Magnetic Curves in Generalized Almost Cosymplectic Manifolds

Abstract

In this paper, we study magnetic curves associated with a contact magnetic field on almost cosymplectic manifolds. In particular, we consider locally conformal almost cosymplectic manifolds and almost $\alpha$-cosymplectic f-manifolds. The magnetic field is defined by the fundamental 2-form $\Omega$, with the corresponding Lorentz force determined by the structure tensor $\phi$. We classify normal magnetic curves in these settings and show that they are either geodesics, Legendre $\phi$-circles, or $\phi$-helices of order three. We further investigate slant curves and derive conditions relating their geometry to the structure functions of the manifold. These results generalize known classifications of magnetic curves in contact and cosymplectic geometries and provide a unified treatment for the considered classes of manifolds.

1. Introduction

The concept of a magnetic curve $\left({\mathcal{M}}_c\right)$ within the framework of a Riemannian manifold $(M,g)$ have been extensively studied in the literature. A closed 2-form F on a smooth manifold M is referred to as a magnetic field $\left(\mathcal{MF}\right)$. The Lorentz force $\left({\mathcal{L}}_f\right)$ generated by F is defined by the endomorphism field $\phi$ associated with it. The Lorentz (or Newton) equation is described as

$${\nabla }_{\dot{\gamma }}\dot{\gamma }=q\phi \dot{\gamma },$$

(1)

where q denotes the field strength, and ∇ represents the Levi–Civita connection.

A curve $\gamma$ that satisfies Lorentz equation is called ${\mathcal{M}}_c$ [1]. Therefore, it follows that ${\mathcal{M}}_c$ corresponds to the path of a charged particle under the influence of ${\mathcal{L}}_f$ generated by F in the framework $(M,g,F)$. When

$$F=0,$$

which denotes the absence of ${\mathcal{L}}_f$, the paths of the particles are geodesics. Thus, ${\mathcal{M}}_c$ can be regarded as a broader concept of geodesics. Moreover, their study can be viewed as the analysis of a nonlinear dynamical system determined by the underlying geometric structure.

During the early 1990s, there were notable advancements in the study of ${\mathcal{M}}_c$ in arbitrary Riemannian manifolds $(M,g)$, although foundational studies have been conducted much earlier. Notably, Arnold’s examinations of charges in a $\mathcal{MF}$ F across various dimensions of $(M,g)$ are significant, as highlighted in [2], along with additional resources cited therein. Adachi’s work in [3] explored the paths of charged particles influenced by F linked to the Kähler form on complex projective spaces. Moreover, how these paths compare to geodesics on the Kähler manifold with negative curvature is given in [4]. In three-dimensional space, the authors analyzed ${\mathcal{M}}_c$ related to a Killing $\mathcal{MF}$ [5,6]. Cabrerizo, in [7], demonstrated that ${\mathcal{M}}_c$ lines form helices in the presence of a Killing vector field in ${\mathbb{S}}^3$. The authors in [8] offered a local classification of Killing ${\mathcal{M}}_c$ in the product space ${\mathbb{S}}^2\times \mathbb{R}$, describing their behavior under different field configurations.

In a three-dimensional framework, the relationship between the $\mathcal{MF}$ and an almost contact metric manifold $\left(\mathcal{ACMM}\right)$ is understood as follows. Let $(M,g)$ possess a volume form $d{v}_g$. The Hodge star operator (*) serves as a connector by transforming the space of all smooth 2-forms into the space of all smooth vector fields. Consider F on $\left(M,g,d{v}_g\right)$, where V denotes the divergence-free vector field, and let $\eta$ be the 1-form metrically dual to V. When V has unit length, it is apparent that $(\phi ,V,\eta )$ constitutes an almost contact structure $\left(\mathcal{ACS}\right)$ on M, congruent with g. In essence, $(M,g,F)$, consisting of $(M,g)$ and F with an equivalent unit length V, qualifies as an $\mathcal{ACMM}$ with a closed fundamental 2-form [6]. This realization shifts the concern towards analyzing ${\mathcal{M}}_c$ associated with F in $\mathcal{ACMM}$.

Within a contact 3-manifold, $\gamma$ is identified as a slant curve $\left({\mathcal{S}}_c\right)$ when its tangent vector has a constant angle with the Reeb vector field $\xi$. ${\mathcal{S}}_c$ is inevitably encountered in the study of Sasakian 3-manifolds within differential geometry. In [9], the authors investigated ${\mathcal{S}}_c$ under the condition of having a proper mean curvature vector field in the contexts of a 3-dimensional f-Kenmotsu manifold and hyperbolic space ${\mathbb{H}}^3$. Furthermore, in [10], ${\mathcal{S}}_c$ is examined in a 3D framework almost f-Kenmotsu. Inoguchi further identified the essential and adequate conditions for a non-geodesic ${\mathcal{S}}_c$ in the same setting to have a proper mean curvature vector field. In [11], the authors investigated ${\mathcal{S}}_c$ in Sasakian 3-manifolds, proving that a non-geodesic curve is ${\mathcal{S}}_c$ iff its ratio of curvature $\left(\kappa \right)$ to torsion $\left(\tau \right)$ is constant.

The complete stratification of $\mathcal{ACMM}$ was achieved in [12]. Among these, the quasi-Sasakian manifolds form a crucial class, presented by Blair in his doctoral thesis [13], they are defined as normal $\mathcal{ACMM}$ possessing a closed fundamental 2-form. The quasi-Sasakian rank, which depends on the powers of $\eta$ and its wedge product with $\eta$ (see [13]), underscores their role as the nearest equivalents of a Kähler manifold in the context of $\mathcal{ACS}$.

Research on ${\mathcal{M}}_c$ in the context of Kähler geometry initially focused on 2-dimensional $(M,g)$, treated as a 1D complex case. Here, the $\mathcal{MF}$ F is often expressed as a fixed, non-zero scaling of the area element. Sunada’s work [14] explored ${\mathcal{M}}_c$ on ${\mathbb{H}}^2$ and compact Riemannian surfaces of genus $\ge 2$ (related discussions appear in [15]). A key scenario arises when F is specified by a scaled Kähler form, termed a Kähler $\mathcal{MF}$. Adachi established early findings in non-flat space forms [3,4,16], noting in [16] that such trajectories form circular paths. Independently, Kalinin [17] studied charged particle motion under this field in constant holomorphic sectional curvature settings using dynamical systems theory. Observations of ${\mathcal{M}}_c$ on this ambient manifold revealing circular trajectories suggest that exploring this analogous phenomenon on the quasi-Sasakian manifold is pertinent.

The Sasakian setting uniquely features a Killing unit vector field $\xi$, which naturally describes a contact $\mathcal{MF}$ F. Studies on such fields in three dimensions were conducted in [18,19]. Subsequently, a complete classification of ${\mathcal{M}}_c$ associated with F was provided in [20], revealing four different trajectory types.

A similar structure arises in cosymplectic geometry, where a closed fundamental 2-form induces F. Here, the parallelism of $\phi$ ensures a parallel ${\mathcal{L}}_f$, leading to a uniform contact $\mathcal{MF}$. Investigations of the broader behavior of ${\mathcal{M}}_c$ in this framework were carried out in [21].

Increasing research attention has been devoted to ${\mathcal{M}}_c$ in generalized geometric settings. For example, Güvenç and Özgür [22] investigated normal ${\mathcal{M}}_c$ in homothetic s-th Sasakian manifolds, showing that such curves are characterized by an osculating order ($r\le 3$). They also constructed homothetic s-th Sasakian manifolds and provided explicit parametric equations for ${\mathcal{M}}_c$.

Furthermore, Calışkan et al. [23] studied ${\mathcal{M}}_c$ in dual space by introducing dual magnetic trajectories via the dual ${\mathcal{L}}_f$. They defined dual-flux ruled surfaces and characterized the conditions under which such surfaces are minimal or developable.

Previous studies on magnetic and slant curves have been conducted in various geometric settings, including Sasakian, quasi-Sasakian, and cosymplectic manifolds. However, these structures belong to different classes of almost contact metric manifolds and are not all contained within almost cosymplectic geometry. In this paper, we restrict our attention to two classes within almost cosymplectic geometry, namely locally conformal almost cosymplectic manifolds $\left(\mathcal{LCACM}\right)$ and almost $\alpha$-cosymplectic f-manifolds $\left(\mathcal{A}\alpha \mathcal{C}f\mathcal{M}\right)$, and investigate magnetic and slant curves in these settings.

In this study, we investigate ${\mathcal{M}}_c$ and ${\mathcal{S}}_c$ arising from a contact $\mathcal{MF}$ on two classes of manifolds, namely $\mathcal{LCACM}$ and $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$. Section 2 presents the necessary preliminaries. Section 3 and Section 4 are devoted to the study of these curves in $\mathcal{LCACM}$, while Section 5 and Section 6 treat the corresponding results in $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$. In both settings, we classify normal ${\mathcal{M}}_c$ into geodesics, Legendre $\phi$-circles, and $\phi$-helices of order three, and show that Legendre helices reduce to $\phi$-circles. We also derive relations connecting the curvature of these curves with the $\mathcal{MF}$ strength and the structural parameters of the manifolds.

2. Preliminaries

This section compiles the essential tools relevant to the study.

2.1. Locally Conformal Almost Cosymplectic Manifold

The concept of $\mathcal{LCACM}$ was first formally introduced and investigated by Z. Olszak in [24]. This seminal paper established the definition and likely explored some of the fundamental characteristics of these manifolds, making it a key starting point for understanding their geometry. Further geometric properties of these manifolds, including the study of normal structures, were developed by Kirichenko and Kharitonova in [25,26].

A smooth manifold $M^{2n+1}$ is classified as $\mathcal{LCACM}$ if its tangent bundle $TM$ admits a $(1,1)$-tensor field $\phi$, a vector field $\xi$, and a 1-form $\eta$, subject to the following conditions:

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{\phi }^2=-I+\eta \otimes \xi ,\hfill \\ \eta \left(\xi \right)=1,\hfill \\ \eta \circ \phi =0,\hfill \\ g(\phi X_1,\phi X_2)=g(X_1,X_2)-\eta \left(X_1\right)\eta \left(X_2\right),\hfill \\ \eta \left(X_1\right)=g(X_1,\xi ),\hfill \end{array}\right.\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \left({\nabla }_{X_1}\phi \right)X_2& =u\left(g(\phi X_1,X_2)\xi -\eta \left(X_2\right)\phi X_1\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\nabla }_{X_1}\xi & =u\left(X_1-\eta \left(X_1\right)\xi \right),\hfill \end{array}$$

(4)

where $X_1,X_2\in \Gamma \left(TM\right)$ and u is the conformal function such that $\omega =u\eta$ (see [24]). A 2-form $\Omega$ is provided by

$$\mathsf{\Omega }(X,Y)=g(X,\phi Y),$$

(5)

referred to as a fundamental 2-form.

If the conformal function takes the values $u=0$ or $u=1$, then in the former case, M is a cosymplectic manifold, while in the latter, it is a Kenmotsu manifold (for further details, see [27,28]).

2.2. Almost $\alpha$-Cosymplectic f-Manifold

Öztürk et al. developed the concept of an $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$ [29], applicable for any real value of $\alpha$. This concept pertains to a metric f-manifold characterized by $(\phi ,{\xi }_i,{\eta }_i,g)$ that meets the conditions

$$d{\eta }_i=0,$$

and

$$d\mathsf{\Omega }=2\alpha \overline{\eta }\wedge \mathsf{\Omega }.$$

Let M be a framed metric manifold with a dimension of $(2n+s)$. As per [30], it possesses $\left(\phi ,{\xi }_i,{\eta }^i,g\right)$, where i spans $\{1,\dots ,s\}$. Here, $\phi$ satisfies

$${\phi }^3+\phi =0,$$

and the rank of $\phi$ is $r=2n$. The ${\xi }_1,\dots ,{\xi }_s$ are vector fields, while ${\eta }^1,\dots ,{\eta }^s$ are 1-forms defined on M, such that

$$\begin{array}{cc}\hfill & {\phi }^2=-I+{\displaystyle \sum _{i=1}^s}{\eta }^i\otimes {\xi }_i,\hfill \\ \hfill & {\eta }^i\left({\xi }_j\right)={\delta }_j^i,{\eta }^i\circ \phi =0,\phi \left({\xi }_i\right)=0,\hfill \\ \hfill & {\eta }^i\left(X\right)=g\left(X,{\xi }_i\right),\hfill \\ \hfill & g(X,\phi Y)+g(\phi X,Y)=0,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & g(\phi X,\phi Y)=g(X,Y)-{\displaystyle \sum _{i=1}^s}{\eta }^i\left(X\right){\eta }^i\left(Y\right),\hfill \end{array}$$

(7)

for all $X,Y\in \Gamma \left(TM\right)$ and $j\in \{1,\dots ,s\}$. If M satisfies the above conditions, it is called a metric f-manifold, denoted by $\left(M,\phi ,{\xi }_i,{\eta }^i,g\right)$ [30]. According to [30], a framed metric structure is considered normal if

$$[\phi ,\phi ]+2d{\eta }^i\otimes {\xi }_i=0,$$

where $[\phi ,\phi ]$ denotes the Nijenhuis tensor field associated with $\phi$.

In this study, we define

$$\overline{\eta }={\eta }^1+{\eta }^2+\dots +{\eta }^s,$$

$$\overline{\xi }={\xi }_1+{\xi }_2+\dots +{\xi }_s,$$

and

$${\overline{\delta }}_i^j={\delta }_i^1+{\delta }_i^2+\dots +{\delta }_i^s.$$

From [29], the following definition is given:

Definition 1.

Let $\left(M,\phi ,{\xi }_i,{\eta }^i,g\right)$ be a manifold of dimension $(2n+s)$. If

$$d{\eta }^i=0,1\le i\le s,$$

and

$$d\mathsf{\Omega }=2\alpha \overline{\eta }\wedge \mathsf{\Omega },$$

then M is called an $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$.

Let M be an $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$ and let $D=ker\left(\overline{\eta }\right)={\bigcap }_{i=1}^{s}ker\left({\eta }_i\right)$ denote the distribution defined by the structure 1-forms. If D is integrable, then the following relations hold:

$$\left\{\begin{array}{c}{L}_{{\xi }_i}{\eta }^j=0,\hfill \\ \left[{\xi }_i,{\xi }_j\right]\in D,\hfill \\ \left[X,{\xi }_j\right]\in D,\forall X\in \Gamma \left(D\right).\hfill \end{array}\right.$$

The connection ∇ is given by

$$\begin{array}{c}\hfill 2g\left(\left({\nabla }_X\phi \right)Y,Z\right)=2\alpha {\displaystyle \sum _{i=1}^s}g\left(g(\phi X,Y){\xi }_i-{\eta }_i\left(Y\right)\phi X,Z\right)+g\left(N(Y,Z),\phi X\right),\end{array}$$

for any $X,Y\in \Gamma \left(TM\right)$, where N denotes the Nijenhuis tensor field.

For $X={\xi }_i$, we obtain

$${\nabla }_{{\xi }_i}\phi =0,$$

which implies ${\tilde{\nabla }}_{{\xi }_i}{\xi }_j\in D^{\perp }$ and therefore ${\nabla }_{{\xi }_i}{\xi }_j={\nabla }_{{\xi }_j}{\xi }_i$, since $\left[{\xi }_i,{\xi }_j\right]=0$. We define

$$A_{i}X=-{\nabla }_X{\xi }_i\mathrm{and}{h}_i={\displaystyle \frac{1}{2}}\left(L_{{\xi }_i}\phi \right),$$

where $L_{{\xi }_i}$ is the Lie derivative along to the vector ${\xi }_i$. If M has Kählerian leaves [31], then

$$\left({\nabla }_X\phi \right)Y={\displaystyle \sum _{i=1}^s}\left[-g\left(\phi A_{i}X,Y\right){\xi }_i+{\eta }^i\left(Y\right)\phi A_{i}X\right].$$

(8)

Proposition 1

([29]). The tensor field $A_i$ is a symmetric endomorphism that satisfies

(i) 

$A_i\left({\xi }_j\right)=0$ for all $i,j\in \{1,\dots ,s\}$,

(ii) 

$A_i\circ \phi +\phi \circ A_i=-2\alpha \phi$,

(iii) 

$tr\left(A_i\right)=-2\alpha n$,

(iv) 

${\nabla }_X{\xi }_i=-\alpha {\phi }^{2}X-\phi h_{i}X$.

Proposition 2

([32]). For each i, $j\in \{1,\dots ,s\}$, $h_i$ is a symmetric operator and satisfies the following:

(i) 

$h_i\left({\xi }_j\right)=0$,

(ii) 

$h_i\circ \phi +\phi \circ h_i=0$,

(iii) 

$tr\left(h_i\right)=0$,

(iv) 

$tr\left(\phi h_i\right)=0$.

2.3. Frenet Curves

Assume that

$$\gamma :I\to \left(M^3,g\right),$$

is parameterized by arc-length $(p.b.a.l)$ in the 3-dimensional manifold $M^3$, with Frenet frame field $(T,N,B)$. Here, T, N, and B denote the tangent, principal normal, and binormal vector fields, respectively.

These vectors satisfy the Frenet–Serret equations with respect to the connection ∇ on $\left(M^3,g\right)$:

$$\left\{\begin{array}{cc}\hfill {\nabla }_{T}T& =\kappa N,\hfill \\ \hfill {\nabla }_{T}N& =-\kappa T+\tau B,\hfill \\ \hfill {\nabla }_{T}B& =-\tau N,\hfill \end{array}\right.$$

(9)

where $\kappa =\left|{\nabla }_{T}T\right|$.

This framework can be extended to a general Riemannian manifold $(M,g)$ of any dimension. A Frenet curve of osculating order $r\ge 1$ (i.e., a curve whose first $r-1$ curvatures are nonzero) admits an orthonormal frame $\left\{T=\dot{\gamma },v_1,\dots ,v_{r-1}\right\}$ along $\gamma$ such that [33]

$$\left\{\begin{array}{c}{\nabla }_{T}T={\kappa }_1{v}_1,\hfill \\ {\nabla }_T{v}_1=-{\kappa }_{1}T+{\kappa }_2{v}_2,\hfill \\ {\nabla }_T{v}_j=-{\kappa }_j{v}_{j-1}+{\kappa }_{j+1}{v}_{j+1},\mathrm{for}j=2,\dots ,r-2,\hfill \\ {\nabla }_T{v}_{r-1}=-{\kappa }_{r-1}{v}_{r-2},\hfill \end{array}\right.$$

(10)

where ${\kappa }_1,{\kappa }_2,\dots ,{\kappa }_{r-1}={\kappa }_i$ are positive smooth functions with respect to s, and each ${\kappa }_j$ is known as the j-th curvature of $\gamma$.

A geodesic corresponds to a Frenet curve with $r=1$, a $\phi$-circle corresponds to $r=2$ with constant ${\kappa }_1$, and a $\phi$-helix of order r has all ${\kappa }_i$ constant.

Now, consider an $\mathcal{ACMM}$ $\left(M^{2n+1},\phi ,\xi ,\eta ,g\right)$, and let $\gamma$ be a Frenet curve of order r in $M^{2n+1}$. Then the following holds:

• For $r\ge 3$, $\gamma$ is called a $\phi$-curve if the subspace spanned by $\{T,v_1,\dots ,v_{r-1}\}$ is invariant under $\phi$.
• For $r=2$, $\gamma$ is a $\phi$-curve if the set $\left\{T,v_1,\xi \right\}$ spans a $\phi$-invariant space.
• A $\phi$-helix of order r is a $\phi$-curve of osculating order r with all ${\kappa }_i$ constant.

The $\phi$-torsions of $\gamma$ are defined by

$${\tau }_{ij}=g\left(\phi v_j,v_i\right),0\le i<j\le r-1,$$

where $v_0=T$, and $v_i=v_i$.

The contact angle $\theta$ of $\gamma$ is the angle between the tangent vector and the Reeb vector field $\xi ,$ given by

$$cos\theta \left(s\right)=g\left(\dot{\gamma }\left(s\right),{\xi }_{\mid \gamma \left(s\right)}\right),$$

where s is the arc length parameter of $\gamma$.

• If $\theta$ is constant, $\gamma$ is called ${\mathcal{S}}_c$.
• For $\theta ={\displaystyle \frac{\pi }{2}}$, ${\mathcal{S}}_c$ is commonly referred to as Legendre $\phi$-curve $\left({\mathcal{L}}_c\right)$.
• For $\theta =0$, ${\mathcal{S}}_c$ is the Reeb flow.

3. Magnetic Curves in a Locally Conformal Almost Cosymplectic Manifold

In physics, a ${\mathcal{M}}_c$ represents the path of a charged particle moving in a Riemannian manifold $(M,g)$ under the influence of a $\mathcal{MF}$ F.

On $(M,g)$, F is defined as a closed 2-form, and the corresponding ${\mathcal{L}}_f$ is represented by the endomorphism $\phi ,$ such that

$$F(X,Y)=g(\phi X,Y),X,Y\in \Gamma \left(TM\right).$$

(11)

A curve $\gamma$ in M that satisfies

$${\nabla }_{\dot{\gamma }}\dot{\gamma }=\phi \left(\dot{\gamma }\right),$$

(12)

is called a ${\mathcal{M}}_c$ of the $\mathcal{MF}$ F.

Equation (12) describes the dynamics of a charged particle under the effect of ${\mathcal{L}}_f$. The curve $\gamma$ must have constant velocity from the skew-symmetry of $\phi$, meaning the particle maintains constant energy along its path.

Unlike the geodesic equation ${\nabla }_{\dot{\gamma }}\dot{\gamma }=0$, the presence of the term $\phi \left(\dot{\gamma }\right)$ introduces curvature-dependent trajectories. The ${\mathcal{M}}_c$ $\gamma \left(s\right)$ is referred to as a normal ${\mathcal{M}}_c$ when it is $p.b.a.l$.

The equation of geodesics $p.b.a.l$:

$${\nabla }_{\dot{\gamma }}\dot{\gamma }=0,$$

is generalized by (12). In this case, ∇ is linked to g.

If

$$\nabla F=0,$$

then F is said to be uniform.

Now, consider an $\mathcal{ACMM}$ $(M^{2n+1},\phi ,\xi ,\eta ,g)$ with the fundamental 2-form $\mathsf{\Omega }$ defined in (5). If $M^{2n+1}$ is a $\mathcal{LCACM}$, then $\mathsf{\Omega }$ is closed, and $F_q$ can be defined on $M^{2n+1}$ by

$$F_q(X,Y)=q\mathsf{\Omega }(X,Y),$$

(13)

for $X,Y\in \Gamma \left(T{M}^{2n+1}\right)$, where $q\ne 0$. We refer to $F_q$ as the contact $\mathcal{MF}$.

$F_q=q\mathsf{\Omega }$ is consistent with the general ${\mathcal{L}}_f$ expression given in (11), due to the skew-symmetry of $\phi$. In fact,

$$F_q(X,Y)=q\mathsf{\Omega }(X,Y)=qg(X,\phi Y)=-qg(\phi X,Y)=-F(X,Y).$$

The negative sign appears due to the skew symmetry of $\phi$, and the sign convention is chosen so that a positive q aligns with the physical interpretation of a charged particle under ${\mathcal{L}}_f$.

Moreover, since $\nabla \phi =0$, then $F_q$ is uniform. The term contact is used here without risk of ambiguity, because the geometric setting is that of a $\mathcal{LCACM}$ $\left(M^{2n+1},\phi ,\xi ,\eta ,g\right)$, which is structurally distinct from the classical contact case.

Equations (5) and (11) can be used to find the expression of the ${\mathcal{L}}_f$, which is

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

As a result, (1) is provided by

$${\nabla }_{\dot{\gamma }}\dot{\gamma }=q\phi \dot{\gamma }.$$

(14)

The Lorentz equation illustrates the acceleration of a charged particle under the influence of $F_q$.

A smooth curve $p.b.a.l$ is represented as

$$\gamma :I\subseteq \mathbb{R}\to M^{2n+1}.$$

The normal ${\mathcal{M}}_c$ for $F_q$ are defined as the solutions of (14) and their categorization in the context of a $\mathcal{LCACM}$ will be presented in the following result.

Theorem 1.

Consider a $\mathcal{LCACM}$ endowed with a $\mathcal{MF}$ $F_q$, where $q\ne 0$. Then a curve γ is a normal ${\mathcal{M}}_c$ relative to $F_q$ if and only if it belongs to one of the following types:

(i) Geodesics represented as integral curve $\left({\mathcal{I}}_c\right)$ associated with ξ.

(ii) Legendre φ-circles with ${\kappa }_1=\left|q\right|$.

(iii) φ-helices of order $r=3$, satisfying

$${\kappa }_1=\left|q\right|sin\theta ,{\kappa }_2=sgn\left(q\right)(qcos\theta -usin\theta ),$$

and

$$sgn\left({\tau }_{01}\right)=-sgn\left(q\right),$$

where $\theta \ne {\displaystyle \frac{\pi }{2}}$.

Proof. 

In Scenario (i), if the ${\mathcal{M}}_c$ $\gamma$ acts as a geodesic, then Equation (14) provides

$$\phi \dot{\gamma }=0,$$

leading to the result that $\gamma$ and $\xi$ are parallel. Given that both are unit vector field, we deduce

$$\dot{\gamma }=\pm \xi ,$$

confirming that $\gamma$ is an ${\mathcal{I}}_c$ of $\xi$.

In what follows, we examine non-geodesic ${\mathcal{M}}_c$, specifically the Frenet curve of $r>1.$ Let

$$\theta =\theta \left(s\right)\in (0,\pi ).$$

It may be demonstrated that $\theta$ remains constant. Therefore,

$$\begin{array}{cc}\hfill 0& =g(q\phi T,\xi )\hfill \\ \hfill & =g({\nabla }_{T}T,\xi )\hfill \\ \hfill & ={\displaystyle \frac{d}{ds}}g(T,\xi )-g(T,{\nabla }_T\xi ).\hfill \end{array}$$

Using (4), it follows that

$${\displaystyle \frac{d}{ds}}g(T,\xi )=0.$$

This implies that $\theta \in (0,\pi )$ within T and $\xi$ remains unchanged. Therefore, we can deduce

$$\eta \left(T\right)=cos\theta .$$

(15)

By integrating the initial of (10) with (14),

$$k_1{v}_1=q\phi T.$$

(16)

It follows that

$${\kappa }_1=\left|q\right|sin\theta .$$

(17)

Using (16) and (17), we obtain

$$\phi T=sgn\left(q\right)sin\theta v_1.$$

(18)

This curvature relation reveals that the particle’s trajectory bends in proportion to the field strength $\left|q\right|$ and $\theta$ between $\dot{\gamma }$ and $\xi$.

We derive the following by differentiating (16) covariantly concerning T and using (3).

$$sin\theta {\nabla }_T{v}_1=-sgn\left(q\right)u\cos \theta \phi T+\left|q\right|[cos\theta \xi -T].$$

(19)

It follows that ${\nabla }_T{v}_1$ is collinear to T if $\theta ={\displaystyle \frac{\pi }{2}}$, where $\gamma$ is a Legendre circle of ${\kappa }_1=\left|q\right|$. Thus, the Theorem’s case (ii) is established.

For ${\mathcal{M}}_c$ with constant $\theta \ne {\displaystyle \frac{\pi }{2}}$, we first establish ${\kappa }_1=\left|q\right|sin\theta$ directly from

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

and the norm

$$\parallel \phi T\parallel =sin\theta ,$$

${\kappa }_2$ is determined by differentiating

$$\phi T=sgn\left(q\right)sin\theta v_1,$$

covariantly, yielding two equivalent expressions for ${\nabla }_T\left(\phi T\right)$.

The Frenet approach gives

$${\nabla }_T\left(\phi T\right)=sgn\left(q\right)sin\theta (-{\kappa }_{1}T+{\kappa }_2{v}_2),$$

while the connection formula yields

$$-ucos\theta sgn\left(q\right)sin\theta v_1+{\kappa }_{1}sgn\left(q\right)(-sin\theta T+ϵcos\theta v_2).$$

Thus, we have

$$-ucos\theta sgn\left(q\right)v_1+k_{1}sgn\left(q\right)(-sin\theta T+ϵcos\theta v_2)=sgn\left(q\right)sin\theta (-k_{1}T+k_2{v}_2).$$

Projecting both onto $v_2$ (noting $v_2\perp v_1$) produces the key equality

$$sgn\left(q\right)sin\theta {\kappa }_2={\kappa }_{1}sgn\left(q\right)ϵcos\theta -sgn\left(q\right)usin\theta .$$

Substituting ${\kappa }_1=\left|q\right|sin\theta$ and solving, we obtain

$${\kappa }_2=ϵ\left|q\right|cos\theta -u=sgn\left(q\right)(qcos\theta -usin\theta ).$$

When $u\ne 0$, the term $-usin\theta$ arises from frame adaptation to balance the conformal distortion $-ucos\theta sin\theta v_1$ in the $v_1$ component. For the cosymplectic case ($u=0$), this reduces to

$${\kappa }_2=sgn\left(q\right)qcos\theta .$$

The Frenet frame of $\gamma$ can be used to express $\xi$ as

$$\xi =cos\theta T+\epsilon sin\theta v_2,$$

(20)

where $\epsilon =sgn(cos\theta )$. Afterwards,

$$\phi v_2=-sgn\left(q\right)\epsilon cos\theta v_1,\eta \left(v_2\right)=\epsilon sin\theta .$$

(21)

Accordingly, (15), (16), and (20) produce

$$\phi v_1=sgn\left(q\right)\left(-sin\theta T+\epsilon cos\theta v_2\right).$$

(22)

The torsion sign constraint follows from

$${\tau }_{01}=g(T,\phi v_1)=-sgn\left(q\right)sin\theta ,$$

ensuring

$$sgn\left({\tau }_{01}\right)=-sgn\left(q\right).$$

Finally,

$${\nabla }_T{v}_2=-{\kappa }_2{v}_1,$$

confirms the curve is a $\phi$-helix of $r=3$ with ${\kappa }_3=0$.

In summary, the normal ${\mathcal{M}}_c$ are Frenet curve of $r=3$, possessing constant

$${\kappa }_1=\left|q\right|sin\theta ,$$

$${\kappa }_2=sgn\left(q\right)(qcos\theta -usin\theta ).$$

Thus, (iii) has been demonstrated. □

In contrast, beginning with a $\phi$-helix $\gamma$ within a $\mathcal{LCACM}$ $\left(M^{2n+1},\phi ,\xi ,\eta ,g\right)$, we determine the prerequisites for which this curve acts as a ${\mathcal{M}}_c$ related to $F_q$ of specified strengths. Initially, we establish another result, which proves helpful in demonstrating the main theorem.

Proposition 3.

Let M be a $\mathcal{LCACM}$. If γ is a non-geodesic Legendre φ-helix in M, then γ is a φ-circle.

Proof. 

Since $\gamma$ is a ${\mathcal{L}}_c$, for such a Legendre-$\phi$ curve $\theta ={\displaystyle \frac{\pi }{2}}$, we have

$$\eta \left(T\right)=0.$$

Subsequently, by performing successive covariant differentiations pertaining to T and applying (10), we obtain from ${\nabla }_T\left(\eta \left(T\right)\right)=0$ and (4):

$$u+{\kappa }_1\eta \left(v_1\right)=u+g({\nabla }_{T}T,\xi )=0.$$

The function u is part of the conformal structure. However, for an ${\mathcal{L}}_c,$ we have $u=0$. Indeed, for an ${\mathcal{L}}_c$ $\gamma$ with $T\in \mathcal{D}=ker\eta$, the covariant derivative ${\nabla }_{T}T={\kappa }_1{v}_1$ must also lie in $\mathcal{D}$, since the contact distribution is preserved under parallel transport in this structure.

From the equation

$${\nabla }_T\xi =u(T-\eta \left(T\right)\xi )=uT,$$

(as $\eta \left(T\right)=0$), any nonzero u would imply ${\nabla }_T\xi \in \mathcal{D}$. However, this is incompatible with the contact geometry: $\xi$ is Reeb-like and $\mathcal{D}$ is $\phi$-invariant, forcing $u=0$ along $\gamma$ to maintain consistency. Thus, $\eta \left(v_1\right)=0$ follows from ${\kappa }_1\ne 0$ and ${\nabla }_T\eta =0$ and

$${\kappa }_{2}g(v_2,\xi )=0.$$

(23)

Suppose ${\kappa }_2\ne 0$. Thus,

$$g(v_2,\xi )=0.$$

The $\phi$-invariance of the system $\mathcal{V}\left(s\right)=span\{T\left(s\right),v_1\left(s\right),v_2\left(s\right)\}$ implies that $\phi T$ is a linear combination of $v_1\left(s\right)$ and $v_2\left(s\right)$. Hence, we can write

$$\phi T=a{v}_1+b{v}_2,$$

(24)

where a and b are smooth functions.

Differentiating covariantly with respect to T, and utilizing (3) alongside the orthogonality of $\phi v_1$, T, and $v_2$ with $v_1$, we obtain

$$u[g(\phi T,T)\xi -\eta \left(T\right)\phi T]+\phi \left(k_1{v}_1\right)=a^{\prime }{v}_1+a{\nabla }_T{v}_1+b^{\prime }{v}_2+b{\nabla }_T{v}_2.$$

Thus,

$${\kappa }_1\phi v_1=-a{\kappa }_{1}T+(a{\kappa }_2+b^{\prime })v_2.$$

(25)

Continuing this process, we differentiate $\phi v_1$ and use (3) to find

$$-b{\kappa }_1^2={(a{\kappa }_2+b^{\prime })}^{\prime },$$

and

$$\phi v_2=-{\displaystyle \frac{a^{\prime }}{{\kappa }_2}}T-{\displaystyle \frac{a{\kappa }_2+b^{\prime }}{{\kappa }_1}}{v}_1.$$

(26)

Now, combining (25) and(26) with

$${\phi }^{2}T=-T+\eta \left(T\right)\xi =-T=a\phi v_1+b\phi v_2,$$

we obtain

$$-T=a\left(-aT+{\displaystyle \frac{a{\kappa }_2+b^{\prime }}{{\kappa }_1}}{v}_2\right)+b\left(-{\displaystyle \frac{a^{\prime }}{{\kappa }_2}}T-{\displaystyle \frac{a{\kappa }_2+b^{\prime }}{{\kappa }_1}}{v}_1\right).$$

Since a and b cannot vanish simultaneously, we deduce that

$$a{\kappa }_2+b=0.$$

From this, it follows that $b=0,a=\pm 1,$ and consequently ${\kappa }_2=0;$ therefore, ${\kappa }_2=0$, which implies that a non-geodesic Legendre $\phi$-helix corresponds to a $\phi$-circle. □

Theorem 2.

Consider γ as a φ-helix with $r\le 3$ in an $\mathcal{LCACM}$, characterized by u and θ.

(i) If

$$cos\theta =\pm 1,$$

then γ is an ${\mathcal{I}}_c$ to ξ, and consequently, it is a normal ${\mathcal{M}}_c$ for any $F_q$, regardless of the choice of q.

(ii) If

$$cos\theta =0,$$

that is, γ is a Legendre φ-curve, then γ represents a magnetic circle produced by $F_{\pm {\kappa }_1}$.

(iii) If

$$cos\theta =\pm {\displaystyle \frac{{\kappa }_2}{\sqrt{{\kappa }_1^2+{\kappa }_2^2}}},$$

then γ is a ${\mathcal{M}}_c$ for $F_{\pm \sqrt{{\kappa }_1^2+{\kappa }_2^2}}$, in which the double indications do not depend on each other.

Proof. 

The condition

$$cos\theta =\pm 1,$$

implies

$$T=\pm \xi ,$$

since, $\eta \left(T\right)=g(T,\xi )=\pm 1$.

For locally conformal manifolds, (4) provides the formula for the covariant derivative of $\xi$. Substituting $T=\pm \xi$ yields

$${\nabla }_T\xi =u(\pm \xi -(\pm 1)\xi )=0.$$

Thus, we have:

$${\nabla }_{T}T={\nabla }_{\pm \xi }(\pm \xi )=0,$$

showing that $\gamma$ is a geodesic.

${\nabla }_{T}T=q\phi T$ is satisfied for each q because

$${\nabla }_{T}T=0,$$

$$\phi T=\phi (\pm \xi )=0.$$

Therefore, the equation reduces to

$$0=q·0,$$

which holds for all $q\in \mathbb{R}$.

$\gamma$ is a Legendre circle according to Proposition 3 if $cos\theta =0$. We obtain

$$\phi T=\pm {\nu }_1,$$

immediately. Thus, $\gamma$ meets (14), which is a ${\mathcal{M}}_c$ analogous to a F of strength $\pm {\kappa }_1$. Assertion (ii) is validated.

We begin by differentiating $cos\theta =g(T,\xi )$ along $\gamma$:

$$0={\displaystyle \frac{d}{ds}}g(T,\xi )=g({\kappa }_1{v}_1,\xi )+g(T,{\nabla }_T\xi ).$$

(27)

For a $\mathcal{LCACM}$, the covariant derivative of $\xi$ takes the form

$${\nabla }_T\xi =u(T-\eta \left(T\right)\xi )=uT(\mathrm{since}\eta \left(T\right)=cos\theta \ne 0).$$

(28)

Substituting (28) into (27) yields

$$0={\kappa }_{1}g(v_1,\xi )+ug(T,T)={\kappa }_{1}g(v_1,\xi )+u,$$

(29)

which implies

$$g(v_1,\xi )=-{\displaystyle \frac{u}{{\kappa }_1}}.$$

(30)

This differs from the cosymplectic case where $g(v_1,\xi )=0$ identically.

Now consider the $\phi$-invariance of $span\{T,v_1,v_2\}$ for a $\phi$-helix of order $r\le 3$:

$$\phi T=a{v}_1+b{v}_2.$$

(31)

The inner product with $\xi$ is taken, and utilizing $\eta \circ \phi =0$ gives

$$0=ag(v_1,\xi )+bg(v_2,\xi ).$$

(32)

Substituting (30) into (32), we obtain

$$bg(v_2,\xi )={\displaystyle \frac{au}{{\kappa }_1}}.$$

(33)

This reveals two cases:

• Case 1 ($u\ne 0$): The curvature constraints become overdetermined because of the following:

  –

  From (33), $g(v_2,\xi )\ne 0$ requires $a,b\ne 0.$

  –

  The Lorentz condition ${\nabla }_{T}T=q\phi T$ would force ${\kappa }_2=0$ (contradicting $cos\theta =\pm {\kappa }_2/\sqrt{{\kappa }_1^2+{\kappa }_2^2}$).

• Case 2 ($u=0$): The system reduces to the cosymplectic case:

  –

  (30) gives $g(v_1,\xi )=0$.

  –

  (32) simplifies to $bg(v_2,\xi )=0$.

  –

  (14) yields $q=\pm \sqrt{{\kappa }_1^2+{\kappa }_2^2}$ as in the original theorem.

Therefore, Statement (iii) holds iff u vanishes along $\gamma$. For general $u\ne 0$, the curvature relations are incompatible with the $\phi$-helix condition. □

Remark 1.

For a φ-helix of order 3 in a $\mathcal{LCACM}$, the φ-torsions ${\tau }_{ij}$ are generally non-constant due to u in the structure equations. However, if

(i) the θ is constant (i.e., ${\tau }_{02}=0$), and

(ii) u is constant along γ (i.e., $T\left(u\right)=0$ where $T=\dot{\gamma }$),

then all three φ-torsions become constant. In this case, the curvature relation modifies to

$${\kappa }_1{\tau }_{12}-{\kappa }_2{\tau }_{01}=u{\kappa }_2(\mathit{reducing}\mathit{to}\mathit{the}\mathit{cosymplectic}{\kappa }_1{\tau }_{12}-{\kappa }_2{\tau }_{01}=0\mathit{when}u=0).$$

Consequently, the ${\mathcal{M}}_c$ condition gains a conformal correction: q becomes

$$q=-{\displaystyle \frac{{\kappa }_1}{{\tau }_{01}}}+{\displaystyle \frac{u{\kappa }_2}{{\tau }_{01}({\kappa }_1^2+{\kappa }_2^2)}},$$

and θ satisfies

$$tan\theta =\pm {\displaystyle \frac{{\kappa }_1}{{\kappa }_2}}{\left(1+{\displaystyle \frac{u}{{\kappa }_1{\tau }_{01}}}\right)}^{-1},$$

where the sign depends on orientation. The original cosymplectic results are recovered when $u\equiv 0$.

4. Slant Curves in Locally Conformal Almost Cosymplectic Manifolds

We now turn our attention to the analysis of ${\mathcal{S}}_c$ in a $\mathcal{LCACM}$. Let $\gamma$ be a smooth curve in an $\mathcal{ACMM}$ $p.b.a.l$. $\theta$ associated to $\gamma$ is provided as

$$cos\theta \left(s\right)=g\left({\gamma }^{\prime }\left(s\right),\xi \right),$$

with $\theta \left(s\right)=[0,\pi ]$. Differentiate the preceding formula along $\gamma$ incorporating ∇ and (4), resulting

$$\begin{array}{cc}\hfill -{\theta }^{\prime }sin\theta & =g(\kappa N,\xi )+g\left(T,{\nabla }_T\xi \right)\hfill \\ & =\kappa \eta \left(N\right)+u(1-{cos}^2\theta )\hfill \end{array}$$

this leads to the result detailed below:

Proposition 4.

If γ is a ${\mathcal{S}}_c$ in a $\mathcal{LCACM}$ with u, then γ fulfills the subsequent indication

$$\eta \left(N\right)=\left(-{\displaystyle \frac{u{sin}^2\theta }{\kappa }}\right).$$

(34)

Proof. 

Utilizing $\{T,N,B\},$ $\xi$ interpreted as

$$\xi =(cos\theta )T+\left(-{\displaystyle \frac{u{sin}^2\theta }{\kappa }}\right)N+\eta \left(B\right)B.$$

(35)

Given that $\xi$ is unitary, (35) provides

$$\eta \left(B\right)={\displaystyle \frac{sin\theta }{\kappa }}\sqrt{{\kappa }^2-u^2{sin}^2\theta }.$$

(36)

□

Remark 2.

In the context of the Frenet frame, ξ associated with ${\mathcal{S}}_c$ γ is expressed as

$$\xi =(cos\theta )T+\left(-{\displaystyle \frac{u{sin}^2\theta }{\kappa }}\right)N+\left({\displaystyle \frac{sin\theta }{\kappa }}\sqrt{{\kappa }^2-u^2{sin}^2\theta }\right)B.$$

(37)

5. Magnetic Curves in an Almost $\mathsf{\alpha }$-Cosymplectic f-Manifold

In this section, we examined ${\mathcal{M}}_c$ within the context of $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$ and derived the subsequent outcomes:

Theorem 3.

Let $\left(M^{2n+1},\phi ,{\xi }_i,{\eta }^i,g\right)$ be an $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$ with $F_q,q\ne 0$. Then γ is a normal ${\mathcal{M}}_c$ related to $F_q$ if and only if γ belongs to the following categories:

(i) geodesics obtained as ${\mathcal{I}}_c$ of ${\xi }_i$;

(ii) Legendre φ-circles with ${\kappa }_1=\left|q\right|$;

(iii) φ-helices of $r=3$, with ${\kappa }_1=\left|q\right|{\sum }_{i=1}^{s}sin{\theta }_i,{\kappa }_2=sgn\left(q\right){\sum }_{i=1}^sϵ(qcos{\theta }_i-g(\phi A_{i}T,T))$ and such that $sgn\left({\tau }_{01}\right)=-sgn\left(q\right),$ where ${\theta }_i\ne {\displaystyle \frac{\pi }{2}}$.

Proof. 

In Scenario (i), if ${\mathcal{M}}_c$ $\gamma$ acts as a geodesic, (14) indicates that $\phi \dot{\gamma }=0,$ leading to the result that $\gamma$ and all ${\xi }_i$ are parallel. Given that both are unit vector fields, we deduce that

$$\dot{\gamma }=\pm {\xi }_i,$$

confirming that $\gamma$ is an ${\mathcal{I}}_c$ of ${\xi }_i$.

In what follows, we examine nongeodesic ${\mathcal{M}}_c$, specifically the Frenet curve of $r>1.$ Assume ${\theta }_i={\theta }_i\left(s\right)\in (0,\pi )$. It can be demonstrated that it remains constant. We find that

$$\begin{array}{ccc}\hfill 0& =& g(q\phi T,{\xi }_i)\hfill \\ & =& g({\nabla }_{T}T,{\xi }_i)\hfill \\ & =& {\displaystyle \frac{d}{ds}}g(T,{\xi }_i)-g(T,{\nabla }_T{\xi }_i).\hfill \end{array}$$

Given that ${\xi }_i$ represents a parallel vector field, it follows that ${\displaystyle \frac{d}{ds}}g(T,{\xi }_i)$ equals zero. This implies that ${\theta }_i\in (0,\pi )$ between T and ${\xi }_i$ remains unchanged. Therefore, we can deduce

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

(38)

By integrating the initial of (10) with (14)

$${\kappa }_1{v}_1=q\phi T.$$

(39)

It follows that

$${\kappa }_1=\left|q\right|sin{\theta }_i.$$

(40)

Using (39) and (40), we have

$$\phi T=sgn\left(q\right)sin{\theta }_i{v}_1.$$

(41)

We have the following by differentiating (39) covariantly with respect to T

$$sin{\theta }_i{\nabla }_T{v}_1=sgn\left(q\right)\left[{\displaystyle \sum _{i=1}^s}[-g(\phi A_{i}T,T){\xi }_i+cos{\theta }_i\phi A_{i}T]\right]+\left|q\right|[-T+{\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_i],$$

(42)

where (8) has been used.

Hence, ${\nabla }_T{v}_1$ is collinear to T iff ${\theta }_i={\displaystyle \frac{\pi }{2}}$ and ${\xi }_i$ vanish, that is,

$${\displaystyle \sum _{i=1}^s}g(\phi A_{i}T,T)=0,$$

when $\gamma$ is a Legendre $\phi$-circle of ${\kappa }_1=\left|q\right|$. Thus, (ii) is established.

Now assume ${\theta }_i\ne {\displaystyle \frac{\pi }{2}}$. Utilizing (41), from (10) one can obtain

$$\begin{array}{cc}\hfill sgn\left(q\right)sin{\theta }_i\left[-\left|q\right|sin{\theta }_{i}T+{\kappa }_2{v}_2\right]=& {\displaystyle \sum _{i=1}^s}\left[-g(\phi A_{i}T,T){\xi }_i+cos{\theta }_i\phi A_{i}T\right]\hfill \\ \hfill & +q\left[-T+{\displaystyle \sum _{i=1}^s}cos{\theta }_i{\xi }_i\right],\hfill \end{array}$$

(43)

and hence

$${\kappa }_2=sgn\left(q\right){\displaystyle \sum _{i=1}^s}ϵ(qcos{\theta }_i-g(\phi A_{i}T,T)).$$

(44)

The Frenet frame of $\gamma$ can be used to express ${\xi }_i$ as

$${\xi }_i=cos{\theta }_{i}T+\epsilon sin{\theta }_i{v}_2,$$

(45)

where $\epsilon =sgn(cos{\theta }_i)$. Then,

$$\left\{\begin{array}{c}\phi v_2=-sgn\left(q\right)\epsilon cos{\theta }_i{v}_1,\hfill \\ \eta \left(v_2\right)=\epsilon sin{\theta }_i.\hfill \end{array}\right.$$

(46)

Then, (38), (39), and (45) produce the following:

$$\phi v_1=sgn\left(q\right)\left(-sin{\theta }_{i}T+\epsilon cos{\theta }_i{v}_2\right).$$

(47)

Estimating

$$\begin{array}{ccc}\hfill {\tau }_{01}& =& g\left(T,\phi v_1\right)\hfill \\ & =& -sgn\left(q\right)sin{\theta }_i,\hfill \end{array}$$

it is evident at once that

$$-sgn\left(q\right)=sgn\left({\tau }_{01}\right).$$

Lastly, we infer by (47) that

$${\nabla }_T{v}_2=-{\kappa }_2{v}_1.$$

Hence,

$${\kappa }_3=0.$$

We determine that the Frenet curve of $r=3$ is the normal ${\mathcal{M}}_c$ with constant

$${\kappa }_1=\left|q\right|{\displaystyle \sum _{i=1}^s}sin{\theta }_i.$$

$${\kappa }_2=sgn\left(q\right){\displaystyle \sum _{i=1}^s}ϵ(qcos{\theta }_i-g(\phi A_{i}T,T)).$$

Hence (iii) is shown. □

On the other hand, beginning with a $\phi$-helix $\gamma$ in an $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$ $\left(M^{2n+1},\phi ,{\xi }_i,{\eta }^i,g\right)$, we determine the conditions under which this curve acts as a ${\mathcal{M}}_c$ related to a $F_q$ of specified strength. Initially, we establish another result, which proves helpful in demonstrating the main theorem.

Proposition 5.

Assume that M is an $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$. If γ is a non-geodesic Legendre φ-helix in M, then γ is a φ-circle.

Proof. 

As $\gamma$ is a ${\mathcal{L}}_c$, for a Legendre-$\phi$ curve ${\theta }_i={\displaystyle \frac{\pi }{2}}$, ${\eta }^i\left(T\right)=0$ holds for every i in the set $\{1,\dots ,s\}$. By performing sequential covariant differentiation with respect to T and applying (10), from

$${\nabla }_T\left({\eta }^i\left(T\right)\right)=0,$$

and the structural equation

$${\nabla }_T{\eta }^i=-\alpha {\eta }^i\circ \phi ,$$

one obtains

$$g(v_1,{\xi }_i)=0\mathrm{for}\mathrm{all}i.$$

Differentiating again and using Proposition 1, we find

$$g({\nabla }_T{v}_1,{\xi }_i)+g(v_1,{\nabla }_T{\xi }_i)=0.$$

We then substitute

$$\left\{\begin{array}{cc}\hfill {\nabla }_T{v}_1& =-{\kappa }_{1}T+{\kappa }_2{v}_2,\hfill \\ \hfill {\nabla }_T{\xi }_i& =-A_{i}T=\alpha {\phi }^{2}T+\phi h_{i}T.\hfill \end{array}\right.$$

We then have

$$g(-{\kappa }_{1}T+{\kappa }_2{v}_2,{\xi }_i)+g(v_1,\alpha {\phi }^{2}T+\phi h_{i}T)=0.$$

Thus,

$${\kappa }_{2}g(v_2,{\xi }_i)+\alpha g(v_1,-T)-g(\phi v_1,h_{i}T)=0,$$

from the Frenet frame orthogonality $g(v_1,T)=0$, and since $\phi v_1\perp {\xi }_i$ (as $\phi D\subset D=ker\left({\eta }^i\right)$), if $h_{i}T\in D$, then

$${\kappa }_{2}g(v_2,{\xi }_i)=0\mathrm{for}\mathrm{all}i.$$

(48)

Now assume ${\kappa }_2\ne 0$. Then

$$g(v_2,{\xi }_i)=0,\forall i.$$

As $\mathcal{V}\left(s\right)=span\{T\left(s\right),v_1\left(s\right),v_2\left(s\right)\}$ is $\phi$-invariant, we can write

$$\phi T=a{v}_1+b{v}_2,$$

(49)

where a and b represent smooth functions. Using (8) to take the covariant derivative regarding T and the knowledge that $\phi v_1,T,v_2$ are orthogonal to $v_1$, it can be concluded that

$$-{\displaystyle \sum _{i=1}^s}g(\phi A_{i}T,T){\xi }_i+{\kappa }_1\phi v_1=-a{\kappa }_{1}T+(a{\kappa }_2+b^{\prime })v_2.$$

The term $-{\sum }_{i=1}^{s}g(\phi A_{i}T,T){\xi }_i$ lies in $span\left\{{\xi }_i\right\}$, while the right-hand side lies in the $span\{T,v_1,v_2\}$. Since ${\xi }_i\perp \{T,v_1,v_2\}$ (from $g(v_1,{\xi }_i)=g(v_2,{\xi }_i)=0$), we must have

$$g(\phi A_{i}T,T)=0,\mathrm{for}\mathrm{all}i.$$

This simplifies the equation to

$${\kappa }_1\phi v_1=-a{\kappa }_{1}T+(a{\kappa }_2+b^{\prime })v_2.$$

(50)

Continuing this process, we differentiate $\phi v_1$ and use the structure equation to find

$$-b{\kappa }_1^2={(a{\kappa }_2+b^{\prime })}^{\prime },$$

and

$$-{\displaystyle \sum _{i=1}^s}g(\phi A_{i}T,v_1){\xi }_i-{\kappa }_1\phi T+{\kappa }_2\phi v_2=-a^{\prime }T-\left(a{\kappa }_1+{\displaystyle \frac{a{\kappa }_2+b^{\prime }}{{\kappa }_1}}{\kappa }_2\right)v_1+{\displaystyle \frac{{(a{\kappa }_2+b^{\prime })}^{\prime }}{{\kappa }_1}}{v}_2.$$

The term $-{\sum }_{i=1}^{s}g(\phi A_{i}T,v_1){\xi }_i$ must vanish because the other terms lie in $span\{T,v_1,v_2\}$. Thus,

$$g(\phi A_{i}T,v_1)=0,\forall i.$$

Hence, we obtain

$$\phi v_2=-{\displaystyle \frac{a^{\prime }}{{\kappa }_2}}T-{\displaystyle \frac{a{\kappa }_2+b^{\prime }}{{\kappa }_1}}{v}_1.$$

(51)

Taking into consideration (50) and (51) with

$$\begin{array}{ccc}\hfill {\phi }^{2}T& =& -T+{\displaystyle \sum _{i=1}^s}{\eta }^i\left(T\right){\xi }_i\hfill \\ & =& -T\hfill \\ & =& a\phi v_1+b\phi v_2,\hfill \end{array}$$

we deduce that

$$-T=a\left(-aT+{\displaystyle \frac{a{\kappa }_2+b^{\prime }}{{\kappa }_1}}{v}_2\right)+b\left(-{\displaystyle \frac{a^{\prime }}{{\kappa }_2}}T-{\displaystyle \frac{a{\kappa }_2+b^{\prime }}{{\kappa }_1}}{v}_1\right).$$

Consequently, the following equations are obtained

(i) Coefficient of T: $-a^2-b{\displaystyle \frac{a^{\prime }}{{\kappa }_2}}=-1$,

(ii) Coefficient of $v_1$: $-b{\displaystyle \frac{a{\kappa }_2+b^{\prime }}{{\kappa }_1}}=0$,

(iii) Coefficient of $v_2$: $a{\displaystyle \frac{a{\kappa }_2+b^{\prime }}{{\kappa }_1}}=0$.

From (iii), since ${\kappa }_1\ne 0$, we obtain

$$a{\kappa }_2+b^{\prime }=0.$$

Then (ii) holds automatically, and (i) simplifies to

$$-a^2=-1\Rightarrow a=\pm 1.$$

Now $a{\kappa }_2+b^{\prime }=0$ with a constant implies $b^{\prime }=-a{\kappa }_2$. However, differentiating $a{\kappa }_2+b^{\prime }=0$ gives $a{\kappa }_2^{\prime }=0$, which forces ${\kappa }_2=0$ (since $a\ne 0$), contradicting our assumption. Therefore, we must have

$${\kappa }_2=0,$$

proving $\gamma$ to be a $\phi$-circle.

Thus, a non-geodesic Legendre $\phi$-helix is a $\phi$-circle. □

Theorem 4.

Assume that γ is a φ-helix with $r\le 3$, represented within the framework of $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$ $(M^{2n+s},\phi ,{\xi }_i,{\eta }^i,g)$. Then the following statement holds:

(i) When $cos{\theta }_i=\pm 1$, γ transforms into ${\mathcal{I}}_c$ associated with $\overline{\xi }={\sum }_{i=1}^s{\xi }_i$ and exhibits a normal ${\mathcal{M}}_c$ for $F_q=qd\overline{\eta }$, where $\overline{\eta }$ denotes the 1-form defined by $\overline{\eta }={\sum }_{i=1}^s{\eta }_i$, with any selected q.

(ii) When $cos{\theta }_i=0$, meaning γ is a Legendre φ-curve, γ acts as a magnetic circle associated with $F_{\pm {\kappa }_1}$.

(iii) In the case $cos{\theta }_i=\pm {\displaystyle \frac{{\kappa }_2}{\sqrt{{\kappa }_1^2+{\kappa }_2^2}}}$, γ is a ${\mathcal{M}}_c$ for $F_{\pm \sqrt{{\kappa }_1^2+{\kappa }_2^2}}$, with both signs being independently applicable.

Proof. 

Case (i): If $cos{\theta }_i=\pm 1$, then $T=\pm \overline{\xi }$. Since ${\nabla }_{{\xi }_i}{\xi }_j=0$ for all $i,j$ and $\phi \overline{\xi }=0$, we have

$$\begin{array}{ccc}\hfill {\nabla }_{T}T& =& {\nabla }_{\pm \overline{\xi }}(\pm \overline{\xi })\hfill \\ & =& 0\hfill \\ & =& q\phi T\hfill \end{array}$$

which holds for every q. Therefore, $\gamma$ is a ${\mathcal{M}}_c$ for arbitrary q.

Case (ii): When $cos{\theta }_i=0$, $\gamma$ is a Legendre curve with $T\perp \overline{\xi }$. Equation (10) provides

$${\nabla }_{T}T={\kappa }_1{v}_1.$$

Since $\phi T$ is a unit vector orthogonal to $\overline{\xi }$, thus from Proposition 5, we have

$$\phi T=\pm v_1.$$

Then (12) becomes

$$\begin{array}{ccc}\hfill {\kappa }_1{v}_1& =& {\nabla }_{T}T\hfill \\ & =& q\phi T\hfill \\ & =& \pm q{v}_1.\hfill \end{array}$$

Thus, $q=\pm {\kappa }_1$.

Case (iii): For the intermediate case, we differentiate $cos{\theta }_i=g(T,\overline{\xi })$ to obtain

$$0=g(T,{\nabla }_T\overline{\xi })+g({\nabla }_{T}T,\overline{\xi }).$$

From Proposition 1, we compute ${\nabla }_T\overline{\xi }=-\alpha {\phi }^{2}T-{\sum }_{i=1}^s\phi h_{i}T$. Since

$${\phi }^{2}T=-T+{\displaystyle \sum _{i=1}^s}{\eta }^i\left(T\right){\xi }_i,$$

and $h_{i}T\perp {\xi }_i,$ we obtain

$$g({\nabla }_{T}T,\overline{\xi })=-\alpha {sin}^2{\theta }_i.$$

Further, ${\nabla }_{T}T={\kappa }_1{v}_1+{\kappa }_2{v}_2$ with $g(v_1,\overline{\xi })=0$ and $g(v_2,\overline{\xi })=sin{\theta }_i$ yields

$${\kappa }_{2}sin{\theta }_i=-\alpha {sin}^2{\theta }_i.$$

Now, for ${\nabla }_{T}T=q\phi T$, one requires

$$\phi T=\pm \left({\displaystyle \frac{{\kappa }_1}{\sqrt{{\kappa }_1^2+{\kappa }_2^2}}}{v}_1+{\displaystyle \frac{{\kappa }_2}{\sqrt{{\kappa }_1^2+{\kappa }_2^2}}}{v}_2\right).$$

Thus, $q=\pm \sqrt{{\kappa }_1^2+{\kappa }_2^2}$. □

Remark 3.

For any φ-helix of $r=3$ within $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$ $(M^{2n+s},\phi ,{\xi }_i,{\eta }^i,g)$, not all φ-torsions ${\tau }_{ij}^k$ are necessarily constant. However, if

(i) ${\theta }_i$ is constant (equivalently, ${\tau }_{02}=0$ where ${\tau }_{02}=g({\nabla }_{T}T,{\phi }^{2}T)$), or

(ii) the Reeb vector fields satisfy ${\nabla }_{{\xi }_i}{\xi }_j=0$ for all $i,j$ (automatic when $\alpha =0$),

then the φ-torsions ${\tau }_{01},{\tau }_{12}$ become constant, and the compatibility condition

$${\kappa }_1{\tau }_{12}-{\kappa }_2{\tau }_{01}=0,$$

holds. Consequently, γ is a ${\mathcal{M}}_c$ for $F_q$ with

$$q=-{\displaystyle \frac{{\kappa }_1}{{\tau }_{01}}}\mathit{and}tan{\theta }_i=\pm {\displaystyle \frac{{\kappa }_1}{{\kappa }_2}},$$

where the sign depends on the orientation relative to $\overline{\xi }={\sum }_{i=1}^s{\xi }_i$.

6. Slant Curves in an Almost $\mathsf{\alpha }$-Cosymplectic f-Manifold

The discussion herein concerns ${\mathcal{S}}_c$ within an $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$. Subsequently, for a smooth $\gamma$ within an almost contact metric 3-manifold $p.b.a.l.$, ${\theta }_i$ specified as

$$cos{\theta }_i\left(s\right)=g\left({\gamma }^{\prime }\left(s\right),{\xi }_i\right),\forall i\in \{1,\dots ,s\},$$

where ${\theta }_i\left(s\right)=[0,\pi ]$. Applying the covariant derivative ∇ along the curve $\gamma$ to the aforementioned formula and utilizing Proposition 1 gives

$$-{\theta }_i^{\prime }sin{\theta }_i=g(\kappa N,{\xi }_i)+g\left(T,{\nabla }_T{\xi }_i\right),$$

which implies

$$\begin{array}{ccc}\hfill \kappa {\eta }^i\left(N\right)+g(T,-\alpha {\phi }^{2}T-\phi h_{i}T)& =& \kappa {\eta }^i\left(N\right)-\alpha g(T,{\phi }^{2}T)-g(T,\phi h_{i}T)\hfill \\ & =& 0.\hfill \end{array}$$

Using Proposition 2, we have $g(T,\phi h_{i}T)=0$, so we obtain

$$\kappa {\eta }^i\left(N\right)+\alpha (1-{cos}^2{\theta }_i)=0.$$

The preceding equation yields the following result:

Proposition 6.

If γ is a ${\mathcal{S}}_c$ in an $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$ with any real value of α, then γ adheres to the following expression:

$${\eta }^i\left(N\right)=\left(-{\displaystyle \frac{\alpha {sin}^2{\theta }_i}{\kappa }}\right).$$

(52)

Proof. 

Employing $\{T,N,B\},\xi$ can be expressed as

$${\xi }_i=(cos{\theta }_i)T+\left(-{\displaystyle \frac{\alpha {sin}^2{\theta }_i}{\kappa }}\right)N+{\eta }^i\left(B\right)B.$$

Since ${\xi }_i$ is unitary, the above equation yields

$${\eta }^i\left(B\right)={\displaystyle \frac{sin{\theta }_i}{\kappa }}\sqrt{{\kappa }^2-{\alpha }^2{sin}^2{\theta }_i}.$$

(53)

□

Remark 4.

The form of ${\xi }_i$ along with ${\mathcal{S}}_c$γ within the Frenet frame field is provided by

$${\xi }_i=(cos{\theta }_i)T+\left(-{\displaystyle \frac{\alpha {sin}^2{\theta }_i}{\kappa }}\right)N+\left({\displaystyle \frac{sin{\theta }_i}{\kappa }}\sqrt{{\kappa }^2-{\alpha }^2{sin}^2{\theta }_i}\right)B.$$

(54)

7. Conclusions

In this paper, we studied ${\mathcal{M}}_c$ and ${\mathcal{S}}_c$ associated with a contact $\mathcal{MF}$ in two classes of manifolds: $\mathcal{LCACM}$ and $\mathcal{A}\alpha \mathcal{C}f\mathcal{M}$.

In both settings, we obtained a classification of normal ${\mathcal{M}}_c$, showing that they are either geodesics, Legendre $\phi$-circles, or $\phi$-helices of order three. In particular, we proved that Legendre helices reduce to $\phi$-circles.

We also investigated ${\mathcal{S}}_c$ and derived explicit conditions relating to their geometric properties to the structure of the manifolds. In particular, we obtained relations involving the curvature and the structural functions that determine the behavior of these curves.

Furthermore, we established explicit relations connecting the curvature of magnetic trajectories with the magnetic field strength and the structural parameters. In the locally conformal case, these depend on the conformal function u, while in the almost $\alpha$-cosymplectic case, they involve the constant $\alpha$.

These results extend known classifications of magnetic curves and provide corresponding insights for slant curves in these geometric settings. Further work may consider similar problems in other classes of almost contact metric manifolds.