On Slant Magnetic Curves in $S$-manifolds

TL;DR

This paper characterizes slant normal magnetic curves in $S$-manifolds, identifying specific types such as geodesics, circles, and helices, and provides explicit constructions and equations for these curves.

Contribution

It provides a complete classification of slant normal magnetic curves in $S$-manifolds and explicit parametric equations for these curves.

Findings

Classification of slant normal magnetic curves as geodesics, circles, and helices.

Explicit parametric equations for slant magnetic curves in $ ext{R}^{2n+s}(-3s)$.

Conditions under which curves are slant normal magnetic in $S$-manifolds.

Abstract

We consider slant normal magnetic curves in $(2n+1)$-dimensional $S$-manifolds. We prove that $\gamma $ is a slant normal magnetic curve in an $% S $-manifold $(M^{2m+s},\varphi ,\xi _{\alpha },\eta ^{\alpha },g)$ if and only if it belongs to a list of slant $\varphi $-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 $\mathbb{R}^{2n+s}(-3s)$ and give the parametric equations of these curves.

Equations

$$g(\Phi X,Y)=F(X,Y),\forall X,Y\in \chi (M),$$

$${\nabla }_{{\gamma }^{\prime }(t)}{\gamma }^{\prime }(t)=\Phi ({\gamma }^{\prime }(t)),$$

$${\phi }^2=-I+\stackrel{s}{\sum _{\alpha =1}}{\eta }^{\alpha }\otimes {\xi }_{\alpha },$$

$${\eta }^{\alpha }\left({\xi }_{\beta }\right)={\delta }_{\beta }^{\alpha },\phi {\xi }_{\alpha }=0,{\eta }^{\alpha }\left(\phi X\right)=0,{\eta }^{\alpha }\left(X\right)=g\left(X,{\xi }_{\alpha }\right),$$

$$g(\phi X,\phi Y)=g(X,Y)-\stackrel{s}{\sum _{\alpha =1}}{\eta }^{\alpha }(X){\eta }^{\alpha }(Y),$$

$$d{\eta }^{\alpha }\left(X,Y\right)=-d{\eta }^{\alpha }\left(Y,X\right)=g(X,\phi Y),$$

$$({\nabla }_X\phi )Y=\underset{\alpha =1}{\sum ^s}\left\{g(\phi X,\phi Y){\xi }_{\alpha }+{\eta }^{\alpha }(Y){\phi }^{2}X\right\},$$

$$\nabla {\xi }_{\alpha }=-\phi ,\alpha \in \left\{1,...,s\right\}.$$

$$\Omega (X,Y)=g(X,\phi Y),$$

$$F_q(X,Y)=q\Omega (X,Y),$$

$${\Phi }_q=-q\phi .$$

$${\nabla }_{T}T=-q\phi T,$$

$$T$$

$${\nabla }_{T}T$$

$${\nabla }_T{v}_2$$

$${\nabla }_T{v}_r$$

$${\eta }^{\alpha }(T)=0,$$

$${\eta }^{\alpha }(T)=\cos \theta ,$$

$$sp\left\{T,v_2,\stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha }\right\}$$

$$\left\{T,\frac{-sgn(q)\phi T}{\sqrt{1-s{\cos }^2\theta }}\right\},$$

$$\left\{T,-sgn(q)\phi T,\frac{-sgn(q)}{\sqrt{s}}\stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha }\right\};$$

$$\left\{T,\frac{-sgn(q)\phi T}{\sqrt{1-s{\cos }^2\theta }},\frac{-\epsilon sgn(q)}{\sqrt{s}\sqrt{1-s{\cos }^2\theta }}\left(-s\cos \theta T+\stackrel{s}{\sum _{\alpha =1}}{\xi }_{\alpha }\right)\right\},$$

$${\nabla }_{T}T=0=-q\phi T$$

$$T\in sp\left\{{\xi }_1,...,{\xi }_s\right\}.$$

$$T=\cos \theta \underset{\alpha =1}{\sum ^s}{\xi }_{\alpha }.$$

$$0=g\left(-q\phi T,{\xi }_{\alpha }\right)=g\left({\nabla }_{T}T,{\xi }_{\alpha }\right)=\frac{d}{dt}g\left(T,{\xi }_{\alpha }\right)-g\left(T,{\nabla }_T{\xi }_{\alpha }\right).$$

$${\nabla }_T{\xi }_{\alpha }=-\phi T.$$

$$\frac{d}{dt}g\left(T,{\xi }_{\alpha }\right)=0,$$

$${\eta }^{\alpha }\left(T\right)=\cos {\theta }_{\alpha }=constant.$$

$${\eta }^{\alpha }\left(T\right)=\cos \theta .$$

$${\nabla }_{T}T={\kappa }_1{v}_2=-q\phi T.$$

$${\kappa }_1=\left|q\right|\Vert \phi T\Vert =\left|q\right|\sqrt{1-s{\cos }^2\theta }.$$

Full text

1. Introduction

Let $(M,g)$ be a Riemannian manifold, $F$ a closed $2$-form and let us denote the Lorentz force on $M$ by $\Phi$.  If $F$ is associated by the relation

[TABLE]

then it is called a* magnetic field* ([1],  [4] and [9]). Let $\nabla$ be the Riemannian connection associated to the metric $g$ and $\gamma:I\rightarrow M$ a smooth curve. If $\gamma$ satisfies the Lorentz equation

[TABLE]

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 $\gamma$ is equal to $1$, then it is called a normal magnetic curve [10].

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 $\left(\tau\pm 1\right)$ and $\kappa$ is constant, where $\tau$ is the geodesic torsion and $\kappa$ 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 $F_{q}=q\Omega$ on a Sasakian $(2n+1)$-manifold $(M^{2n+1},\varphi,\xi,\eta,g)$, where $\Omega$ is the fundamental $2$-form. The main objective of [11] is the study of trajectories for particles moving under the influence 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 $S^{2n+1}$ lies on a totally geodesic sphere $S^{3}$. They also considered two particular magnetic fields on three-dimensional torus obtained from two different contact forms on the Euclidean space $E^{3}$ 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. Vanzura studied almost $r$-structures in [24]. A differentiable manifold with this structure is the same as a framed $f$-manifold as defined by Nakagawa. On the other hand, Hasegawa, Okuyama and Abe defined a $p$th 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. In Section 2, we give brief information on $S$-manifolds and magnetic curves. In Section 3, we prove that $\gamma$ is a slant normal magnetic curve in an $S$-manifold $(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ if and only if it belongs to a list of slant $\varphi$-curves. This list consists of some specific geodesics, slant circles, Legendre and slant helices of order $3$. Finally, in Section 4, we construct slant normal magnetic curves in $\mathbb{R}^{2n+s}(-3s)$ 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 $\left(M^{2n+s},g\right)$ be a differentiable manifold, $\varphi$ a $(1,1)$-type tensor field, $\eta^{\alpha}$ 1-forms, $\xi_{\alpha}$ vector fields for $\alpha=1,...,s$, satisfying

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $X,Y\in TM$. Then $(\varphi,\xi_{\alpha},\eta^{\alpha},g)$ is called framed $\varphi$-structure and $(M^{2n+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ is called framed $\varphi$-manifold [20]. $(M^{2n+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ is also called framed metric manifold [25] or almost r-contact metric manifold [24]. If the Nijenhuis tensor of $\varphi$ is equal to $-2d\eta^{\alpha}\otimes\xi_{\alpha}$, then $(M^{2n+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ 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]:

[TABLE]

[TABLE]

Let $M^{2n+s}=(M^{2n+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ be an $S$-manifold and $\Omega$ the fundamental $2$-form of $M^{2n+s}$ defined by

[TABLE]

(see [20] and [24]). From the definition of framed $\varphi$-structure, we have $\Omega=d\eta^{\alpha}$. Hence, the fundamental 2-form $\Omega$ on M2n+s is closed. The magnetic field $F_{q}$ on $M^{2n+s}$ can be defined by

[TABLE]

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 $q\neq 0$ (see [6] and [10]).

From (1.1) and (2.5), the Lorentz force $\Phi$ associated to the contact magnetic field $F_{q}$ can be written as

[TABLE]

So the Lorentz equation (1.2) can be written as

[TABLE]

where $\gamma:I\subseteq R\rightarrow M^{2n+s}$ is a smooth unit-speed curve and $T=\gamma^{\prime}$ (see [10] and [15]).

3. Slant magnetic curves in $S$-manifolds

Let $\left(M^{n},g\right)$ be a Riemannian manifold. A unit-speed curve $\gamma:I\rightarrow M$ is said to be a Frenet curve of osculating order $r$, if there exists positive functions $\kappa_{1},...,\kappa_{r-1}$ on $I$ satisfying

[TABLE]

where $1\leq r\leq n$ and $T,v_{2},...,v_{r}$ are a $g$-orthonormal vector fields along the curve. The positive functions $\kappa_{1},...,\kappa_{r-1}$ are called* curvature functions *and $\left\{T,v_{2},...,v_{r}\right\}$ 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 $\kappa_{1}$. A helix of order $r$ is a Frenet curve of osculating order $r$ with constant curvature functions $\kappa_{1},...,\kappa_{r-1}$. A helix of order $3$ is simply called a helix.

Let $(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ be an $S$-manifold. For a unit-speed curve $\gamma:I\rightarrow M$, if

[TABLE]

for all $\alpha=1,...,s$, then $\gamma$ is called a Legendre curve of $M$ [21]. More generally, if there exists a constant angle $\theta$ such that

[TABLE]

for all $\alpha=1,...,s$, then $\gamma$ is called a *slant curve *and $\theta$ is called the contact angle of $\gamma$, where $\left|\cos\theta\right|$ $\leq 1/\sqrt{s}$ [12].

Let $(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ be an $S$-manifold. A Frenet curve of osculating order $r\geq 3$ is called a $\varphi$-curve in $M$ if its Frenet vector fields $T,v_{2},...,v_{r}$ span a $\varphi$-invariant space. A $\varphi$-curve of osculating order $r$ with constant curvature functions $\kappa_{1},...,\kappa_{r-1}$ is called a $\varphi$-helix of order $r$. A curve of osculating order 2 is called a $\varphi$*-curve *if

[TABLE]

is a $\varphi$-invariant space.

Throughout  the paper, when we state ”slant magnetic curve”, we mean ”slant curves which satisfy equation (2.6)”. For magnetic curves, $\eta^{\alpha}(T)=\cos\theta_{\alpha}$ does not have to be equal for all $\alpha=1,...,s.$ By taking the curve as slant, we only study the equality case of the slant angles $\theta_{\alpha}$ 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 1**.**

Let $(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ be an $S$-manifold and consider the contact magnetic field $F_{q}$ for $q\neq 0$. Then $\gamma$ is a slant normal magnetic curve associated to $F_{q}$ in $M^{2m+s}$ if and only if $\gamma$ belongs to the following list:

$a)$* geodesics obtained as integral curves of $\left(\pm\frac{1}{\sqrt{s}}\underset{\alpha=1}{\overset{s}{\sum}}\xi_{\alpha}\right)$;*

$b)$* non-geodesic slant circles with the curvature $\kappa_{1}=\sqrt{q^{2}-s}$, having the contact angle $\theta=\arccos\left(\frac{1}{q}\right)$ and the Frenet frame field*

[TABLE]

where $\left|q\right|>\sqrt{s}$ ;

$c)$* Legendre helices with curvatures $\kappa_{1}=\left|q\right|$ and $\kappa_{2}=\sqrt{s}$, having the Frenet frame field*

[TABLE]

i.e., a class of 1-dimensional integral submanifolds of the contact distribution;

$d)$* slant helices with curvatures $\kappa_{1}=\left|q\right|\sqrt{1-s\cos^{2}\theta}$ and $\kappa_{2}=\sqrt{s}\left|1-q\cos\theta\right|$, having the Frenet frame field*

[TABLE]

where $\theta\neq\frac{\pi}{2}$ is the contact angle satisfying $\left|\cos\theta\right|<\frac{1}{\sqrt{s}}$ and $\varepsilon=sgn(1-q\cos\theta)$.

Proof.

Let $\gamma$ be a normal magnetic curve. If the magnetic curve is a geodesic, then

[TABLE]

gives us

[TABLE]

If $\gamma$ is slant, then we can write

[TABLE]

Since $\gamma$ is unit speed, we have $\cos\theta=\pm\frac{1}{\sqrt{s}}$. So the proof of a) is complete.

From now on, we suppose that $\gamma$ is a non-geodesic Frenet curve of osculating order $r>1$. Let us choose an $\alpha\in\left\{1,...,s\right\}$. Applying $\xi_{\alpha}$ to $\nabla_{T}T=-q\varphi T$, we obtain

[TABLE]

From (2.4), we also have

[TABLE]

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

[TABLE]

that is,

[TABLE]

Let us assume $\theta_{\alpha}=\theta$ for all $\alpha=1,...,s$, i.e., $\gamma$ is slant. So, we have

[TABLE]

Equations (2.6) and (3) give us

[TABLE]

Then we get

[TABLE]

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

[TABLE]

which gives us

[TABLE]

If $\kappa_{2}=0$, then the magnetic curve is a Frenet curve of osculating order $r=2$. Since $\kappa_{1}$ is a constant, $\gamma$ is a circle. From (3.7), we have

[TABLE]

that is,

[TABLE]

If we differentiate the last equation along the curve $\gamma$, we obtain

[TABLE]

So, we calculate

[TABLE]

Since $r=2$, we find

[TABLE]

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

[TABLE]

Since $\gamma$ is non-geodesic, we have

[TABLE]

Then equation (3.6) becomes

[TABLE]

where $\left|q\right|>\sqrt{s}$. So the proof of b) is complete.

Let $\kappa_{2}\neq 0$. From (2.1) and (3.4), we find

[TABLE]

Using (2.3) and (3.4), we have

[TABLE]

which gives us

[TABLE]

Differentiating (3.7), we also find

[TABLE]

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

[TABLE]

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

[TABLE]

Let us denote $\varepsilon=sgn(1-q\cos\theta)$. If we write (3.13) in (3.12), we obtain

[TABLE]

Applying $\varphi$ to (3.14), we find

[TABLE]

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

[TABLE]

Let us choose a $\beta\in\left\{1,...,s\right\}.$ From (3.15), we calculate

[TABLE]

If we differentiate (3.14) along the curve $\gamma$, we get

[TABLE]

which gives us

[TABLE]

Since $\varphi T\parallel v_{2}$, we find $\kappa_{3}=0$. This proves d) of the theorem.

Let us examine Legendre case separately, that is, $\theta=\frac{\pi}{2}$. Then we have $\varepsilon=1$, $\kappa_{1}=\left|q\right|$, $\kappa_{2}=\sqrt{s}$, $\kappa_{3}=0$ and equation (3.14) gives us

[TABLE]

This completes the proof of c).

Conversely, let $\gamma$ satisfy one of $a)$, $b)$, $c)$ or $d)$. Using the Frenet frame field and Frenet equations, it is straightforward to show that $\nabla_{T}T=-q\varphi T$, i.e., $\gamma$ 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. The order of a slant magnetic curve in an $S$-manifold is still $r\leq 3$, 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 $r\leq 3.$

Theorem 2**.**

Let $(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ be an $S$-manifold and consider the contact magnetic field $F_{q}$ for $q\neq 0$. If $\gamma$ is a normal magnetic curve associated to $F_{q}$ in $M^{2m+s}$, then the osculating order $r\leq 3$.

Proof.

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

[TABLE]

If we differentiate this equation along the curve, we have

[TABLE]

for all $\alpha=1,...,s.$ From the Frenet equations (3), we obtain

[TABLE]

From the definition of framed $\varphi$-structure, we calculate

[TABLE]

where we denote

[TABLE]

Then, we have

[TABLE]

and

[TABLE]

Thus, $\varphi T$ can be rewritten as

[TABLE]

Again, from the definition of framed $\varphi$-structure, we have

[TABLE]

where we denote

[TABLE]

After some calculations, we get

[TABLE]

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

[TABLE]

From equation (3.16), we also find

[TABLE]

which corresponds to equation (3.11). In this last equation, we can replace $\kappa_{1}=\left|q\right|\sqrt{1-A}$. Finally, we have

[TABLE]

So, if we denote the norm of the right hand side of equation (3) by $C$, we find

[TABLE]

which is a constant. Hence, we obtain

[TABLE]

From equation (3), we also have $v_{3}\in span\left\{T,\xi_{1},...,\xi_{s}\right\}.$ The angles between $v_{3}$ and $T,\xi_{1},...,\xi_{s}$ are all constants since all the coefficients in equation (3) are constants. Then, we can write

[TABLE]

for some constants $c_{0},...,c_{s}.$ If we differentiate equation (3.18), we get

[TABLE]

Since $\varphi T$ is parallel to $v_{2}$, if we take the inner product of the last equation with $v_{4}$, we find $\kappa_{3}=0$. This proves the theorem. ∎

In particular, if $\gamma$ is slant, i.e. $\theta_{\alpha}=\theta$ for all $\alpha=1,...,s,$ then we obtain the following corollary:

Corollary 1**.**

If $\theta_{\alpha}=\theta,$ for all $\alpha=1,...,s,$ then

[TABLE]

[TABLE]

and $\kappa_{2}=\sqrt{s}\left|1-q\cos\theta\right|.$

Now, let us state the following proposition:

Proposition 1**.**

Let $\gamma$ be a slant $\varphi$-helix of order 3 in an $S$-manifold $(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ with contact angle $\theta$. Then

[TABLE]

where $\rho=g\left(v_{3},\underset{\alpha=1}{\overset{s}{\sum}}\xi_{\alpha}\right)=s\eta^{\alpha}\left(v_{3}\right)$ is a real constant such that $\rho^{2}=s-s^{2}\cos^{2}\theta$. Hence, $\gamma$ has the Frenet frame field

[TABLE]

Proof.

From the assumption, the Frenet frame field $\left\{T,v_{2},v_{3}\right\}$ is $\varphi$-invariant and

[TABLE]

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

[TABLE]

If we differentiate once again, we have

[TABLE]

which means the value of $\eta^{\alpha}(v_{3})$ does not depend on $\alpha.$

Firstly, let us assume that $\theta\neq\frac{\pi}{2}$. Since the space spanned by the Frenet frame field is $\varphi$-invariant, then $\varphi^{2}T$ is in the set. Using (3.8) and (3.20), we can write

[TABLE]

that is,

[TABLE]

If we take the norm of both sides, we find $\rho^{2}=s-s^{2}\cos^{2}\theta$. Since the value of $\eta^{\alpha}(v_{3})$ does not depend on $\alpha$, we obtain

[TABLE]

If we apply $\varphi T$ to (3.22), we get $g(\varphi T,v_{3})=0$. Since $\varphi T\perp T$ , $\varphi T\perp v_{3}$ and $sp\left\{T,v_{2},v_{3}\right\}$ is $\varphi$-invariant, we have $\varphi T\parallel v_{2}$. As a result, we find

[TABLE]

Now let us consider the Legendre case, i.e., $\theta=\frac{\pi}{2}$. From (3.21), we find

[TABLE]

Using (2.1) and (2.3), we calculate

[TABLE]

Using the last equation, we obtain

[TABLE]

Equations (3.23) and (3) give us $g\left(\varphi T,v_{3}\right)=0$, that is, $\varphi T\parallel v_{2}$. Thus, we have $\varphi T=\pm v_{2}$. Consequently, the Frenet frame field becomes $\left\{T,\pm\varphi T,v_{3}\right\}$. Now, we must show that $v_{3}$ is parallel to $\underset{\alpha=1}{\overset{s}{\sum}}\xi_{\alpha}$. Since the space spanned by the Frenet frame field is $\varphi$-invariant, from orthonormal expansion, we can write

[TABLE]

which reduces to

[TABLE]

If we apply $\varphi$ to equation (3.26) and use (2.1), we find

[TABLE]

Applying $\varphi T$ to (3.27) and using the Frenet frame field, we have $g(\varphi v_{3},T)=0$. As a result, we get $\varphi v_{3}=0$ and equation (3.27) becomes

[TABLE]

We have already shown that the value of $\eta^{\alpha}(v_{3})$ does not depend on $\alpha$; so, we can write

[TABLE]

Since $v_{3}$ and $\xi_{\alpha}$ are unit for all $\alpha=1,...,s,$ we find $\eta^{\alpha}(v_{3})=\frac{\pm 1}{\sqrt{s}}.$ Finally, for $\theta=\frac{\pi}{2},$ we have $\rho^{2}=s$ and $\underset{\alpha=1}{\overset{s}{\sum}}\xi_{\alpha}=\rho v_{3}$, which completes the proof. ∎

Corollary 2**.**

Let $\gamma$ be a Legendre $\varphi$-helix of order 3 in an $S$-manifold $(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$. Then $\kappa_{2}=\sqrt{s}$, $v_{2}=\pm\varphi T$ and $v_{3}=\pm\frac{1}{\sqrt{s}}\underset{\alpha=1}{\overset{s}{\sum}}\xi_{\alpha}$.

Proof.

From equation (3.28), we already have

[TABLE]

If we differentiate this equation and use (3), we obtain

[TABLE]

Using equations (2.4) and (3.29), we find that $\kappa_{2}=\sqrt{s}$ and $v_{2}=\pm\varphi T$. ∎

Finally, we can give the following theorem:

Theorem 3**.**

Let $\gamma$ be a slant $\varphi$-helix of order $r\leq 3$ on an $S$-manifold $(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$. Let $\theta$ denote the contact angle of $\gamma$. Then we have

i. If $\cos\theta=\pm\frac{1}{\sqrt{s}}$, then $\gamma$ is an integral curve of  $\pm\frac{1}{\sqrt{s}}\underset{\alpha=1}{\overset{s}{\sum}}\xi_{\alpha}$, hence it is a normal magnetic curve for $F_{q}$ with an arbitrary $q$.

ii. If $\cos\theta=0$ and $\kappa_{1}\neq 0$ (i.e. $\gamma$ is a non-geodesic Legendre curve), then $\gamma$ is a magnetic curve for $F_{\pm\kappa_{1}}$.

iii. If $\cos\theta=\frac{\varepsilon}{\sqrt{\kappa_{1}^{2}+s}}$, then $\gamma$ is a magnetic curve for $F_{\varepsilon\sqrt{\kappa_{1}^{2}+s}}$, where $\varepsilon=-sgn(g(\varphi T,v_{2}))$. In this case, $\gamma$ is a slant $\varphi$-circle.

iv. If $\cos\theta=\frac{\varepsilon\sqrt{s}\pm\kappa_{2}}{\sqrt{s}\sqrt{\kappa_{1}^{2}+\left(\varepsilon\sqrt{s}\pm\kappa_{2}\right)^{2}}}$, then $\gamma$ is a magnetic curve for $F_{\varepsilon\sqrt{\kappa_{1}^{2}+\left(\varepsilon\sqrt{s}\pm\kappa_{2}\right)^{2}}}$, where $\varepsilon=-sgn(g(\varphi T,v_{2}))$ and the sign $\pm$ corresponds to the sign of $\eta^{\alpha}(v_{3})$.

v. Except the above cases, $\gamma$ is not a magnetic curve for any $F_{q}$.

Proof.

Let $\cos\theta=\pm\frac{1}{\sqrt{s}}$. Then we have

[TABLE]

which gives us $\nabla_{T}T=0$. We also have $\varphi T=0$. So $\gamma$ satisfies $\nabla_{T}T=-q\varphi T$ for any $q$, which proves i.

Now let $\cos\theta=0$ and $\kappa_{1}\neq 0$. Using Corollary 2, we have

[TABLE]

which gives us $q=\pm\kappa_{1}$. This completes the proof of ii.

From Proposition 1, we have the Frenet frame field

[TABLE]

when $r=3$ and

[TABLE]

when $r=2$. If we differentiate $v_{2}$ along the curve, after some calculations, in both cases, we find

[TABLE]

(taking $\kappa_{2}=0$, when $r=2$).

Next, let us assume $\cos\theta=\frac{\varepsilon}{\sqrt{\kappa_{1}^{2}+s}}$, where we denote $\varepsilon=-sgn(g(\varphi T,v_{2}))$. Then the left side of equation (3.30) vanishes. Thus we get $\kappa_{2}=0$. From the assumption, we also have $\kappa_{1}=constant$, that is, $\gamma$ is a slant $\varphi$-circle. Using the Frenet frame field, we find $\nabla_{T}T=-q\varphi T=\kappa_{1}v_{2}$, where $q=\varepsilon\sqrt{\kappa_{1}^{2}+s}$. So, we have just completed the proof of iii.

Finally, let us assume $\cos\theta=\frac{\varepsilon\sqrt{s}\pm\kappa_{2}}{\sqrt{s}\sqrt{\kappa_{1}^{2}+\left(\varepsilon\sqrt{s}\pm\kappa_{2}\right)^{2}}}$, where $\varepsilon=-sgn(g(\varphi T,v_{2}))$ and the sign $\pm$ corresponds to the sign of $\eta^{\alpha}(v_{3})$. In this case, let us take $\kappa_{2}\neq 0$, since we have already investigated order $r=2$. Using the Frenet frame field, after some calculations, we obtain $\nabla_{T}T=-q\varphi T=\kappa_{1}v_{2}$, where $q=\varepsilon\sqrt{\kappa_{1}^{2}+\left(\varepsilon\sqrt{s}\pm\kappa_{2}\right)^{2}}$. Hence, the proof of iv is complete.

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

From the proof of Theorem 1, we can give the following proposition:

Proposition 2**.**

Let $(M^{2m+s},\varphi,\xi_{\alpha},\eta^{\alpha},g)$ be an $S$-manifold. There exist no non-geodesic slant $\varphi$-circles as magnetic curves corresponding to $F_{q}$ for $0<\left|q\right|\leq\sqrt{s}$.

Theorem 3 and Proposition 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 $\mathbb{R}^{2n+s}(-3s)$

In this section, we find parametric equations of slant normal magnetic curves in $\mathbb{R}^{2n+s}(-3s)$. As a start, we recall structures defined on this $S$-manifold. Let us take $M=\mathbb{R}^{2n+s}$ with coordinate functions $\left\{x_{1},...x_{n},y_{1},...,y_{n},z_{1},...,z_{s}\right\}$ and define

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

It is well-known that $\left(\mathbb{R}^{2n+s},\varphi,\xi_{\alpha},\eta^{\alpha},g\right)$ is an $S$-space form with constant $\varphi$-sectional curvature $-3s.$ Hence it is denoted by $\mathbb{R}^{2n+s}(-3s)$ [13]. The following vector fields

[TABLE]

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

[TABLE]

[TABLE]

(see [13]). Let $\gamma:I\rightarrow\mathbb{R}^{2n+s}(-3s)$ be a unit-speed slant curve with contact angle $\theta$. Let us denote

[TABLE]

where $t$ is the arc-length parameter. Then $\gamma$ has the tangent vector field

[TABLE]

which can be written as

[TABLE]

Since $\gamma$ is slant curve, we have

[TABLE]

for all $\alpha=1,...,s$. So, we obtain

[TABLE]

Since $\gamma$ is a unit-speed, we can write

[TABLE]

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 $\gamma:I\rightarrow\mathbb{R}^{2n+s}(-3s)$ is a normal magnetic curvature. From the Lorentz equation, we have

[TABLE]

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

[TABLE]

and

[TABLE]

From equations (4.3), (4) and (4.5), we have

[TABLE]

which is equivalent to

[TABLE]

where we denote $\lambda=-q+2s\cos\theta$. Firstly, let us assume $\lambda\neq 0$. From equation (4.6), if we select pairs

[TABLE]

solving ODEs, we have

[TABLE]

where $c_{1},...,c_{n}$ are arbitrary constants. Thus, we can write

[TABLE]

where $f_{1},...,f_{n}$ are differentiable functions on $I$. From (4.6) and (4.7), we find

[TABLE]

which gives us

[TABLE]

where $a_{1},...,a_{n}$ are arbitrary constants. Now, if we integrate (4.7), we have

[TABLE]

[TABLE]

where $b_{i}$ and $d_{i}$ are arbitrary constants $(i=1,...,n)$. Thus, we get

[TABLE]

Using the last equation with (4.1), we obtain

[TABLE]

where $\alpha=1,...,s$. If we integrate this last equation, we find

[TABLE]

for $\alpha=1,...,s$ and $h_{1},...,h_{s}$ are arbitrary constants. Moreover, from (4.2) and (4.7), we have

[TABLE]

Thus, we have just finished the case $\lambda\neq 0$.

Secondly, let $\lambda=0$. In this case, we have

[TABLE]

which gives us

[TABLE]

for $i=1,...,2n,$ where $c_{i}$ and $d_{i}$ are arbitrary constants. Using the last equation, we calculate

[TABLE]

So, equation (4.1) becomes

[TABLE]

which gives us

[TABLE]

where $h_{\alpha}$ are arbitrary constants for $\alpha=1,...,s$. Since $\gamma$ is unit-speed, from (4.2), we have

[TABLE]

To sum up, we give the following Theorem:

Theorem 4**.**

The slant normal magnetic curves on $\mathbb{R}^{2n+s}(-3s)$ satisfying the Lorentz equation $\nabla_{T}T=-q\varphi T$ have the parametric equations

a)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $a_{i},$ $b_{i},$ $c_{i},$ $d_{i}$ and $h_{\alpha}$ are arbitrary constants such that $c_{i}$ satisfies

[TABLE]

or

b)

[TABLE]

[TABLE]

[TABLE]

where $c_{i},$ $d_{i}$ and $h_{\alpha}$ are arbitrary constants such that $c_{i}$ satisfies

[TABLE]

In both cases, $q\neq 0$ is a constant and $\theta$ denotes the constant contact angle satisfying $\left|\cos\theta\right|\leq\frac{1}{\sqrt{s}}.$

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

Acknowledgements. This work is financially supported by Balikesir Research Grant no. BAP 2018/016.