Magnetic Curves in Homothetic s-th Sasakian Manifolds

Abstract

We investigate normal magnetic curves in $(2n+s)$-dimensional homothetic s-th Sasakian manifolds as a generalization of S-manifolds. We show that a curve $\gamma$ is a normal magnetic curve in a homothetic s-th Sasakian manifold if and only if its osculating order satisfies $r\le 3$ and it belongs to a family of ${\theta }_i$-slant helices. Additionally, we construct a homothetic s-th Sasakian manifold using generalized D-homothetic transformations and present the parametric equations of normal magnetic curves in this manifold.

1. Introduction

Magnetic curves are trajectories of charged particles in a Riemannian or pseudo-Riemannian manifold $\left(M,g\right)$ under the influence of a magnetic field. Formally, they are critical points of a variational problem defined by a magnetic Lagrangian, which incorporates both the kinetic energy and the interaction with the magnetic field [1]. Let $\left(M,g\right)$ be a Riemannian manifold, and let F be a closed 2-form representing the magnetic field. A curve $\gamma :I\to M$ is called a magnetic curve if it satisfies the following equation:

$${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }=\mathrm{\Phi }\left({\gamma }^{\prime }\right),$$

(1)

where ∇ is the Levi-Civita connection of g and $\mathrm{\Phi }:\chi \left(M\right)\to \chi \left(M\right)$ is a $\left(1,1\right)$-type tensor field determined by $F,$ defined as [2,3,4]:

$$g(\mathrm{\Phi }U,V)=F(U,V),\forall U,V\in \chi \left(M\right).$$

(2)

In Equation (1), the left-hand term, ${\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime }$, represents the geodesic acceleration and the right-hand term, $\mathrm{\Phi }\left({\gamma }^{\prime }\right)$, represents the Lorentz force associated to the magnetic field. The equation itself is well-known as the Lorentz equation.

Magnetic curves generalize the concept of geodesics to include the effect of a magnetic field [5]. When $F=0$, the equation reduces to the geodesic equation, and $\gamma$ describes free motion in the manifold. Nonzero F introduces a deviation due to the magnetic force. From the preservation of energy, the speed $∥{\gamma }^{\prime }∥$ of a magnetic curve is constant because the magnetic force does no work [1]:

$${\displaystyle \frac{d}{dt}}{\displaystyle \frac{1}{2}}{∥{\gamma }^{\prime }∥}^2=g\left({\nabla }_{{\gamma }^{\prime }}{\gamma }^{\prime },{\gamma }^{\prime }\right)=g\left(\mathrm{\Phi }\left({\gamma }^{\prime }\right),{\gamma }^{\prime }\right)=F({\gamma }^{\prime },{\gamma }^{\prime })=0.$$

Moreover, if $∥{\gamma }^{\prime }∥=1$, $\gamma$ is called a normal magnetic curve [6].

In ref. [7], Nakagawa introduced the concept of framed f-structures, extending the idea of almost contact structures. Later, in ref. [8], Hasegawa, Okuyama, and Abe defined the notion of p-th Sasakian manifolds, providing illustrative examples to deepen the understanding of these structures. In [9], Alegre, Fernandez, and Prieto-Martin introduced a new class of metric f-manifolds, expanding the study of almost contact metric structures. Their work explored the foundational properties of these manifolds and provided several examples to demonstrate their geometric significance.

Subsequently, Adachi [2] explored the bounds of curvature and the behavior of magnetic trajectories on Hadamard surfaces. His findings revealed that under certain curvature constraints, every normal trajectory in a 2-dimensional, complete, and simply connected Riemannian manifold extends unboundedly in both directions.

In the realm of contact geometry, Baikoussis and Blair [10] investigated Legendre curves in 3-dimensional contact manifolds, demonstrating that the torsion of these curves is invariably 1 in Sasakian manifolds. Building on these foundations, Cho, Inoguchi, and Lee [11] defined and studied slant curves. Extending this idea, the first author introduced ${\theta }_i$-slant curves in S-manifolds [12], broadening the framework with innovative examples and applications in framed metric f-structures.

Cabrerizo, Fernandez, and Gomez [13] developed an elegant approach for constructing almost contact metric structures compatible with given metrics on 3-dimensional oriented Riemannian manifolds. Subsequently, Druta-Romaniuc et al. [6] investigated magnetic trajectories in Sasakian $(2n+1)$-manifolds under contact magnetic fields $F_q=q\omega$, where $\omega$ is the fundamental 2-form. Their research paved the way for further explorations, such as particle trajectories in cosymplectic manifolds [14] and closed magnetic paths on 3-dimensional Berger spheres [15]. In ref. [16], Jleli, Munteanu, and Nistor advanced these studies by examining magnetic curves in almost contact metric manifolds and concluded that normal magnetic curves correspond to helices of order 5 or less.

In para-Kaehler manifolds, Jleli and Munteanu [17] analyzed spacelike and timelike normal magnetic trajectories associated to para-Kaehler 2-forms, establishing their circular nature. Their earlier works [18,19] provided classifications of unit-speed Killing magnetic curves and examined normal magnetic trajectories on Sasakian spheres $S^{2n+1}$, showing their restriction to totally geodesic subspheres $S^3$. This line of research culminated in a study of closed normal trajectories on 3-dimensional tori derived from various contact forms on ${\mathbb{E}}^3$ [20].

Further developments included the introduction of T-magnetic, N-magnetic, and B-magnetic curves in 3-dimensional semi-Riemannian manifolds [21], as well as the classification of magnetic trajectories generated by Killing vector fields in normal paracontact metric 3-manifolds [22]. The second author also contributed by examining magnetic curves in the 3-dimensional Heisenberg group [23]. More recently, the present authors focused on slant magnetic curves in S-manifolds [24], delving into their geometric characteristics under specific magnetic influences. These contributions highlight the interplay between curvatures and contact structures in shaping the behavior of magnetic curves. For a deeper understanding of the foundational concepts underlying these advancements, readers are encouraged to consult [25,26,27].

Motivated by recent studies, this paper investigates normal magnetic curves within the context of $(2n+s)$-dimensional homothetic s-th Sasakian manifolds, which serve as a generalization of S-manifolds. We obtain that a curve $\gamma$ qualifies as a normal magnetic curve in a homothetic s-th Sasakian manifold if and only if its osculating order satisfies $r\le 3$ and it belongs to a family of ${\theta }_i$-slant helices. Moreover, we construct a homothetic s-th Sasakian manifold using generalized D-homothetic transformations and provide the parametric equations describing normal magnetic curves within this manifold.

2. Preliminaries

A Riemannian manifold $\left(M,g\right)$ is called a homothetic s-th Sasakian manifold if it satisfies the following properties for all $U,V\in \chi \left(M\right)$:

$$f^{2}U=-U+\stackrel{s}{\sum _{i=1}}{\eta }_i\left(U\right){\xi }_i,$$

(3)

$${\eta }_i\left({\xi }_j\right)={\delta }_{ij},f{\xi }_i=0,{\eta }_i\left(fU\right)=0,{\eta }_i\left(U\right)=g\left(U,{\xi }_i\right),$$

$$g(fU,fV)=g(U,V)-\stackrel{s}{\sum _{i=1}}{\eta }_i\left(U\right){\eta }_i\left(V\right),$$

(4)

$$d{\eta }_i\left(U,V\right)=-d{\eta }_i\left(V,U\right)={\alpha }_{i}g(U,fV),$$

where f is a $\left(1,1\right)$-type tensor field, ${\xi }_i$$\left(i=1,2,\dots ,s\right)$ are Killing characteristic vector fields, ${\eta }_i$$\left(j=1,2,\dots ,s\right)$ are 1-forms, ${\alpha }_i$$\left(i=1,2,\dots ,s\right)$ are nonzero constants, and this f-structure is normal [8]. It is denoted in short by $M=(M^{2n+s},f,{\xi }_i,{\eta }_i,g).$ If ${\alpha }_i=1$$\left(i=1,2,\dots ,s\right),$ then M becomes an S-manifold. It is important to mention that these manifolds are a subclass of trans-S-manifolds [9]. In a homothetic s-th Sasakian manifold, we have

$$\left({\nabla }_{U}f\right)V=\underset{i=1}{\sum ^s}{\alpha }_i\left\{g(fU,fV){\xi }_i+{\eta }_i\left(V\right)f^{2}U\right\},$$

and

$${\nabla }_U{\xi }_i=-{\alpha }_{i}fU,i\in \left\{1,\dots ,s\right\},$$

where ∇ is the Levi-Civita connection associated to g. The fundamental 2-form on M is given by

$$\omega (U,V)=g(U,fV).$$

(5)

One can easily show that $\omega$ is closed because

$$d\omega =d\left({\displaystyle \frac{1}{{\alpha }_i}}d{\eta }_i\right)={\displaystyle \frac{1}{{\alpha }_i}}{d}^2{\eta }_i=0.$$

As a result, we can define the magnetic field $F_q$ with strength q as

$$F_q(U,V)=q\omega (U,V),$$

where $U,V\in \chi \left(M\right)$, and q is a real constant [16]. By using Equations (2) and (5), the Lorentz force $\mathrm{\Phi }$ is calculated as

$${\mathrm{\Phi }}_q=-qf.$$

Consequently, we can rewrite the Lorentz equation in (1) as

$${\nabla }_{T}T=-qfT,$$

(6)

where $\gamma :I\to M$ is an arc-length parameterized smooth curve, and $T={\gamma }^{\prime }$ (see [6,16]).

3. Main Results

Let $\left(M,g\right)$ be a Riemannian manifold and $\gamma :I\to M$ a smooth curve. Then, the set of vector fields $\left\{T=E_1,E_2,\dots ,E_r\right\}$ is called the Frenet frame field of $\gamma ,$ which satisfies

$$\begin{array}{ccc}\hfill T& =& E_1={\gamma }^{\prime },\hfill \\ \hfill {\nabla }_{T}T& =& {\kappa }_1{E}_2,\hfill \\ \hfill {\nabla }_T{E}_2& =& -{\kappa }_{1}T+{\kappa }_2{E}_3,\hfill \\ & & \dots \hfill \\ \hfill {\nabla }_T{E}_j& =& -{\kappa }_{j-1}{E}_{j-1}+{\kappa }_j{E}_{j+1},\left(2<j<r\right),\hfill \\ & & \dots \hfill \\ \hfill {\nabla }_T{E}_r& =& -{\kappa }_{r-1}{E}_{r-1},\hfill \end{array}$$

(7)

where ∇ denotes the Levi-Civita connection. In this case, we call the positive integer $r\le n$ the osculating order and ${\kappa }_1,{\kappa }_2,\dots ,{\kappa }_{r-1}$ the curvatures of $\gamma$. Consequently, $\gamma$ is called a Frenet curve of osculating order r.

Curves are classified depending on their curvatures as follows: A Frenet curve of osculating order $r=1$ is a geodesic. A Frenet curve of osculating order $r=2$ with constant curvature ${\kappa }_1$ is a circle. A Frenet curve of osculating order $r\le n$ with constant curvatures ${\kappa }_1,{\kappa }_2,\dots ,{\kappa }_{r-1}$ is a helix of order $r.$ We call a helix of order $r=3$ shortly as helix.

Let $M=(M^{2n+s},f,{\xi }_i,{\eta }_i,g)$ be a homothetic s-th Sasakian manifold and $\gamma :I\to M$ a unit-speed curve. We call the functions ${\theta }_i={\theta }_i\left(t\right)$ the contact angles between T and  ${\xi }_i$, that is,

$$cos{\theta }_i\left(t\right)=g(T,{\xi }_i).$$

$\gamma$ is called a ${\theta }_i$-slant curve if all ${\theta }_i$ are constants. If these constant contact angles are all equal to the same value, we call $\gamma$ a slant curve. Additionally, if the contact angles are all equal to ${\displaystyle \frac{\pi }{2}},$ then it is called a Legendre curve and it becomes a 1-dimensional integral submanifold of the contact distribution (see [12]).

For a ${\theta }_i$-slant curve of osculating order r in a homothetic s-th Sasakian manifold, the following calculations are direct:

$$\begin{array}{ccc}\hfill {\nabla }_{T}fT& =& \left({\nabla }_{T}f\right)T+f{\nabla }_{T}T\hfill \\ & =& \left(1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i\right)\left({\displaystyle \sum _{i=1}^s}{\alpha }_i{\xi }_i\right)\hfill \\ & & +\left({\displaystyle \sum _{i=1}^s}{\alpha }_{i}cos{\theta }_i\right)\left(-T+{\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_i\right)+{\kappa }_{1}f{E}_2\hfill \end{array}$$

and

$${\nabla }_T{\xi }_i=-{\alpha }_{i}fT,i=1,2,\dots ,s.$$

By differentiating ${\eta }_i\left(T\right)=cos{\theta }_i,$ we find

$${\eta }_i\left(E_2\right)=0,i=1,2,\dots ,s.$$

Now we can state our first proposition, constructing a one-way bridge from normal magnetic curves to ${\theta }_i$-slant curves:

Proposition 1.

Let $(M^{2n+s},f,{\xi }_i,{\eta }_i,g)$ be a homothetic s-th Sasakian manifold and consider the contact magnetic field $F_q$ for $q\ne 0$. If $\gamma :I\to M$ is a normal magnetic curve associated to $F_q$ in M, then its contact angles are constants, i.e., γ is a ${\theta }_i$-slant curve.

Proof. 

Let $\gamma :I\to M$ be a normal magnetic curve associated to $F_q$ in $M.$ Then, using

$${\nabla }_{T}T=-qfT,$$

we obtain

$$\begin{array}{ccc}\hfill g\left({\nabla }_{T}T,{\xi }_i\right)& =& g\left(-qfT,{\xi }_i\right)=0\hfill \\ & =& {\displaystyle \frac{d}{dt}}g\left(T,{\xi }_i\right)-g\left(T,{\nabla }_T{\xi }_i\right)\hfill \\ & =& {\displaystyle \frac{d}{dt}}g\left(T,{\xi }_i\right)-g\left(T,-{\alpha }_{i}fT\right)\hfill \\ & =& {\displaystyle \frac{d}{dt}}g\left(T,{\xi }_i\right).\hfill \end{array}$$

As a result, we have

$$g\left(T,{\xi }_i\right)={\eta }_i\left(T\right)=cos{\theta }_i=\mathrm{constant}.$$

□

After this proposition, we can present the following theorem, which is the main theorem of the paper.

Theorem 1.

Let $(M^{2n+s},f,{\xi }_i,{\eta }_i,g)$ be a homothetic s-th Sasakian manifold and consider the contact magnetic field $F_q$ for $q\ne 0$. Then γ is a normal magnetic curve associated to $F_q$ in M if and only if γ belongs to the following list:

(a) 

non-Legendre ${\theta }_i$-slant geodesics as integral curves of ${\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_i,$ where ${\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i=1;$

(b) 

non-Legendre ${\theta }_i$-slant circles with the curvature

$${\kappa }_1=\sqrt{q^2-\stackrel{s}{\sum _{i=1}}{\alpha }_i^2},$$

having contact angles

$${\theta }_i=arccos\left({\displaystyle \frac{{\alpha }_i}{q}}\right),i=1,2,\dots ,s,$$

and the Frenet frame field

$$\left\{T,{\displaystyle \frac{-qfT}{\sqrt{q^2-{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}}}\right\},$$

where $\left|q\right|>\sqrt{{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}$;

(c) 

Legendre helices with curvatures ${\kappa }_1=\left|q\right|$ and ${\kappa }_2=\sqrt{{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}$, having the Frenet frame field

$$\left\{T,-sgn\left(q\right)fT,{\displaystyle \frac{-sgn\left(q\right)}{\sqrt{{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}}}{\displaystyle \sum _{i=1}^s}{\alpha }_i{\xi }_i\right\};$$

(d) 

${\theta }_i$-slant helices with

$${\kappa }_1=\left|q\right|\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i},$$

$${\kappa }_2=\sqrt{\mathrm{\Lambda }};$$

having the Frenet frame field

$$\left\{T,{\displaystyle \frac{-sgn\left(q\right)fT}{\sqrt{1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i}}},E_3\right\};$$

where we denote

$$\begin{array}{ccc}\hfill \mathrm{\Lambda }& =& \left({\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i\right)q^2-2\left({\displaystyle \sum _{i=1}^s}{\alpha }_{i}cos{\theta }_i\right)q+{\left(\stackrel{s}{\sum _{i=1}}{\alpha }_{i}cos{\theta }_i\right)}^2\hfill \\ & & +\left(1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i\right)\left({\displaystyle \sum _{i=1}^s}{\alpha }_i^2\right),\hfill \end{array}$$

(8)

$$E_3={\displaystyle \frac{-sgn\left(q\right)}{\sqrt{\mathrm{\Lambda }}\sqrt{1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i}}}\left[\begin{array}{c}\left(q{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i-{\displaystyle \sum _{i=1}^s}{\alpha }_{i}cos{\theta }_i\right)T+\left(1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i\right)\left({\displaystyle \sum _{i=1}^s}{\alpha }_i{\xi }_i\right)\\ +\left(-q+{\displaystyle \sum _{i=1}^s}{\alpha }_{i}cos{\theta }_i\right)\left({\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_i\right)\end{array}\right]$$

(9)

and ${\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i<1.$

Proof. 

Let $\gamma$ be a normal magnetic curve for $F_q$ in $M.$ Then $\gamma$ is a ${\theta }_i$-slant curve with constant contact angles ${\theta }_i,$ $i=1,2,\dots ,s$. From Frenet equations and the Lorentz equation, we find

$${\nabla }_{T}T={\kappa }_1{E}_2=-qfT.$$

If ${\kappa }_1=0,$ then $-qfT=0$ gives us $fT=0.$ Thus, we find

$$f^{2}T=-T+{\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_1=0,$$

which results in

$$T={\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_1.$$

From $g\left(T,T\right)=1$, we obtain ${\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i=1.$ Hence, $\gamma$ belongs to (a) in the list.

Let ${\kappa }_1\ne 0.$ In the expression ${\kappa }_1{E}_2=-qfT$, by taking the norm of both sides, we find

$$\begin{array}{ccc}\hfill {\kappa }_1& =& ∥-qfT∥=\left|q\right|.∥fT∥\hfill \\ & =& \left|q\right|\sqrt{g\left(fT,fT\right)}\hfill \\ & =& \left|q\right|\sqrt{g\left(T,T\right)-\stackrel{s}{\sum _{i=1}}{\left[{\eta }_i\left(T\right)\right]}^2}\hfill \\ & =& \left|q\right|\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i},\hfill \end{array}$$

(10)

which is a constant.

As a result, we can write

$$fT=-sgn\left(q\right)\sqrt{1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i}{E}_2.$$

(11)

If ${\kappa }_2=0,$ then $\gamma$ is a circle, since it has constant ${\kappa }_1.$ For ${\theta }_i$-slant curves, differentiating ${\eta }_i\left(T\right)=cos{\theta }_i=$ constant, we also have ${\eta }_i\left(E_2\right)=0.$ If we differentiate once again,

$$\begin{array}{ccc}\hfill {\nabla }_T{\eta }_i\left(E_2\right)& =& 0\hfill \\ & =& g\left({\nabla }_T{E}_2,{\xi }_i\right)+g\left(E_2,{\nabla }_T{\xi }_i\right)\hfill \\ & =& g\left(-{\kappa }_{1}T,{\xi }_i\right)+g\left(E_2,-{\alpha }_{i}fT\right)\hfill \\ & =& -\left|q\right|\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i}cos{\theta }_i+{\alpha }_i.sgn\left(q\right)\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i}\hfill \\ & =& sgn\left(q\right)\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i}\left(-qcos{\theta }_i+{\alpha }_i\right).\hfill \end{array}$$

Thus, we deduce that $-qcos{\theta }_i+{\alpha }_i=0,$ or else ${\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i=1$ would be the same as (a) in the list and $\gamma$ would be a geodesic. This gives us

$${\theta }_i=arccos\left({\displaystyle \frac{{\alpha }_i}{q}}\right),i=1,2,\dots ,s.$$

From the fact that ${\alpha }_i>0,$$\gamma$ cannot be Legendre. So, ${\kappa }_1$ becomes

$$\begin{array}{ccc}\hfill {\kappa }_1& =& \left|q\right|\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i}\hfill \\ & =& \left|q\right|\sqrt{1-\stackrel{s}{\sum _{i=1}}{\displaystyle \frac{{\alpha }_i^2}{q^2}}}\hfill \\ & =& \sqrt{q^2-\stackrel{s}{\sum _{i=1}}{\alpha }_i^2}>0,\hfill \end{array}$$

that is, $\left|q\right|>\sqrt{{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}.$ In this case, $\gamma$ belongs to (b) in the list.

Let ${\kappa }_2\ne 0.$ For ${\theta }_i$-slant curves, we calculate

$$f^{2}T=-T+{\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_i$$

(12)

and

$$\begin{array}{ccc}\hfill {\nabla }_{T}fT& =& \left(1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right)\left(\stackrel{s}{\sum _{i=1}}{\alpha }_i{\xi }_i\right)\hfill \\ & & +\left(-q+\stackrel{s}{\sum _{i=1}}{\alpha }_{i}cos{\theta }_i\right)\left(-T+\stackrel{s}{\sum _{i=1}}cos{\theta }_i{\xi }_i\right).\hfill \end{array}$$

(13)

If we differentiate (11), we also have

$$\begin{array}{ccc}\hfill {\nabla }_{T}fT& =& -sgn\left(q\right)\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i}{\nabla }_T{E}_2\hfill \\ & =& -sgn\left(q\right)\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i}\left(-{\kappa }_{1}T+{\kappa }_2{E}_3\right)\hfill \\ & =& q\left(1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right)T-{\kappa }_{2}sgn\left(q\right)\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i}{E}_3.\hfill \end{array}$$

(14)

From Equations (13) and (14), it follows that

$$\begin{array}{ccc}\hfill -{\kappa }_{2}sgn\left(q\right)\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i}{E}_3& =& \left(q\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i-\stackrel{s}{\sum _{i=1}}{\alpha }_{i}cos{\theta }_i\right)T+\left(1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right)\left(\stackrel{s}{\sum _{i=1}}{\alpha }_i{\xi }_i\right)\hfill \\ & & +\left(-q+\stackrel{s}{\sum _{i=1}}{\alpha }_{i}cos{\theta }_i\right)\left(\stackrel{s}{\sum _{i=1}}cos{\theta }_i{\xi }_i\right).\hfill \end{array}$$

By taking the norm of both sides, we find ${\kappa }_2=\sqrt{\mathrm{\Lambda }},$ where $\mathrm{\Lambda }$ is given in (8) and $E_3$ is given in (14). Notice that ${\kappa }_2$ is also a constant. Thus, $\gamma$ is a ${\theta }_i$-slant helix that belongs to (d) in the list.

Let us consider the Legendre case separately. In this case, since $cos{\theta }_i=0,$ $\forall i,$ we have

$${\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i=0,{\displaystyle \sum _{i=1}^s}{\alpha }_{i}cos{\theta }_i=0,\stackrel{s}{\sum _{i=1}}cos{\theta }_i{\xi }_i=0,$$

which gives us

$${\kappa }_1=\left|q\right|,{\kappa }_2=\sqrt{{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}$$

using (10) and calculating $\mathrm{\Lambda }={\displaystyle \sum _{i=1}^s}{\alpha }_i^2$. Thus, $\gamma$ is a Legendre helix that belongs to (c) in the list. One can easily see that $E_3\in sp\left\{T,{\xi }_1,{\xi }_2,\dots ,{\xi }_s\right\}$ and the coefficients are constants. If we write

$$E_3=c_{0}T+{\displaystyle \sum _{i=1}^s}{c}_i{\xi }_i$$

for some constants $c_0,\dots ,c_s,$ we obtain

$$\begin{array}{ccc}\hfill {\nabla }_T{E}_3& =& -{\kappa }_2{E}_2+{\kappa }_3{E}_4\hfill \\ & =& c_0{\nabla }_{T}T+{\displaystyle \sum _{i=1}^s}{c}_i{\nabla }_T{\xi }_i\hfill \\ & =& c_0{\kappa }_1{E}_2-\left(\stackrel{s}{\sum _{i=1}}{c}_i{\alpha }_i\right)fT.\hfill \end{array}$$

For a normal magnetic curve, since $fT\Vert E_2,$ we deduce that ${\kappa }_3=0$. So, the list is complete. □

The proof of this theorem also leads to a remarkable result that bounds the osculating order, making its inclusion here both meaningful and well-placed.

Corollary 1.

The osculating order of a normal magnetic curve in a homothetic s-th Sasakian manifold is at most 3.

Proof. 

From the previous proof, The Gram–Schmidt process definitively concludes after we differentiate $E_3$ and find ${\kappa }_3=0$, if it has not already. If it concludes earlier, it would imply $r<3$. In either case, r cannot exceed 3. □

In the next proposition, we present a nice result for Legendre helices in homothetic s-th Sasakianmanifolds:

Proposition 2.

Let γ be a unit-speed Legendre helix of order 3 with $fT\Vert E_2$ in a homothetic s-th Sasakian manifold $(M^{2n+s},f,{\xi }_i,{\eta }_i,g)$. Then, we have

$${\kappa }_2=\sqrt{\underset{i=1}{\sum ^s}{\alpha }_i^2},E_2=\pm fT,E_3={\displaystyle \frac{\pm 1}{\sqrt{{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}}}\underset{i=1}{\sum ^s}{\alpha }_i{\xi }_i.$$

Proof. 

Since $\gamma$ is a Legendre helix, we have $cos{\theta }_i=0,$ $i=1,2,\dots ,s$, and ${\kappa }_1,$${\kappa }_2$ are constants. $fT\Vert E_2$ gives us

$$fT=\lambda E_2$$

for some real valued differentiable function $\lambda$. Taking the norm of both sides, we have

$$\sqrt{g\left(fT,fT\right)}=\left|\lambda \right|∥E_2∥,$$

which is calculated as

$$\sqrt{g\left(T,T\right)-\underset{i=1}{\sum ^s}{cos}^2{\theta }_i}=\left|\lambda \right|.$$

So, we obtain $\lambda =\pm 1$ and

$$E_2=\pm fT.$$

(15)

If we differentiate Equation (15), we find

$$\begin{array}{ccc}\hfill -{\kappa }_{1}T+{\kappa }_2{E}_3& =& \pm {\nabla }_{T}fT\hfill \\ & =& \pm \left(\underset{i=1}{\sum ^s}{\alpha }_i{\xi }_i+{\kappa }_{1}f{E}_2\right).\hfill \end{array}$$

(16)

From Equation (15), if we apply $f,$ we obtain

$$f{E}_2=\pm f^{2}T=\pm \left(-T+\underset{i=1}{\sum ^s}cos{\theta }_i{\xi }_i\right)=\mp T.$$

(17)

Equations (16) and (17) give us

$$\begin{array}{ccc}\hfill -{\kappa }_{1}T+{\kappa }_2{E}_3& =& \pm \left(\underset{i=1}{\sum ^s}{\alpha }_i{\xi }_i\mp {\kappa }_{1}T\right)\hfill \\ & =& \pm {\displaystyle \sum _{i=1}^s}{\alpha }_i{\xi }_i-{\kappa }_{1}T,\hfill \end{array}$$

that is,

$${\kappa }_2{E}_3=\pm \underset{i=1}{\sum ^s}{\alpha }_i{\xi }_i.$$

The norm of this last equation concludes

$${\kappa }_2=\sqrt{\underset{i=1}{\sum ^s}{\alpha }_i^2}.$$

Then, $E_3$ becomes

$$E_3={\displaystyle \frac{\pm 1}{\sqrt{{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}}}{\displaystyle \sum _{i=1}^s}{\alpha }_i{\xi }_i.$$

□

With the following theorem, we provide the criteria for the contact angles and the strength of the magnetic field that determine when ${\theta }_i$-slant helices with $fT\Vert E_2$ will be normal magnetic curves:

Theorem 2.

Let γ be a unit-speed ${\theta }_i$-slant helix of order $r\le 3$ satisfying $fT\Vert E_2$, with given curvatures ${\kappa }_1,$${\kappa }_2$ and contact angles ${\theta }_i$$\left(i=1,2,\dots ,s\right)$ in a homothetic s-th Sasakian manifold $(M^{2n+s},f,{\xi }_i,{\eta }_i,g)$. Then,

(i) 

If ${\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i=1$, then γ is a geodesic as an integral curve of ${\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_i$; therefore, it is a normal magnetic curve for $F_q$ with any q.

(ii) 

If ${\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i=0$ and ${\kappa }_1\ne 0$ (namely, γ is a non-geodesic Legendre curve), then γ is a normal magnetic curve for $F_{\mp {\kappa }_1}$.

(iii) 

If $cos{\theta }_i={\displaystyle \frac{\epsilon {\alpha }_i}{\sqrt{{\kappa }_1^2+{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}}},$$\left(i=1,2,\dots ,s\right)$, then γ is a normal magnetic curve for $F_{\epsilon \sqrt{{\kappa }_1^2+{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}}$, where $\epsilon =-sgn\left(g(fT,E_2)\right)$. Given this situation, γ is a ${\theta }_i$-slant circle.

(iv) 

If

$$\underset{i=1}{\sum ^s}{cos}^2{\theta }_i=1-{\displaystyle \frac{{\kappa }_1^2}{{\mathrm{\Psi }}^2}},$$

(18)

then γ is a normal magnetic curve for $F_{\epsilon \mathrm{\Psi }}$, where we denote $\epsilon =-sgn\left(g(fT,E_2)\right)$ and

$$\mathrm{\Psi }=\sqrt{\begin{array}{c}{\left({\kappa }_1+\epsilon {\displaystyle \frac{{\displaystyle \sum _{i=1}^s}{\alpha }_{i}cos{\theta }_i}{\sqrt{1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i}}}\right)}^2+{\kappa }_2^2+\left({\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i-2\right){\displaystyle \frac{{\left({\displaystyle \sum _{i=1}^s}{\alpha }_{i}cos{\theta }_i\right)}^2}{1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i}}\\ +\left({\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i-1\right){\displaystyle \sum _{i=1}^s}{\alpha }_i^2\end{array}}.$$

(19)

(v) 

If none of the above is satisfied, γ is not qualified as a normal magnetic curve for any $F_q.$

Proof. 

Since $\gamma$ is a unit-speed ${\theta }_i$-slant helix, it is given that $cos{\theta }_i$$\left(i=1,2,\dots ,s\right)$ and ${\kappa }_1,$${\kappa }_2$ are all constants. Furthermore, from $fT\Vert E_2$, we can write

$$fT=\lambda E_2$$

(20)

for some differentiable function $\lambda$. Equation (20) gives us

$$g\left(fT,fT\right)={\lambda }^{2}g\left(E_2,E_2\right),$$

which is equivalent to

$$\lambda =\pm \sqrt{1-\underset{i=1}{\sum ^s}{cos}^2{\theta }_i}.$$

As a result, we obtain

$$fT=-\epsilon \sqrt{1-\underset{i=1}{\sum ^s}{cos}^2{\theta }_i}{E}_2,$$

(21)

where we denote $\epsilon =-sgn\left(g(fT,E_2)\right).$

(i) If ${\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i=1,$ then Equation (21) becomes

$$fT=0.$$

Applying $f,$ we find

$$f^{2}T=-T+\underset{i=1}{\sum ^s}cos{\theta }_i{\xi }_i=0,$$

that is, $T={\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_i.$ Then, we calculate

$$\begin{array}{ccc}\hfill {\nabla }_{T}T& =& {\nabla }_{\left({\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_i\right)}\left(\underset{k=1}{\sum ^s}cos{\theta }_k{\xi }_k\right)\hfill \\ & =& \underset{i,k=1}{\sum ^s}cos{\theta }_{i}cos{\theta }_k{\nabla }_{{\xi }_i}{\xi }_k=0\hfill \\ & =& {\kappa }_1{E}_2.\hfill \end{array}$$

Hence, we obtain ${\kappa }_1=0,$ i.e., $\gamma$ is a geodesic. We also have ${\nabla }_{T}T=0=-qfT$ for any q. Thus, $\gamma$ is a normal magnetic curve for $F_q$ with any $q.$

(ii) If ${\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i=0$ and ${\kappa }_1\ne 0$, then $cos{\theta }_i=0,$$\left(i=1,2,\dots ,s\right);$ that is, $\gamma$ is a Legendre curve. From Proposition 2, we have $E_2=\pm fT.$ Using Frenet equations, we calculate

$${\nabla }_{T}T={\kappa }_1{E}_2=\pm {\kappa }_{1}fT.$$

As a result, $\gamma$ becomes a normal magnetic curve for $F_{\mp {\kappa }_1}.$

(iii) Let

$$cos{\theta }_i={\displaystyle \frac{\epsilon {\alpha }_i}{\sqrt{{\kappa }_1^2+{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}}},\left(i=1,2,\dots ,s\right).$$

So, we find

$${\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i={\displaystyle \frac{{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}{{\kappa }_1^2+{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}},$$

$$1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i={\displaystyle \frac{{\kappa }_1^2}{{\kappa }_1^2+{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}},$$

and

$$\sqrt{1-\underset{i=1}{\sum ^s}{cos}^2{\theta }_i}={\displaystyle \frac{{\kappa }_1}{\sqrt{{\kappa }_1^2+{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}}}.$$

From Equation (21), we can write

$$fT={\displaystyle \frac{-\epsilon {\kappa }_1}{\sqrt{{\kappa }_1^2+{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}}}{E}_2.$$

Then, it is easy to see that

$$\begin{array}{ccc}\hfill {\nabla }_{T}T& =& {\kappa }_1{E}_2\hfill \\ & =& -\epsilon \sqrt{{\kappa }_1^2+\underset{i=1}{\sum ^s}{\alpha }_i^2}fT.\hfill \end{array}$$

As a result, $\gamma$ becomes a normal magnetic curve for $F_{\epsilon \sqrt{{\kappa }_1^2+{\displaystyle \sum _{i=1}^s}{\alpha }_i^2}}.$ Additionally, if we use Equation (8) for $q=\epsilon \sqrt{{\kappa }_1^2+{\displaystyle \sum _{i=1}^s}{\alpha }_i^2},$ we find ${\kappa }_2=\sqrt{\mathrm{\Lambda }}=0.$ So, $\gamma$ is a ${\theta }_i$-slant circle.

(iv) Finally, let

$$\underset{i=1}{\sum ^s}{cos}^2{\theta }_i=1-{\displaystyle \frac{{\kappa }_1^2}{{\mathrm{\Psi }}^2}},$$

where $\mathrm{\Psi }$ is as given in (19). From Equation (21), we have

$$\begin{array}{ccc}\hfill fT& =& -\epsilon \sqrt{1-\left(1-{\displaystyle \frac{{\kappa }_1^2}{{\mathrm{\Psi }}^2}}\right)}{E}_2\hfill \\ & =& -\epsilon {\displaystyle \frac{{\kappa }_1}{\mathrm{\Psi }}}{E}_2.\hfill \end{array}$$

Using the Frenet equations, we can write

$$\begin{array}{ccc}\hfill {\nabla }_{T}T& =& {\kappa }_1{E}_2={\kappa }_1\left(-\epsilon {\displaystyle \frac{\mathrm{\Psi }}{{\kappa }_1}}\right)fT\hfill \\ & =& -\epsilon \mathrm{\Psi }fT.\hfill \end{array}$$

As a result, $\gamma$ becomes a normal magnetic curve for $F_{\epsilon \mathrm{\Psi }}.$ Now, let us see how $\mathrm{\Psi }$ is calculated. Here, our aim is to determine $q=\epsilon \mathrm{\Psi }$ in terms of ${\kappa }_1$ and ${\kappa }_2.$ We have already shown that $\gamma$ is a normal magnetic curve for $F_{\epsilon \mathrm{\Psi }}.$ So, we can use Theorem 1 (d). We can write

$${\kappa }_1=\left|q\right|\sqrt{1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i},$$

(22)

$${\kappa }_2=\sqrt{\mathrm{\Lambda }}.$$

(23)

Equation (22) can be rewritten as

$$\left(\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right)q^2=q^2-{\kappa }_1^2.$$

(24)

Then, Equation (23) gives us

$$\begin{array}{ccc}\hfill {\kappa }_2^2& =& \mathrm{\Lambda }=\left(\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right)q^2-2\left(\stackrel{s}{\sum _{i=1}}{\alpha }_{i}cos{\theta }_i\right)q+{\left(\stackrel{s}{\sum _{i=1}}{\alpha }_{i}cos{\theta }_i\right)}^2\hfill \\ & & +\left(1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right)\left(\stackrel{s}{\sum _{i=1}}{\alpha }_i^2\right),\hfill \\ & =& \left(q^2-{\kappa }_1^2\right)-2\left(\stackrel{s}{\sum _{i=1}}{\alpha }_{i}cos{\theta }_i\right)q+{\left(\stackrel{s}{\sum _{i=1}}{\alpha }_{i}cos{\theta }_i\right)}^2\hfill \\ & & +\left(1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right)\left(\stackrel{s}{\sum _{i=1}}{\alpha }_i^2\right).\hfill \end{array}$$

(25)

Now, we assign

$${\displaystyle \sum _{i=1}^s}{\alpha }_{i}cos{\theta }_i=\sqrt{1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i}D$$

for some constant D. From Equation (22), we also know that

$$\epsilon {\kappa }_1=q\sqrt{1-{\displaystyle \sum _{i=1}^s}{cos}^2{\theta }_i},$$

where $\epsilon =-sgn\left(g(fT,E_2)\right).$ We rearrange the terms in (25) as

$$\begin{array}{ccc}\hfill -2\left(\stackrel{s}{\sum _{i=1}}{\alpha }_{i}cos{\theta }_i\right)q& =& -2\sqrt{1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i}Dq,\hfill \\ & =& -2\epsilon {\kappa }_{1}D\hfill \end{array}$$

(26)

and

$${\left({\displaystyle \sum _{i=1}^s}{\alpha }_{i}cos{\theta }_i\right)}^2=\left(1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right)D^2.$$

(27)

Finally, we write (26) and (27) in (25). Then, after completing the square to obtain ${\left({\kappa }_1+\epsilon D\right)}^2,$ we leave the first term $q^2$ on the right-hand side of (25). As a result, we find $\mathrm{\Psi }$ as given in (19).

Since the list in Theorem 1 includes all cases where $\gamma$ is a normal magnetic curve, then there does not exist any other normal magnetic curve in M, as stated in (v). □

4. Parameterization of Magnetic Curves in ${\mathbb{R}}^{\mathbf{2}\mathit{n}+\mathit{s}}$ as a Homothetic $\mathit{s}$-th Sasakian Manifold

${\mathbb{R}}^{2n+s}\left(-3s\right)$ is a well-known S-manifold [8], which is a specific kind of trans-S-manifold with ${\alpha }_i=1,$ ${\beta }_i=0,$ $i=1,2,\dots ,s$ [9]. Using generalized D-homothetic transformations, from Theorem 4.4 of [9], we can produce the following homothetic s-th Sasakian manifold using the structures of ${\mathbb{R}}^{2n+s}\left(-3s\right)$. The newly generated manifold is denoted in short by $M=({\mathbb{R}}^{2n+s},f,{\xi }_i,{\eta }_i,g).$

Let us consider ${\mathbb{R}}^{2n+s}$ and its coordinate functions $\left\{x_1,\dots ,x_n,y_1,\dots ,y_n,z_1,\dots ,z_s\right\}.$ Let a and b be positive real numbers. One can define

$${\xi }_i={\displaystyle \frac{2}{a}}{\displaystyle \frac{\partial }{\partial z_i}},i=1,\dots ,s,$$

$${\eta }_i={\displaystyle \frac{a}{2}}\left(d{z}_i-\stackrel{n}{\sum _{j=1}}{y}_{j}d{x}_j\right),i=1,\dots ,s,$$

$$fU=\stackrel{n}{\sum _{j=1}}{V}_j{\displaystyle \frac{\partial }{\partial x_j}}-\stackrel{n}{\sum _{j=1}}{U}_j{\displaystyle \frac{\partial }{\partial y_j}}+\left(\stackrel{n}{\sum _{j=1}}{V}_j{y}_j\right)\left({\displaystyle \sum _{i=1}^s}{\displaystyle \frac{\partial }{\partial z_i}}\right),$$

$$g=\stackrel{s}{\sum _{i=1}}{\eta }_i\otimes {\eta }_i+{\displaystyle \frac{b}{4}}\stackrel{n}{\sum _{j=1}}\left(d{x}_j\otimes d{x}_j+d{y}_j\otimes d{y}_j\right),$$

where

$$U=\stackrel{n}{\sum _{j=1}}\left(U_j{\displaystyle \frac{\partial }{\partial x_j}}+V_j{\displaystyle \frac{\partial }{\partial y_j}}\right)+\stackrel{s}{\sum _{i=1}}\left(W_i{\displaystyle \frac{\partial }{\partial z_i}}\right)\in \chi \left(M\right).$$

Let us denote $M=({\mathbb{R}}^{2n+s},f,{\xi }_i,{\eta }_i,g)$. Using Theorem 4.4 of [9], M becomes a trans-S-manifold with

$${\alpha }_i={\displaystyle \frac{a}{b}},{\beta }_i=0,i=1,2,\dots ,s,$$

that is, M is a homothetic s-th Sasakian manifold. The vector fields

$$U_j={\displaystyle \frac{2}{\sqrt{b}}}{\displaystyle \frac{\partial }{\partial y_j}},U_{n+j}=f{U}_j={\displaystyle \frac{2}{\sqrt{b}}}\left({\displaystyle \frac{\partial }{\partial x_j}}+y_j{\displaystyle \sum _{i=1}^s}{\displaystyle \frac{\partial }{\partial z_i}}\right),{\xi }_i={\displaystyle \frac{2}{a}}{\displaystyle \frac{\partial }{\partial z_i}}$$

are g-orthonormal. The Riemannian connection associated to g can be calculated directly from equation (4.16) of [9] as

$${\nabla }_{U_j}{U}_k={\nabla }_{U_{n+j}}{U}_{n+k}=0,{\nabla }_{U_j}{U}_{n+k}={\displaystyle \frac{a}{b}}{\delta }_{jk}\stackrel{s}{\sum _{i=1}}{\xi }_i,{\nabla }_{U_{n+j}}{U}_k={\displaystyle \frac{-a}{b}}{\delta }_{jk}\stackrel{s}{\sum _{i=1}}{\xi }_i,$$

$${\nabla }_{U_j}{\xi }_i={\nabla }_{{\xi }_i}{U}_j={\displaystyle \frac{-a}{b}}{U}_{n+j},{\nabla }_{U_{n+j}}{\xi }_i={\nabla }_{{\xi }_i}{U}_{n+j}={\displaystyle \frac{a}{b}}{U}_j.$$

Now, we can give the following theorem:

Theorem 3.

Normal magnetic curves on $M=({\mathbb{R}}^{2n+s},f,{\xi }_i,{\eta }_i,g)$ that satisfy the Lorentz equation ${\nabla }_{T}T=-qfT$ are described by the following parametric equations:

(a) 

$${\gamma }_j\left(t\right)={\displaystyle \frac{c_j}{-\lambda }}sin{f}_j\left(t\right)+b_j,$$

(28)

$${\gamma }_{n+j}\left(t\right)={\displaystyle \frac{c_j}{\lambda }}cos{f}_j\left(t\right)+d_j,$$

(29)

$${\gamma }_{2n+i}\left(t\right)={\displaystyle \frac{2cos{\theta }_i}{a}}t-\stackrel{n}{\sum _{j=1}}\left\{{\displaystyle \frac{c_j^2}{4{\lambda }^2}}\left[sin\left(2f_j\left(t\right)\right)+2f_j\left(t\right)\right]+{\displaystyle \frac{c_j{d}_j}{\lambda }}sin{f}_j\left(t\right)\right\}+h_i,$$

(30)

$$f_j\left(t\right)=-\lambda t+a_j,$$

$$i=1,\dots ,s,j=1,2,\dots ,n,$$

$$\lambda =-q+{\displaystyle \frac{2a}{b}}\left(\stackrel{s}{\sum _{i=1}}cos{\theta }_i\right)\ne 0,$$

where $a_j,$ $b_j,$ $c_j,$ $d_j$, and $h_i$ are arbitrary constants such that $c_j$ satisfies

$$\stackrel{n}{\sum _{j=1}}{c}_j^2={\displaystyle \frac{4}{b}}\left(1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right);$$

(31)

or

(b) 

$${\gamma }_j\left(t\right)=c_{j}t+d_j,$$

$${\gamma }_{n+j}\left(t\right)=c_{n+j}t+d_{n+j},$$

$${\gamma }_{2n+i}\left(t\right)={\displaystyle \frac{2cos{\theta }_i}{a}}t+\stackrel{n}{\sum _{j=1}}{c}_j\left({\displaystyle \frac{c_{n+j}}{2}}{t}^2+d_{n+j}t\right)+h_i,$$

$$i=1,\dots ,s,j=1,\dots ,n,$$

$$q={\displaystyle \frac{2a}{b}}\left(\stackrel{s}{\sum _{i=1}}cos{\theta }_i\right),$$

where $c_j,$ $c_{n+j},$ $d_j,$ $d_{n+j}$, and $h_i$ are arbitrary constants such that $c_j$ and $c_{n+j}$ satisfy

$$\stackrel{n}{\sum _{j=1}}\left(c_j^2+c_{n+j}^2\right)={\displaystyle \frac{4}{b}}\left(1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right).$$

Proof. 

Let $\gamma :I\to M$ be a normal magnetic curve. Then, Proposition 1 gives us that $\gamma$ is a ${\theta }_i$-slant curve. Moreover, from Corollary 1, its osculating order is at most 3. Let

$$\gamma \left(t\right)=\left({\gamma }_1\left(t\right),\dots ,{\gamma }_n\left(t\right),{\gamma }_{n+1}\left(t\right),\dots ,{\gamma }_{2n}\left(t\right),{\gamma }_{2n+1}\left(t\right),\dots ,{\gamma }_{2n+s}\left(t\right)\right)$$

denote the parameterization of $\gamma ,$ where t is the arc-length parameter. As a result, its tangential vector field T becomes

$$T=\stackrel{n}{\sum _{j=1}}{\gamma }_j^{\prime }{\displaystyle \frac{\partial }{\partial x_j}}+\stackrel{n}{\sum _{j=1}}{\gamma }_{n+j}^{\prime }{\displaystyle \frac{\partial }{\partial y_j}}+{\displaystyle \sum _{i=1}^s}{\gamma }_{2n+i}^{\prime }{\displaystyle \frac{\partial }{\partial z_i}}.$$

It is more useful to write T in terms of the g-orthonormal basis as

$$\begin{array}{ccc}\hfill T& =& {\displaystyle \frac{\sqrt{b}}{2}}\left(\stackrel{n}{\sum _{j=1}}{\gamma }_{n+j}^{\prime }{U}_j+\stackrel{n}{\sum _{j=1}}{\gamma }_j^{\prime }{U}_{n+j}\right)\hfill \\ & & +{\displaystyle \frac{a}{2}}\left[\stackrel{s}{\sum _{i=1}}\left({\gamma }_{2n+i}^{\prime }-\stackrel{n}{\sum _{j=1}}{\gamma }_j^{\prime }{\gamma }_{n+j}\right){\xi }_i\right].\hfill \end{array}$$

Using the fact that $\gamma$ is a ${\theta }_i$-slant curve, we obtain

$${\eta }_i\left(T\right)=cos{\theta }_i,i=1,2,\dots ,s,$$

which is equivalent to

$${\gamma }_{2n+i}^{\prime }={\displaystyle \frac{2cos{\theta }_i}{a}}+\stackrel{n}{\sum _{j=1}}{\gamma }_j^{\prime }{\gamma }_{n+j}.$$

(32)

We also have $g\left(T,T\right)=1,$ so one can easily calculate

$$\stackrel{n}{\sum _{j=1}}{\left({\gamma }_j^{\prime }\right)}^2+\stackrel{n}{\sum _{j=1}}{\left({\gamma }_{n+j}^{\prime }\right)}^2={\displaystyle \frac{4}{b}}\left(1-\stackrel{s}{\sum _{i=1}}{cos}^2{\theta }_i\right).$$

(33)

So now, we need the Lorentz equation

$${\nabla }_{T}T=-qfT$$

(34)

to be satisfied by $\gamma ,$ since it is a normal magnetic curve. ${\nabla }_{T}T$ and $fT$ can be calculated as

$${\nabla }_{T}T={\displaystyle \frac{\sqrt{b}}{2}}\left\{\begin{array}{c}{\displaystyle \sum _{j=1}^n}\left[{\gamma }_{n+j}^{\prime \prime }+{\displaystyle \frac{2a}{b}}\left({\displaystyle \sum _{i=1}^s}cos{\theta }_i\right){\gamma }_j^{\prime }\right]U_j\\ +{\displaystyle \sum _{j=1}^n}\left[{\gamma }_j^{\prime \prime }-{\displaystyle \frac{2a}{b}}\left({\displaystyle \sum _{i=1}^s}cos{\theta }_i\right){\gamma }_{n+j}^{\prime }\right]U_{n+j}\end{array}\right\},$$

(35)

and

$$fT={\displaystyle \frac{\sqrt{b}}{2}}\left[\stackrel{n}{\sum _{j=1}}\left(-{\gamma }_j^{\prime }\right)U_j+\stackrel{n}{\sum _{j=1}}{\gamma }_{n+j}^{\prime }{U}_{n+j}\right].$$

(36)

From Equation (34), we deduce that ${\nabla }_{T}T\Vert$$fT$, i.e., the corresponding coefficients of their unique representations in terms of the g-orthonormal basis must be proportional, and it is easy to see that the proportionality constant is $\left(-q\right)$. Simplifying by canceling $\sqrt{b}/2,$ we obtain

$${\displaystyle \frac{{\gamma }_{n+j}^{\prime \prime }+\frac{2a}{b}\left({\displaystyle \sum _{i=1}^s}cos{\theta }_i\right){\gamma }_j^{\prime }}{-{\gamma }_j^{\prime }}}={\displaystyle \frac{{\gamma }_j^{\prime \prime }-\frac{2a}{b}\left({\displaystyle \sum _{i=1}^s}cos{\theta }_i\right){\gamma }_{n+j}^{\prime }}{{\gamma }_{n+j}^{\prime }}}=-q,j=1,2,\dots ,n,$$

which can be rewritten as

$${\displaystyle \frac{{\gamma }_{n+j}^{\prime \prime }}{-{\gamma }_j^{\prime }}}={\displaystyle \frac{{\gamma }_j^{\prime \prime }}{{\gamma }_{n+j}^{\prime }}}=\lambda ,j=1,2,\dots ,n,$$

(37)

where we denote the corresponding proportionality constant by $\lambda$ as

$$\lambda =-q+{\displaystyle \frac{2a}{b}}\left(\stackrel{s}{\sum _{i=1}}cos{\theta }_i\right).$$

Firstly, let $\lambda \ne 0.$ Then we have the following ODEs from (37) for all $j=1,2,\dots ,n$:

$${\gamma }_{n+j}^{\prime \prime }{\gamma }_{n+j}^{\prime }+{\gamma }_j^{\prime \prime }{\gamma }_j^{\prime }=0,$$

which are integrated to find

$${\left({\gamma }_j^{\prime }\right)}^2+{\left({\gamma }_{n+j}^{\prime }\right)}^2=c_j^2,$$

for some arbitrary constant $c_j$. These circular equations are to be solved by using

$${\gamma }_j^{\prime }=c_{j}cos{f}_j,{\gamma }_{n+j}^{\prime }=c_{j}sin{f}_j,$$

(38)

where $f_j:I\to \mathbb{R},$ $\left(j=1,2,\dots ,s\right)$ are functions of the arc-length parameter t. From Equations (37) and (38), we find

$$f_j^{\prime }=-\lambda ,$$

that is,

$$f_j\left(t\right)=-\lambda t+a_i,$$

for some arbitrary constant $a_i$. If we replace $f_j$ in (38) and integrate, we obtain (28) and (29). Then, we use these in (32) and find (30). Equation (31) is calculated by using the fact that $\gamma$ is unit-speed, i.e., $g\left(T,T\right)=1.$ The proof of (a) is now complete.

Finally, let $\lambda =0.$ Following the same procedure, we obtain the linear equations of ${\gamma }_j,$${\gamma }_{n+j}$, and the parabolic equations of ${\gamma }_{2n+i}$ as given in (b). In fact, from Equation (37) and $\lambda =0,$ we obtain

$${\gamma }_j^{\prime \prime }={\gamma }_{n+j}^{\prime \prime }=0,$$

from which the proof of (b) is straightforward. □

We conclude our study with two explicit examples:

Example 1.

Let $n=1$, $s=2$, $a=2$ and $b=4$. Then $\gamma :I\to M,$

$$\gamma \left(t\right)=\left({\displaystyle \frac{1}{2}}t,{\displaystyle \frac{1}{2}}t,{\displaystyle \frac{1}{8}}{t}^2+{\displaystyle \frac{1}{2}}t,{\displaystyle \frac{1}{8}}{t}^2+{\displaystyle \frac{\sqrt{2}}{2}}t\right)$$

is a normal magnetic curve for $F_{{\displaystyle \frac{1+\sqrt{2}}{2}}}.$ It is a ${\theta }_i$-slant curve with contact angles ${\theta }_1={\displaystyle \frac{\pi }{3}}$ and ${\theta }_2={\displaystyle \frac{\pi }{4}}.$ It satisfies Theorem 3 (b).

Example 2.

Let $n=1$, $s=3$, $a=3$ and $b=2.$ Then $\gamma :I\to M,$

$$\gamma \left(t\right)=\left(sint,-cost,{\displaystyle \frac{-1}{6}}t-{\displaystyle \frac{1}{4}}sin2t,{\displaystyle \frac{-5}{6}}t-{\displaystyle \frac{1}{4}}sin2t,{\displaystyle \frac{-1}{2}}t-{\displaystyle \frac{1}{4}}sin2t\right)$$

is a normal magnetic curve for $F_1.$ It is a ${\theta }_i$-slant curve with contact angles ${\theta }_1={\displaystyle \frac{\pi }{3}}$, ${\theta }_2={\displaystyle \frac{2\pi }{3}}$, and ${\theta }_3={\displaystyle \frac{\pi }{2}}.$ It satisfies Theorem 3 (a).

5. Discussion

If we select ${\alpha }_i=1,$ $i=1,2,\dots ,s,$ then a homothetic s-th Sasakian manifold becomes an S-manifold. Therefore, our new results not only complete our previous study [24] for normal magnetic curves in S-manifolds when the contact angles do not necessarily need to be equal but also generalize those prior results to a broader class of manifolds. Our parameterization Theorem 3 can be considered in a similar manner, since $a=b=1$ gives $M={\mathbb{R}}^{2n+s}\left(-3s\right)$, and then $\gamma$ represents all of the normal magnetic curves in ${\mathbb{R}}^{2n+s}\left(-3s\right)$ without any requirement on the contact angles.