Journal of Nonlinear Mathematical Physics

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
https://doi.org/10.1080/14029251.2019.1640463How 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 (M2m+s, φ, ξα, ηα, 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 ℝ2n+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,Yχ(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 : IM 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 Fq = 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 (M2m+s, φ, ξα, ηα, 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 ℝ2n+s(−3s) and its structures, we construct slant normal magnetic curves in ℝ2n+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 (M2n+s, g) be a differentiable manifold, φ a (1, 1)-type tensor field, ηα 1-forms, ξα vector fields for α = 1,...,s, satisfying

ϕ2=I+α=1sηαξα,(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, YTM. Then (φ, ξα, ηα, g) is called framed φ-structure and (M2n+s, φ, ξα, ηα, g) is called framed φ-manifold [20]. (M2n+s, φ, ξα, ηα, g) is also called framed metric manifold [25] or almost r-contact metric manifold [24]. If the Nijenhuis tensor of φ is equal to −2αξα, then (M2n+s, φ, ξα, ηα, 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 M2n+s = (M2n+s, φ, ξα, ηα, g) be an S-manifold and Ω the fundamental 2-form of M2n+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 = α. Hence, the fundamental 2-form Ω on M2n+s is closed. The magnetic field Fq on M2n+s can be defined by
Fq(X,Y)=qΩ(X,Y),
where X and Y are vector fields on M2n+s and q is a real constant. Fq is called the contact magnetic field with strength q [15]. If q = 0 then the magnetic curves are geodesics of M2n+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 Fq can be written as

Φq=qϕ.

So the Lorentz equation (1.2) can be written as

TT=qϕT,(2.6)
where γ : IRM2n+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 : IM is said to be a Frenet curve of osculating order r, if there exists positive functions κ1,...,κr−1 on I satisfying

T=v1=γ,TT=k1v2,Tv2=k1T+k2v3,Tvr=kr1vr1,(3.1)
where 1 ≤ rn and T, v2,...,vr are a g-orthonormal vector fields along the curve. The positive functions κ1,...,κr−1 are called curvature functions and {T, v2,...,vr} 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,...,κr−1. A helix of order 3 is simply called a helix.

Let (M2m+s, φ, ξα, ηα, g) be an S-manifold. For a unit-speed curve γ : IM, 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 (M2m+s, φ, ξα, ηα, 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,...,vr span a φ-invariant space. A φ-curve of osculating order r with constant curvature functions κ1,...,κr−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, ηα(T) = cosθα 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 θα 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 (M2m+s, φ, ξα, ηα, g) be an S-manifold and consider the contact magnetic field Fq for q ≠ 0. Then γ is a slant normal magnetic curve associated to Fq in M2m+s if and only if γ belongs to the following list:

  1. a)

    geodesics obtained as integral curves of (±1sα=1sξα);

  2. b)

    non-geodesic slant circles with the curvature κ1=q2s, having the contact angle θ=arccos(1q) and the Frenet frame field

    {T,sgn(q)ϕT1scos2θ},
    where |q|>s;

  3. c)

    Legendre helices with curvatures κ1 = |q| and κ2=s, having the Frenet frame field

    {T,sgn(q)ϕT,sgn(q)sα=1sξα};
    i.e., a class of 1-dimensional integral submanifolds of the contact distribution;

  4. d)

    slant helices with curvatures κ1=|q|1scos2θ and κ2=s|1qcosθ|, having the Frenet frame field

    {T,sgn(q)ϕT1scos2θ,εsgn(q)s1scos2θ(scosθT+α=1sξα)},
    where θπ2 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
Tsp{ξ1,,ξs}.

If γ is slant, then we can write

T=cosθα=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 ξα 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 θα = 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|1scos2θ.(3.6)

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

qϕT=κ1v2=|q|1scos2θv2,
which gives us
ϕT=|q|q1scos2θv2=sgn(q)1scos2θv2.(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)1scos2θηα(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)1scos2θv2)=0.

Since r = 2, we find

κ1cosθ+sgn(q)1scos2θ=0.

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

|q|1scos2θ(cosθ+1q)=0.

Since γ is non-geodesic, we have

cosθ=1q.

Then equation (3.6) becomes

κ1=|q|1scos2θ=q2s,
where |q|>s. So the proof of b) is complete.

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

ϕ2T=T+cosθα=1sξα.(3.8)

Using (2.3) and (3.4), we have

(Tϕ)T=scosθT+α=1sξα,(3.9)
which gives us
TϕT=(Tϕ)T+ϕTT=scosθT+α=1sξα+ϕ(qϕT)=scosθT+α=1sξαq(T+cosθα=1sξα).(3.10)

Differentiating (3.7), we also find

TϕT=sqn(q)1scos2θ(κ1T+κ2v3).(3.11)

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

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

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

κ2=s|1qcosθ|.(3.13)

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

α=1sξα=scosθTεsgn(q)s1scos2θv3.(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)1scos2θT+εcosθsv3.(3.15)

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

ηβ(v3)=εsgn(q)1scos2θs.

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

α=1sTξα=scosθTTεsgn(q)s1scos2θTv3,
which gives us
s(1cosθ)ϕT=εsgn(q)s1scos2θ(κ2v2+κ3v4).

Since φTv2, 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)sα=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 (M2m+s, φ, ξα, ηα, g) be an S-manifold and consider the contact magnetic field Fq for q ≠ 0. If γ is a normal magnetic curve associated to Fq in M2m+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)=1A,
where we denote
A=α=1scos2θα.

Then, we have

ϕT=1A
and
κ1=|q|1A.

Thus, φT can be rewritten as

ϕT=sgn(q)1Av2.(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=(qB)T+(1A)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)1A(κ1T+κ2v3),
which corresponds to equation (3.11). In this last equation, we can replace κ1=|q|1A. Finally, we have
sgn(q)1Aκ2v3=(1A)α=1sξα+(q+B)V+(qAB)T.(3.17)

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

C=(1A)(Aq2As+B22Bq+s),
which is a constant. Hence, we obtain
κ2=C1A=Aq2As+B22Bq+s=constant.

From equation (3.17), we also have v3 ∈ span{T, ξ1,...,ξs}. The angles between v3 and T, ξ1,..., ξs 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,...,cs. If we differentiate equation (3.18), we get
κ2v2+κ3v4=c0κ1v2c1ϕTcsϕT.

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. θα = θ for all α = 1,...,s, then we obtain the following corollary:

Corollary 3.1.

If θα = θ, for all α = 1,...,s, then

A=scos2θ,B=scosθ,V=cosθα=1sξα,C=(1scos2θ)s(1qcosθ)2
and κ2=s|1qcosθ|.

Now, let us state the following proposition:

Proposition 3.1.

Let γ be a slant φ-helix of order 3 in an S-manifold (M2m+s, φ, ξα, ηα, 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 = ss2 cos2 θ. Hence, γ has the Frenet frame field
{T,±ϕT1scos2θ,±1s1scos2θ(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 ηα(v3) does not depend on α.

Firstly, let us assume that θπ2. 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=1sξαsp{T,v3},
that is,
α=1sξα=scosθT+ρv3.(3.22)

If we take the norm of both sides, we find ρ2 = ss2 cos2 θ. Since the value of ηα(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 φTT, φTv3 and sp{T, v2, v3} is φ-invariant, we have φTv2. As a result, we find

v2=±ϕTϕT=±ϕT1scos2θ.

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+ϕTT=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, φTv2. 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 ηα(v3) does not depend on α; so, we can write

v3=ηα(v3)α=1sξα.(3.28)

Since v3 and ξα 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 (M2m+s, φ, ξα, ηα, g). Then κ2=s, v2 = ±φT and v3=±1sα=1sξα.

Proof.

From equation (3.28), we already have

v3=±1sα=1sξα.

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

κ2v2=±1sα=1sTξα.(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 (M2m+s, φ, ξα, ηα, g). Let θ denote the contact angle of γ. Then we have

  1. i.

    If cosθ=±1s, then γ is an integral curve of ±1sα=1sξα, hence it is a normal magnetic curve for Fq with an arbitrary q.

  2. 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.

  3. 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.

  4. 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 ηα(v3).

  5. v.

    Except the above cases, γ is not a magnetic curve for any Fq.

Proof.

Let cosθ=±1s. Then we have

T=±1sα=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,±ϕT1scos2θ,±1s1scos2θ(scosθT+α=1sξα)}
when r = 3 and
{T,±ϕT1scos2θ}
when r = 2. If we differentiate v2 along the curve, after some calculations, in both cases, we find
(1±κ1cosθ1scos2θ)(scosθT+α=1sξα)=±κ21scos2θv3,(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 ηα(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 (M2m+s, φ, ξα, ηα, g) be an S-manifold. There exist no non-geodesic slant φ-circles as magnetic curves corresponding to Fq 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 ℝ2n+s(−3s)

In this section, we find parametric equations of slant normal magnetic curves in ℝ2n+s(−3s). As a start, we recall structures defined on this S-manifold. Let us take M = ℝ2n+s with coordinate functions {x1,...,xn, y1,...,yn, z1,...,zs} and define

ξα=2zα,α=1,,s,ηα=12(dzαi=1nyidxi),α=1,,s,ϕX=i=1nYixii=1nXiyi+(i=1nYiyi)(α=1szα),g=α=1sηαηα+14i=1n(dxidxi+dyidyi),
where
X=i=1n(Xixi+Yiyi)+α=1s(Zαzα)χ(M).

It is well-known that (ℝ2n+s, φ, ξα, ηα, g) is an S-space form with constant φ-sectional curvature −3s. Hence it is denoted by ℝ2n+s(−3s) [13]. The following vector fields

Xi=2yi,Xn+i=ϕXi=2(xi+yiα=1szα),ξα=2zα
form a g-orthonormal basis and the Levi-Civita connection is
XiXj=Xn+iXn+j=0,XiXn+j=δijα=1sξα,Xn+iXj=δijα=1sξα,Xiξα=ξαXi=Xn+i,Xn+iξα=ξαXn+i=Xi.
(see [13]). Let γ : I → ℝ2n+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=γ1x1++γnxn+γn+1y1++γ2nyn+γ2n+1z1++γ2n+szs,
which can be written as
T=12[γn+1X1++γ2nXn+γ1Xn+1++γnX2n+(γ2n+1γ1γn+1γnγ2n)ξ1++(γ2n+sγ1γn+1γnγ2n)ξs].

Since γ is slant curve, we have

ηα(T)=12(γ2n+αγ1γn+1γnγ2n)=cosθ
for all α = 1,...,s. So, we obtain
γ2n+1==γ2n+s=2cosθ+γ1γn+1++γnγ2n.(4.1)

Since γ is a unit-speed, we can write

(γ1)2++(γ2n)2=4(1scos2θ).(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 → ℝ2n+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+(γ12scosθγn+1)Xn+1++(γn2scosθγ2n)X2n}(4.4)
and
ϕT=12{γ1X1γnXn+γn+1Xn+1++γ2nX2n}.(4.5)

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

γn+1+2scosθγ1γ1==γ2n+2scosθγnγn=γ12scosθγn+1γn+1==γn2scosθγ2nγ2n=q,
which is equivalent to
γn+1γ1=γ2nγn=γ1γn+1==γnγ2n=λ,(4.6)
where we denote λ = −q + 2scosθ. Firstly, let us assume λ ≠ 0. From equation (4.6), if we select pairs
γn+1γ1=γ1γn+1,,γ2nγn=γnγ2n,
solving ODEs, we have
(γ1)2+(γn+1)2=c12,,(γn)2+(γ2n)2=cn2,
where c1,...,cn are arbitrary constants. Thus, we can write
γ1=c1cosf1,,γn=cncosfn,γn+1=c1sinf1,,γ2n=cnsinfn,(4.7)
where f1,..., fn 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,...,an are arbitrary constants. Now, if we integrate (4.7), we have
γ1=c1λsinf1+b1,,γn=cnλsinfn+bn,
γn+1=c1λcosf1+d1,,γ2n=cnλcosfn+dn,
where bi and di are arbitrary constants (i = 1,...,n). Thus, we get
γ1γn+1++γnγ2n=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=2tcosθi=1n{ci24λ2[sin(2fi)+2fi]+cidiλsinfi}+hα,
for α = 1,...,s and h1,...,hs are arbitrary constants. Moreover, from (4.2) and (4.7), we have
c12++cn2=4(1scos2θ).(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 ci and di are arbitrary constants. Using the last equation, we calculate
γ1γn+1++γnγ2n=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 hα are arbitrary constants for α = 1,...,s. Since γ is unit-speed, from (4.2), we have
c12++c2n2=4(1scos2θ).

To sum up, we give the following Theorem:

Theorem 4.1.

The slant normal magnetic curves on2n+s(−3s) satisfying the Lorentz equationT T = −qφT have the parametric equations

  1. a)

    γi(t)=ciλsinfi(t)+bi,γn+i(t)=ciλcosfi(t)+di,γ2n+α(t)=2tcosθi=1n{ci24λ2[sin(2fi(t))+2fi(t)]+cidiλsinfi(t)}+hα,fi(t)=λt+ai,α=1,,s,i=1,2,,n,λ=q+2scosθ0
    where ai, bi, ci, di and hα are arbitrary constants such that ci satisfies
    c12++cn2=4(1scos2θ);
    or

  2. b)

    γi(t)=cit+di,γ2n+α(t)=2tcosθ+i=1nci(cn+i2t2+dn+it)+hα,α=1,,s,i=1,,2n,
    where ci, di and hα are arbitrary constants such that ci satisfies
    c12++c2n2=4(1scos2θ).
    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.

References

[12]Ş. Güvenç and C. Özgür, On slant curves in S-manifolds, Commun. Korean Math. Soc., Vol. 33, No. 1, 2018, pp. 293-303.
[13]I. Hasegawa, Y. Okuyama, and T. Abe, On p-th Sasakian manifolds, J. Hokkaido Univ. Ed. Sect. II A, Vol. 37, No. 1, 1986, pp. 1-16.
[21]C. Özgür and Ş. Güvenç, On biharmonic Legendre curves in S-space forms, Turkish J. Math., Vol. 38, No. 3, 2014, pp. 454-461.
[22]C. Özgür, On magnetic curves in the 3-dimensional Heisenberg group, in Proceedings of the Institute of Mathematics and Mechanics (Azerbaijan, 2017), pp. 278-286.
[24]J. Vanzura, Almost r-contact structures, Ann. Scuola Norm. Sup. Pisa, Vol. 26, No. 3, 1972, pp. 97-115.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
26 - 4
Pages
536 - 554
Publication Date
2019/07
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2019.1640463How to use a DOI?
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/).

Cite this article

TY  - JOUR
AU  - Güvenç Şaban
AU  - Cihan Özgür
PY  - 2019
DA  - 2019/07
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
DO  - https://doi.org/10.1080/14029251.2019.1640463
ID  - Şaban2019
ER  -