(X,ξα),g(ϕX,ϕY)=g(X,Y)−∑α=1sηα(X)ηα(Y),dηα(X,Y)=−dηα(Y,X)=g(X,ϕY),(2.2)

where X, Y ∈ TM. Then (φ, ξ_α, η^α, g) is called framed φ-structure and (M^2n+s, φ, ξ_α, η^α, g) is called framed φ-manifold [20]. (M^2n+s, φ, ξ_α, η^α, g) is also called framed metric manifold [25] or almost r-contact metric manifold [24]. If the Nijenhuis tensor of φ is equal to −2dη^α ⊗ ξ_α, then (M^2n+s, φ, ξ_α, η^α, g) is called an S-manifold [3]. For s = 1, an S-structure becomes a Sasakian structure. For an S-structure, the following properties are satisfied [3]:

(∇Xϕ)Y=∑α=1s{g(ϕX,ϕY)ξα+ηα(Y)ϕ2X},(2.3)

∇ξα=−ϕ, α∈{1,…,s}.(2.4)

Let M^2n+s = (M^2n+s, φ, ξ_α, η^α, g) be an S-manifold and Ω the fundamental 2-form of M^2n+s defined by

Ω(X,Y)=g(X,ϕY),(2.5)

(see [20] and [24]). From the definition of framed φ-structure, we have Ω = dη^α. Hence, the fundamental 2-form Ω on M^2n+s is closed. The magnetic field F_q on M^2n+s can be defined by

Fq(X,Y)=qΩ(X,Y),

where X and Y are vector fields on M^2n+s and q is a real constant. F_q is called the contact magnetic field with strength q [15]. If q = 0 then the magnetic curves are geodesics of M^2n+s. Because of this reason we shall consider q ≠ 0 (see [6] and [10]).

From (1.1) and (2.5), the Lorentz force Φ associated to the contact magnetic field F_q can be written as

Φq=−qϕ.

So the Lorentz equation (1.2) can be written as

∇TT=−qϕT,(2.6)

where γ : I ⊆ R → M^2n+s is a smooth unit-speed curve and T = γ′ (see [10] and [15]).

3. Slant magnetic curves in S-manifolds

Let (Mⁿ, g) be a Riemannian manifold. A unit-speed curve γ : I → M is said to be a Frenet curve of osculating order r, if there exists positive functions κ₁,...,κ_r−1 on I satisfying

T=v1=γ′,∇TT=k1v2,∇Tv2=−k1T+k2v3,⋯∇Tvr=−kr−1vr−1,(3.1)

where 1 ≤ r ≤ n and T, v₂,...,v_r are a g-orthonormal vector fields along the curve. The positive functions κ₁,...,κ_r−1 are called curvature functions and {T, v₂,...,v_r} is called the Frenet frame field. A geodesic is a Frenet curve of osculating order r = 1. A circle is a Frenet curve of osculating order r = 2 with a constant curvature function κ₁. A helix of order r is a Frenet curve of osculating order r with constant curvature functions κ₁,...,κ_r−1. A helix of order 3 is simply called a helix.

Let (M^2m+s, φ, ξ_α, η^α, g) be an S-manifold. For a unit-speed curve γ : I → M, if

ηα(T)=0,

for all α = 1,...,s, then γ is called a Legendre curve of M [21]. More generally, if there exists a constant angle θ such that

ηα(T)=cosθ,

for all α = 1,...,s, then γ is called a slant curve and θ is called the contact angle of γ, where |cosθ|≤1/s [12].

Let (M^2m+s, φ, ξ_α, η^α, g) be an S-manifold. A Frenet curve of osculating order r ≥ 3 is called a φ-curve in M if its Frenet vector fields T, v₂,...,v_r span a φ-invariant space. A φ-curve of osculating order r with constant curvature functions κ₁,...,κ_r−1 is called a φ-helix of order r. A curve of osculating order 2 is called a φ-curve if

sp{T,v2,∑α=1sξα}

is a φ-invariant space.

Throughout the paper, when we state “slant magnetic curve”, we mean “slant curves which satisfy equation (2.6)”. For magnetic curves, η^α(T) = cosθ_α does not have to be equal for all α = 1,...,s. By taking the curve as slant, we only study the equality case of the slant angles θ_α in the present paper. The complete classification of magnetic curves in S-manifolds is still an open problem.

Firstly, we state the following theorem:

Theorem 3.1.

Let (M^2m+s, φ, ξ_α, η^α, g) be an S-manifold and consider the contact magnetic field F_q for q ≠ 0. Then γ is a slant normal magnetic curve associated to F_q in M^2m+s if and only if γ belongs to the following list:

a)
geodesics obtained as integral curves of (±1s∑α=1sξα);
b)
non-geodesic slant circles with the curvature κ1=q2−s, having the contact angle θ=arccos(1q) and the Frenet frame field
{T,−sgn(q)ϕT1−scos2θ},
where |q|>s;
c)
Legendre helices with curvatures κ₁ = |q| and κ2=s, having the Frenet frame field
{T,−sgn(q)ϕT,−sgn(q)s∑α=1sξα};
i.e., a class of 1-dimensional integral submanifolds of the contact distribution;
d)
slant helices with curvatures κ1=|q|1−scos2θ and κ2=s|1−qcosθ|, having the Frenet frame field
{T,−sgn(q)ϕT1−scos2θ,−εsgn(q)s1−scos2θ(−scosθT+∑α=1sξα)},
where θ≠π2 is the contact angle satisfying |cosθ|<1s and ε = sgn(1 − qcosθ).

Proof.

Let γ be a normal magnetic curve. If the magnetic curve is a geodesic, then

∇TT=−qϕT,

gives us

T∈sp{ξ1,…,ξs}.

If γ is slant, then we can write

T=cosθ∑α=1sξα.

Since γ is unit speed, we have cosθ=±1s. So the proof of a) is complete.

From now on, we suppose that γ is a non-geodesic Frenet curve of osculating order r > 1. Let us choose an α ∈ {1,...,s}. Applying ξ_α to ∇_TT = −qφT, we obtain

0=g(−qϕT,ξα)=g(∇TT,ξα)=ddtg(T,ξα)−g(T,∇Tξα).(3.2)

From (2.4), we also have

∇Tξα=−ϕT.(3.3)

Using equations (3.2) and (3.3), we find

ddtg(T,ξα)=0,

that is,

ηα(T)=cosθα=constant.

Let us assume θ_α = θ for all α = 1,...,s, i.e., γ is slant. So, we have

ηα(T)=cosθ.(3.4)

Equations (2.6) and (3.1) give us

∇TT=κ1v2=−qϕT.(3.5)

Then we get

κ1=|q|‖ϕT‖=|q|1−scos2θ.(3.6)

If we write (3.6) in (3.5), we find

−qϕT=κ1v2=|q|1−scos2θv2,

which gives us

ϕT=−|q|q1−scos2θv2=−sgn(q)1−scos2θv2.(3.7)

If κ₂ = 0, then the magnetic curve is a Frenet curve of osculating order r = 2. Since κ₁ is a constant, γ is a circle. From (3.7), we have

ηα(ϕT)=0=−sgn(q)1−scos2θηα(v2),

that is,

ηα(v2)=0.

If we differentiate the last equation along the curve γ, we obtain

∇Tηα(v2)=0=g(∇Tv2,ξα)+g(v2,∇Tξα).

So, we calculate

g(−κ1T,ξα)+g(v2,sgn(q)1−scos2θv2)=0.

Since r = 2, we find

−κ1cosθ+sgn(q)1−scos2θ=0.

Using equation (3.6) in the last equation, it is easy to see that

|q|1−scos2θ(−cosθ+1q)=0.

Since γ is non-geodesic, we have

cosθ=1q.

Then equation (3.6) becomes

κ1=|q|1−scos2θ=q2−s,

where |q|>s. So the proof of b) is complete.

Let κ₂ ≠ 0. From (2.1) and (3.4), we find

ϕ2T=−T+cosθ∑α=1sξα.(3.8)

Using (2.3) and (3.4), we have

(∇Tϕ)T=−scosθT+∑α=1sξα,(3.9)

which gives us

∇TϕT=(∇Tϕ)T+ϕ∇TT=−scosθT+∑α=1sξα+ϕ(−qϕT)=−scosθT+∑α=1sξα−q(−T+cosθ∑α=1sξα).(3.10)

Differentiating (3.7), we also find

∇TϕT=−sqn(q)1−scos2θ(−κ1T+κ2v3).(3.11)

By the use of (3.6), (3.10) and (3.11), after some calculations, we obtain

(1−qcosθ)(−scosθT+∑α=1sξα)=−sgn(q)1−scos2θκ2v3.(3.12)

If we find the norm of both sides in (3.12), we get

κ2=s|1−qcosθ|.(3.13)

Let us denote ε = sgn(1 − qcosθ). If we write (3.13) in (3.12), we obtain

∑α=1sξα=scosθT−εsgn(q)s1−scos2θv3.(3.14)

Applying φ to (3.14), we find

ϕv3=−εscosθv2.

If we apply φ to (3.7) and then use equations (3.4) and (3.14) together, we have

ϕv2=sgn(q)1−scos2θT+εcosθsv3.(3.15)

Let us choose a β ∈ {1,...,s}. From (3.15), we calculate

ηβ(v3)=−εsgn(q)1−scos2θs.

If we differentiate (3.14) along the curve γ, we get

∑α=1s∇Tξα=scosθ∇TT−εsgn(q)s1−scos2θ∇Tv3,

which gives us

−s(1−cosθ)ϕT=−εsgn(q)s1−scos2θ(−κ2v2+κ3v4).

Since φT ‖ v₂, we find κ₃ = 0. This proves d) of the theorem.

Let us examine Legendre case separately, that is, θ=π2. Then we have ε = 1, κ₁ = |q|, κ2=s, κ₃ = 0 and equation (3.14) gives us

v3=−sgn(q)s∑α=1sξα.

This completes the proof of c).

Conversely, let γ satisfy one of a), b), c) or d). Using the Frenet frame field and Frenet equations, it is straightforward to show that ∇_T T = −qφT, i.e., γ is a slant normal magnetic curve.

The above theorem is a generalization of Theorem 3.1 of [10] (by Simona Luiza Druta-Romaniuc et al.) for S-manifolds. If we choose s = 1, since an S-manifold becomes a Sasakian manifold, we find their results.

Remark 3.1.

The order of a slant magnetic curve in an S-manifold is still r ≤ 3, as in the case of a magnetic curve of a Sasakian manifold, which was considered in [10].

Now, let us remove the slant condition from the hypothesis and show that the osculating order is still r ≤ 3.

Theorem 3.2.

Let (M^2m+s, φ, ξ_α, η^α, g) be an S-manifold and consider the contact magnetic field F_q for q ≠ 0. If γ is a normal magnetic curve associated to F_q in M^2m+s, then the osculating order r ≤ 3.

Proof.

Let γ be a normal magnetic curve. Then, the Lorentz equation (2.6) gives us

ηα(T)=cosθα,α=1,…,s.

If we differentiate this equation along the curve, we have

ηα(E2)=0

for all α = 1,...,s. From the Frenet equations (3.1), we obtain

−qϕT=κ1v2.

From the definition of framed φ-structure, we calculate

g(ϕT,ϕT)=1−A,

where we denote

A=∑α=1scos2θα.

Then, we have

‖ϕT‖=1−A

and

κ1=|q|1−A.

Thus, φT can be rewritten as

ϕT=−sgn(q)1−Av2.(3.16)

Again, from the definition of framed φ-structure, we have

ϕ2T=−T+V,

where we denote

V=∑α=1scosθαξα.

After some calculations, we get

∇TϕT=(q−B)T+(1−A)∑α=1sξα+(−q+B)V,

which corresponds to equation (3.10). Here, we denote

B=∑α=1scosθα.

From equation (3.16), we also find

∇TϕT=−sqn(q)1−A(−κ1T+κ2v3),

which corresponds to equation (3.11). In this last equation, we can replace κ1=|q|1−A. Finally, we have

−sgn(q)1−Aκ2v3=(1−A)∑α=1sξα+(−q+B)V+(qA−B)T.(3.17)

So, if we denote the norm of the right hand side of equation (3.17) by C, we find

C=(1−A)(Aq2−As+B2−2Bq+s),

which is a constant. Hence, we obtain

κ2=C1−A=Aq2−As+B2−2Bq+s=constant.

From equation (3.17), we also have v₃ ∈ span{T, ξ₁,...,ξ_s}. The angles between v₃ and T, ξ₁,..., ξ_s are all constants since all the coefficients in equation (3.17) are constants. Then, we can write

v3=c0T+c1ξ1+⋯+csξs(3.18)

for some constants c₀,...,c_s. If we differentiate equation (3.18), we get

−κ2v2+κ3v4=c0κ1v2−c1ϕT−⋯−csϕT.

Since φT is parallel to v₂, if we take the inner product of the last equation with v₄, we find κ₃ = 0. This proves the theorem.

In particular, if γ is slant, i.e. θ_α = θ for all α = 1,...,s, then we obtain the following corollary:

Corollary 3.1.

If θ_α = θ, for all α = 1,...,s, then

A=scos2θ, B=scosθ, V=cosθ∑α=1sξα,C=(1−scos2θ)s(1−qcosθ)2

and κ2=s|1−qcosθ|.

Now, let us state the following proposition:

Proposition 3.1.

Let γ be a slant φ-helix of order 3 in an S-manifold (M^2m+s, φ, ξ_α, η^α, g) with contact angle θ. Then

∑α=1sξα=scosθT+ρv3,(3.19)

where ρ=g(v3,∑α=1sξα)=sηα(v3) is a real constant such that ρ² = s − s² cos² θ. Hence, γ has the Frenet frame field

{T,±ϕT1−scos2θ,±1s1−scos2θ(−scosθT+∑α=1sξα)}.

Proof.

From the assumption, the Frenet frame field {T, v₂, v₃} is φ-invariant and

ηα(T)=cosθ.

Differentiating the last equation along the curve, it is easy to see that

ηα(v2)=0.(3.20)

If we differentiate once again, we have

g(ϕT,v2)=−κ1cosθ+κ2ηα(v3),(3.21)

which means the value of η^α(v₃) does not depend on α.

Firstly, let us assume that θ≠π2. Since the space spanned by the Frenet frame field is φ-invariant, then φ²T is in the set. Using (3.8) and (3.20), we can write

∑α=1sξα∈sp{T,v3},

that is,

∑α=1sξα=scosθT+ρv3.(3.22)

If we take the norm of both sides, we find ρ² = s − s² cos² θ. Since the value of η^α(v₃) does not depend on α, we obtain

ρ=g(v3,∑α=1sξα)=sηα(v3).

If we apply φT to (3.22), we get g(φT, v₃) = 0. Since φT ⊥ T, φT ⊥ v₃ and sp{T, v₂, v₃} is φ-invariant, we have φT ‖ v₂. As a result, we find

v2=±ϕT‖ϕT‖=±ϕT1−scos2θ.

Now let us consider the Legendre case, i.e., θ=π2. From (3.21), we find

∇Tg(ϕT, v2)=−κ2g(ϕT,v3).(3.23)

Using (2.1) and (2.3), we calculate

∇TϕT=(∇Tϕ)T+ϕ∇TT=∑α=1sξα+κ1ϕv2.(3.24)

Using the last equation, we obtain

∇Tg(ϕT,v2)=g(∇TϕT,v2)+g(ϕT,∇Tv2)=κ2g(ϕT,v3).(3.25)

Equations (3.23) and (3.25) give us g(φT, v₃) = 0, that is, φT ‖ v₂. Thus, we have φT = ±v₂. Consequently, the Frenet frame field becomes {T, ±φT, v₃}. Now, we must show that v₃ is parallel to ∑α=1sξα. Since the space spanned by the Frenet frame field is φ-invariant, from orthonormal expansion, we can write

ϕv3=g(ϕv3,T)T±g(ϕv3,±ϕT)ϕT+g(ϕv3,v3)v3,

which reduces to

ϕv3=g(ϕv3,T)T.(3.26)

If we apply φ to equation (3.26) and use (2.1), we find

−v3+∑α=1sηα(v3)ξα=g(ϕv3,T)ϕT.(3.27)

Applying φT to (3.27) and using the Frenet frame field, we have g(φv₃, T) = 0. As a result, we get φv₃ = 0 and equation (3.27) becomes

−v3+∑α=1sηα(v3)ξα=0.

We have already shown that the value of η^α(v₃) does not depend on α; so, we can write

v3=ηα(v3)∑α=1sξα.(3.28)

Since v₃ and ξ_α are unit for all α = 1,...,s, we find ηα(v3)=±1s. Finally, for θ=π2, we have ρ² = s and ∑α=1sξα=ρv3, which completes the proof.

Corollary 3.2.

Let γ be a Legendre φ-helix of order 3 in an S-manifold (M^2m+s, φ, ξ_α, η^α, g). Then κ2=s, v₂ = ±φT and v3=±1s∑α=1sξα.

Proof.

From equation (3.28), we already have

v3=±1s∑α=1sξα.

If we differentiate this equation and use (3.1), we obtain

−κ2v2=±1s∑α=1s∇Tξα.(3.29)

Using equations (2.4) and (3.29), we find that κ2=s and v₂ = ±φT.

Finally, we can give the following theorem:

Theorem 3.3.

Let γ be a slant φ-helix of order r ≤ 3 on an S-manifold (M^2m+s, φ, ξ_α, η^α, g). Let θ denote the contact angle of γ. Then we have

i.
If cosθ=±1s, then γ is an integral curve of ±1s∑α=1sξα, hence it is a normal magnetic curve for F_q with an arbitrary q.
ii.
If cosθ = 0 and κ₁ ≠ 0 (i.e. γ is a non-geodesic Legendre curve), then γ is a magnetic curve for F_±κ₁.
iii.
If cosθ=εκ12+s, then γ is a magnetic curve for Fεκ12+s, where ε = −sgn(g(φT, v₂)). In this case, γ is a slant φ-circle.
iv.
If cosθ=εs±κ2sκ12+(εs±κ2)2, then γ is a magnetic curve for Fεκ12+(εs±κ2)2, where ε = −sgn(g(φT, v₂)) and the sign ± corresponds to the sign of η^α(v₃).
v.
Except the above cases, γ is not a magnetic curve for any F_q.

Proof.

Let cosθ=±1s. Then we have

T=±1s∑α=1sξα,

which gives us ∇_TT = 0. We also have φT = 0. So γ satisfies ∇_TT = −qφT for any q, which proves i.

Now let cosθ = 0 and κ₁ ≠ 0. Using Corollary 3.2, we have

∇TT=κ1(±ϕT)=−qϕT,

which gives us q = ±κ₁. This completes the proof of ii.

From Proposition 3.1, we have the Frenet frame field

{T,±ϕT1−scos2θ,±1s1−scos2θ(−scosθT+∑α=1sξα)}

when r = 3 and

{T,±ϕT1−scos2θ}

when r = 2. If we differentiate v₂ along the curve, after some calculations, in both cases, we find

(1±κ1cosθ1−scos2θ)(−scosθT+∑α=1sξα)=±κ21−scos2θv3,(3.30)

(taking κ₂ = 0, when r = 2).

Next, let us assume cosθ=εκ12+s, where we denote ε = −sgn(g(φT, v₂)). Then the left side of equation (3.30) vanishes. Thus we get κ₂ = 0. From the assumption, we also have κ₁ = constant, that is, γ is a slant φ-circle. Using the Frenet frame field, we find ∇_TT = −qφT = κ₁v₂, where q=εκ12+s. So, we have just completed the proof of iii.

Finally, let us assume cosθ=εs±κ2sκ12+(εs±κ2)2, where ε = −sgn(g(φT, v₂)) and the sign ± corresponds to the sign of η^α(v₃). In this case, let us take κ₂ ≠ 0, since we have already investigated order r = 2. Using the Frenet frame field, after some calculations, we obtain ∇_T T = −qφT = κ₁v₂, where q=εκ12+(εs±κ2)2. Hence, the proof of iv is complete.

Since we have considered all cases, we can state that there exist no other slant magnetic φ-helices in M.

From the proof of Theorem 3.1. we can give the following proposition:

Proposition 3.2.

Let (M^2m+s, φ, ξ_α, η^α, g) be an S-manifold. There exist no non-geodesic slant φ-circles as magnetic curves corresponding to F_q for 0<|q|≤s.

Theorem 3.3 and Proposition 3.2 generalize Theorem 3.2 and Proposition 3.2 in [10] to S-manifolds, respectively. Under the condition s = 1, we obtain their results.

4. Construction of slant normal magnetic curves in ℝ^2n+s(−3s)

In this section, we find parametric equations of slant normal magnetic curves in ℝ^2n+s(−3s). As a start, we recall structures defined on this S-manifold. Let us take M = ℝ^2n+s with coordinate functions {x₁,...,x_n, y₁,...,y_n, z₁,...,z_s} and define

ξα=2∂∂zα, α=1,…,s,ηα=12(dzα−∑i=1nyidxi), α=1,…,s,ϕX=∑i=1nYi∂∂xi−∑i=1nXi∂∂yi+(∑i=1nYiyi)(∑α=1s∂∂zα),g=∑α=1sηα⊗ηα+14∑i=1n(dxi⊗dxi+dyi⊗dyi),

where

X=∑i=1n(Xi∂∂xi+Yi∂∂yi)+∑α=1s(Zα∂∂zα)∈χ(M).

It is well-known that (ℝ^2n+s, φ, ξ_α, η^α, g) is an S-space form with constant φ-sectional curvature −3s. Hence it is denoted by ℝ^2n+s(−3s) [13]. The following vector fields

Xi=2∂∂yi, Xn+i=ϕXi=2(∂∂xi+yi∑α=1s∂∂zα), ξα=2∂∂zα

form a g-orthonormal basis and the Levi-Civita connection is

∇XiXj=∇Xn+iXn+j=0, ∇XiXn+j=δij∑α=1sξα, ∇Xn+iXj=−δij∑α=1sξα,∇Xiξα=∇ξαXi=−Xn+i, ∇Xn+iξα=∇ξαXn+i=Xi.

(see [13]). Let γ : I → ℝ^2n+s(−3s) be a unit-speed slant curve with contact angle θ. Let us denote

γ(t)=(γ1(t),…,γn(t),γn+1(t),…,γ2n(t),γ2n+1(t),…,γ2n+s(t)),

where t is the arc-length parameter. Then γ has the tangent vector field

T=γ1′∂∂x1+⋯+γn′∂∂xn+γn+1′∂∂y1+⋯+γ2n′∂∂yn+γ2n+1′∂∂z1+⋯+γ2n+s′∂∂zs,

which can be written as

T=12[γn+1′X1+⋯+γ2n′Xn+γ1′Xn+1+⋯+γn′X2n+(γ2n+1′−γ1′γn+1′−⋯−γn′γ2n′)ξ1+⋯+(γ2n+s′−γ1′γn+1′−⋯−γn′γ2n)ξs].

Since γ is slant curve, we have

ηα(T)=12(γ2n+α′−γ1′γn+1−⋯−γn′γ2n)=cosθ

for all α = 1,...,s. So, we obtain

γ2n+1′=⋯=γ2n+s′=2cosθ+γ1′γn+1′+⋯+γn′γ2n.(4.1)

Since γ is a unit-speed, we can write

(γ1′)2+⋯+(γ2n′)2=4(1−scos2θ).(4.2)

These equations were obtained in our paper [12].

Now, our aim is to find parametric equations for slant normal magnetic curves. So, let us assume that γ : I → ℝ^2n+s(−3s) is a normal magnetic curve. From the Lorentz equation, we have

∇TT=−qϕT,(4.3)

where q ≠ 0 is a constant. Using the Levi-Civita connection, we calculate

∇TT=12{(γn+1″+2scosθγ1′)X1+⋯+(γ2n″+2scosθγn′)Xn+(γ1″−2scosθγn+1′)Xn+1+⋯+(γn″−2scosθγ2n′)X2n}(4.4)

and

ϕT=12{−γ1′X1−⋯−γn′Xn+γn+1′Xn+1+⋯+γ2n′X2n}.(4.5)

From equations (4.3), (4.4) and (4.5), we have

γn+1″+2scosθγ1′−γ1′=⋯=γ2n″+2scosθγn′−γn′=γ1″−2scosθγn+1′γn+1′=⋯=γn″−2scosθγ2n′γ2n′=−q,

which is equivalent to

γn+1″−γ1′=⋯γ2n″−γn′=γ1″γn+1′=⋯=γn″γ2n′=λ,(4.6)

where we denote λ = −q + 2scosθ. Firstly, let us assume λ ≠ 0. From equation (4.6), if we select pairs

γn+1″−γ1′=γ1″γn+1′,⋯,γ2n″−γn′=γn″γ2n′,

solving ODEs, we have

(γ1′)2+(γn+1′)2=c12,…,(γn′)2+(γ2n′)2=cn2,

where c₁,...,c_n are arbitrary constants. Thus, we can write

γ1′=c1cosf1,…,γn′=cncosfn,γn+1′=c1sinf1,…,γ2n′=cnsinfn,(4.7)

where f₁,..., f_n are differentiable functions on I. From (4.6) and (4.7), we find

f1′=⋯=fn′=−λ,

which gives us

fi=−λt+ai, i=1,2,…,n

where a₁,...,a_n are arbitrary constants. Now, if we integrate (4.7), we have

γ1=c1−λsinf1+b1,…,γn=cn−λsinfn+bn,

γn+1=c1λcosf1+d1,…,γ2n=cnλcosfn+dn,

where b_i and d_i are arbitrary constants (i = 1,...,n). Thus, we get

γ1′γn+1+⋯+γn′γ2n=∑i=1n(ci2λcos2fi+cidicosfi).

Using the last equation with (4.1), we obtain

γ2n+α′=2cosθ+∑i=1n(ci2λcos2fi+cidicosfi),

where α = 1,...,s. If we integrate this last equation, we find

γ2n+α=2tcosθ−∑i=1n{ci24λ2[sin(2fi)+2fi]+cidiλsinfi}+hα,

for α = 1,...,s and h₁,...,h_s are arbitrary constants. Moreover, from (4.2) and (4.7), we have

c12+⋯+cn2=4(1−scos2θ).(4.8)

Thus, we have just finished the case λ ≠ 0.

Secondly, let λ = 0. In this case, we have

γ1″=γ2″=⋯=γ2″=0,

which gives us

γi=cit+di,

for i = 1,...,2n, where c_i and d_i are arbitrary constants. Using the last equation, we calculate

γ1′γn+1+⋯+γn′γ2n=∑i=1nci(cn+it+dn+i).

So, equation (4.1) becomes

γ2n+α′=2cosθ+∑i=1nci(cn+it+dn+i),

which gives us

γ2n+α=2tcosθ+∑i=1nci(cn+i2t2+dn+it)+hα,

where h_α are arbitrary constants for α = 1,...,s. Since γ is unit-speed, from (4.2), we have

c12+⋯+c2n2=4(1−scos2θ).

To sum up, we give the following Theorem:

Theorem 4.1.

The slant normal magnetic curves on ℝ^2n+s(−3s) satisfying the Lorentz equation ∇_T T = −qφT have the parametric equations

a)
γi(t)=ci−λsinfi(t)+bi,γn+i(t)=ciλcosfi(t)+di,γ2n+α(t)=2tcosθ−∑i=1n{ci24λ2[sin(2fi(t))+2fi(t)]+cidiλsinfi(t)}+hα,fi(t)=−λt+ai,α=1,…,s, i=1,2,…,n,λ=−q+2scosθ≠0
where a_i, b_i, c_i, d_i and h_α are arbitrary constants such that c_i satisfies
c12+⋯+cn2=4(1−scos2θ);
or
b)
γi(t)=cit+di,γ2n+α(t)=2tcosθ+∑i=1nci(cn+i2t2+dn+it)+hα,α=1,…,s, i=1,…,2n,
where c_i, d_i and h_α are arbitrary constants such that c_i satisfies
c12+⋯+c2n2=4(1−scos2θ).
In both cases, q ≠ 0 is a constant and θ denotes the constant contact angle satisfying |cosθ|≤1s.

In particular, if s = 1, we obtain Theorem 3.5 in [10].

Acknowledgements

The authors would like to thank the referee for his/her useful remarks which helped improving the quality of the paper. This work is financially supported by Balikesir Research Grant no. BAP 2018/016.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 26 - 4
Pages: 536 - 554
Publication Date: 2019/07/09
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AU  - Güvenç Şaban
AU  - Cihan Özgür
PY  - 2019
DA  - 2019/07/09
TI  - On Slant Magnetic Curves in S-manifolds
JO  - Journal of Nonlinear Mathematical Physics
SP  - 536
EP  - 554
VL  - 26
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1640463
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Journal of Nonlinear Mathematical Physics

On Slant Magnetic Curves in S-manifolds

1. Introduction

2. Preliminaries

3. Slant magnetic curves in S-manifolds

Theorem 3.1.

Proof.

Remark 3.1.

Theorem 3.2.

Proof.

Corollary 3.1.

Proposition 3.1.

Proof.

Corollary 3.2.

Proof.

Theorem 3.3.

Proof.

Proposition 3.2.

4. Construction of slant normal magnetic curves in ℝ2n+s(−3s)

Theorem 4.1.

Acknowledgements

References

Cite this article

4. Construction of slant normal magnetic curves in ℝ^2n+s(−3s)