aa r X i v : . [ m a t h . G M ] M a r ON SLANT MAGNETIC CURVES IN S -MANIFOLDS S¸ABAN G ¨UVENC¸ AND C˙IHAN ¨OZG ¨UR
Abstract.
We consider slant normal magnetic curves in (2 n +1)-dimensional S -manifolds. We prove that γ is a slant normal magnetic curve in an S -manifold ( M m + s , ϕ, ξ α , η α , g ) if and only if it belongs to a list of slant ϕ -curves satisfying some special curvature equations. This list consists of somespecific geodesics, slant circles, Legendre and slant helices of order 3. Weconstruct slant normal magnetic curves in R n + s ( − s ) and give the parametricequations of these curves. Introduction
Let (
M, g ) be a Riemannian manifold, F a closed 2-form and let us denote theLorentz force on M by Φ. If F is associated by the relation g (Φ X, Y ) = F ( X, Y ) , ∀ X, Y ∈ χ ( M ) , (1.1)then it is called a magnetic field ([1], [4] and [9]). Let ∇ be the Riemannianconnection associated to the metric g and γ : I → M a smooth curve. If γ satisfiesthe Lorentz equation ∇ γ ′ ( t ) γ ′ ( t ) = Φ( γ ′ ( t )) , (1.2)then it is called a magnetic curve or a trajectory for the magnetic field F . TheLorentz equation is a generalization of the equation for geodesics. A curve whichsatisfies the Lorentz equation is called magnetic trajectory . Magnetic trajectorieshave constant speed. If the speed of the magnetic curve γ is equal to 1, then it iscalled a normal magnetic curve [10].In [1], Adachi studied curvature bound and trajectories for magnetic fields on aHadamard surface. He showed that every normal trajectory is unbounded in bothdirections in a 2-dimensional complete simply connected Riemannian manifold sat-isfying some special curvature conditions. In [5], Baikoussis and Blair consideredLegendre curves in contact 3-manifolds and they proved that the torsion of a Le-gendre curve in a 3-dimensional Sasakian manifold is equal to 1. Moreover, in [8],Cho, Inoguchi and Lee proved that a non-geodesic curve in a Sasakian 3-manifoldis a slant curve if and only if the ratio of ( τ ±
1) and κ is constant, where τ is thegeodesic torsion and κ is the geodesic curvature. Cabrerizo, Fernandez and Gomezgave a nice geometric construction of an almost contact metric structure compat-ible with an assigned metric on a 3-dimensional oriented Riemannian manifold in[6]. In the paper [10], Drut¸˘a-Romaniuc, Inoguchi, Munteanu and Nistor studiedthe magnetic trajectories of the contact magnetic field F q = q Ω on a Sasakian(2 n + 1)-manifold ( M n +1 , ϕ, ξ, η, g ), where Ω is the fundamental 2-form. Themain objective of [11] is the study of trajectories for particles moving under the Mathematics Subject Classification.
Key words and phrases.
Magnetic curve, slant curve, S -manifold. influence of a contact magnetic curve in a cosymplectic manifold. The paper [14]is concerned with closed magnetic trajectories on 3-dimensional Berger spheres. In[15], the authors studied magnetic trajectories in an almost contact metric man-ifold. They proved that normal magnetic curves are helices of maximum order5. Moreover, in [16], Jleli and Munteanu worked in the context of a para-Kaehlermanifold, showing that spacelike and timelike normal magnetic curves correspond-ing to the para-Kaehler 2-forms are circles. In [17], the authors gave a completeclassification of Killing magnetic curves with unit speed. Furthermore, in [18], thesame authors proved that a normal magnetic curve on the Sasakian sphere S n +1 lies on a totally geodesic sphere S . They also considered two particular mag-netic fields on three-dimensional torus obtained from two different contact formson the Euclidean space E and studied their closed normal magnetic trajectoriesin their recent paper [19]. In [23], the authors investigated some special curves in3-dimensional semi-Riemannian manifolds, such as T -magnetic curves, N -magneticcurves and B -magnetic curves, that are defined by means of their Frenet elements.Calvaruso, Munteanu and Perrone provided a complete classification of the mag-netic trajectories of a Killing characteristic vector field on an arbitrary normalparacontact metric manifold of dimension 3 in [7]. The present authors consideredbiharmonic Legendre curves of S -space forms in [21]. The second author studiedmagnetic curves in the 3-dimensional Heisenberg group in [22]. In [20], Nakagawaintroduced the notion of framed f -structures, which is a generalization of almostcontact structures. Vanzura studied almost r -structures in [24]. A differentiablemanifold with this structure is the same as a framed f -manifold as defined by Nak-agawa. On the other hand, Hasegawa, Okuyama and Abe defined a p th Sasakianmanifold and gave some typical examples in [13].Motivated by the above studies, in the present paper, we consider slant normalmagnetic curves in (2 n + s )-dimensional S -manifolds. In Section 2, we give briefinformation on S -manifolds and magnetic curves. In Section 3, we prove that γ is aslant normal magnetic curve in an S -manifold ( M m + s , ϕ, ξ α , η α , g ) if and only if itbelongs to a list of slant ϕ -curves. This list consists of some specific geodesics, slantcircles, Legendre and slant helices of order 3. Finally, in Section 4, we constructslant normal magnetic curves in R n + s ( − s ) and give the parametric equations ofthese curves in two cases. 2. Preliminaries
In this section, we give brief information on S -manifolds and magnetic curves.Let (cid:0) M n + s , g (cid:1) be a differentiable manifold, ϕ a (1 , η α ξ α vector fields for α = 1 , ..., s , satisfying ϕ = − I + s X α =1 η α ⊗ ξ α , (2.1) η α ( ξ β ) = δ αβ , ϕξ α = 0 , η α ( ϕX ) = 0 , η α ( X ) = g ( X, ξ α ) ,g ( ϕX, ϕY ) = g ( X, Y ) − s X α =1 η α ( X ) η α ( Y ) , (2.2) dη α ( X, Y ) = − dη α ( Y, X ) = g ( X, ϕY ) , where X, Y ∈ T M . Then ( ϕ, ξ α , η α , g ) is called framed ϕ -structure and ( M n + s , ϕ, ξ α , η α , g )is called framed ϕ -manifold [20] . ( M n + s , ϕ, ξ α , η α , g ) is also called framed metric N SLANT MAGNETIC CURVES IN S -MANIFOLDS 3 manifold [25] or almost r-contact metric manifold [24]. If the Nijenhuis tensor of ϕ is equal to − dη α ⊗ ξ α , then ( M n + s , ϕ, ξ α , η α , g ) is called an S -manifold [3].For s = 1, an S -structure becomes a Sasakian structure. For an S -structure, thefollowing properties are satisfied [3]:( ∇ X ϕ ) Y = s X α =1 (cid:8) g ( ϕX, ϕY ) ξ α + η α ( Y ) ϕ X (cid:9) , (2.3) ∇ ξ α = − ϕ, α ∈ { , ..., s } . (2.4)Let M n + s = ( M n + s , ϕ, ξ α , η α , g ) be an S -manifold and Ω the fundamental -form of M n + s defined by Ω( X, Y ) = g ( X, ϕY ) , (2.5)(see [20] and [24]). From the definition of framed ϕ -structure, we have Ω = dη α .Hence, the fundamental 2-form Ω on M n + s is closed. The magnetic field F q on M n + s can be defined by F q ( X, Y ) = q Ω( X, Y ) , where X and Y are vector fields on M n + s and q is a real constant. F q is calledthe contact magnetic field with strength q [15]. If q = 0 then the magnetic curvesare geodesics of M n + s . Because of this reason we shall consider q = 0 (see [6] and[10]).From (1.1) and (2.5), the Lorentz force Φ associated to the contact magneticfield F q can be written as Φ q = − qϕ. So the Lorentz equation (1.2) can be written as ∇ T T = − qϕT, (2.6)where γ : I ⊆ R → M n + s is a smooth unit-speed curve and T = γ ′ (see [10] and[15]). 3. Slant magnetic curves in S -manifolds Let ( M n , g ) be a Riemannian manifold. A unit-speed curve γ : I → M is said tobe a Frenet curve of osculating order r , if there exists positive functions κ , ..., κ r − on I satisfying T = v = γ ′ , ∇ T T = k v , ∇ T v = − k T + k v , (3.1) ... ∇ T v r = − k r − v r − , where 1 ≤ r ≤ n and T, v , ..., v r are a g -orthonormal vector fields along the curve.The positive functions κ , ..., κ r − are called curvature functions and { T, v , ..., v r } is called the Frenet frame field . A geodesic is a Frenet curve of osculating order r = 1 . A circle is a Frenet curve of osculating order r = 2 with a constant curvaturefunction κ . A helix of order r is a Frenet curve of osculating order r with constantcurvature functions κ , ..., κ r − . A helix of order 3 is simply called a helix . S¸ABAN G¨UVENC¸ AND C˙IHAN ¨OZG¨UR
Let ( M m + s , ϕ, ξ α , η α , g ) be an S -manifold. For a unit-speed curve γ : I → M ,if η α ( T ) = 0 , for all α = 1 , ..., s , then γ is called a Legendre curve of M [21]. More generally, ifthere exists a constant angle θ such that η α ( T ) = cos θ, for all α = 1 , ..., s , then γ is called a slant curve and θ is called the contact angleof γ , where | cos θ | ≤ / √ s [12].Let ( M m + s , ϕ, ξ α , η α , g ) be an S -manifold. A Frenet curve of osculating order r ≥ ϕ -curve in M if its Frenet vector fields T, v , ..., v r span a ϕ -invariant space. A ϕ -curve of osculating order r with constant curvature functions κ , ..., κ r − is called a ϕ -helix of order r . A curve of osculating order 2 is called a ϕ -curve if sp ( T, v , s X α =1 ξ α ) is a ϕ -invariant space.Throughout the paper, when we state ”slant magnetic curve”, we mean ”slantcurves which satisfy equation (2.6)”. For magnetic curves, η α ( T ) = cos θ α does nothave to be equal for all α = 1 , ..., s. By taking the curve as slant, we only study theequality case of the slant angles θ α in the present paper. The complete classificationof magnetic curves in S -manifolds is still an open problem.Firstly, we state the following theorem: Theorem 1.
Let ( M m + s , ϕ, ξ α , η α , g ) be an S -manifold and consider the contactmagnetic field F q for q = 0 . Then γ is a slant normal magnetic curve associated to F q in M m + s if and only if γ belongs to the following list: a ) geodesics obtained as integral curves of (cid:18) ± √ s s P α =1 ξ α (cid:19) ; b ) non-geodesic slant circles with the curvature κ = p q − s , having the contactangle θ = arccos (cid:16) q (cid:17) and the Frenet frame field (cid:26) T, − sgn ( q ) ϕT √ − s cos θ (cid:27) , where | q | > √ s ; c ) Legendre helices with curvatures κ = | q | and κ = √ s , having the Frenetframe field ( T, − sgn ( q ) ϕT, − sgn ( q ) √ s s X α =1 ξ α ) ; i.e., a class of 1-dimensional integral submanifolds of the contact distribution; d ) slant helices with curvatures κ = | q | √ − s cos θ and κ = √ s | − q cos θ | ,having the Frenet frame field ( T, − sgn ( q ) ϕT √ − s cos θ , − εsgn ( q ) √ s √ − s cos θ − s cos θT + s X α =1 ξ α !) , where θ = π is the contact angle satisfying | cos θ | < √ s and ε = sgn (1 − q cos θ ) . N SLANT MAGNETIC CURVES IN S -MANIFOLDS 5 Proof.
Let γ be a normal magnetic curve. If the magnetic curve is a geodesic, then ∇ T T = 0 = − qϕT gives us T ∈ sp { ξ , ..., ξ s } . If γ is slant, then we can write T = cos θ s X α =1 ξ α . Since γ is unit speed, we have cos θ = ± √ s . So the proof of a) is complete.From now on, we suppose that γ is a non-geodesic Frenet curve of osculatingorder r >
1. Let us choose an α ∈ { , ..., s } . Applying ξ α to ∇ T T = − qϕT , weobtain 0 = g ( − qϕT, ξ α ) = g ( ∇ T T, ξ α ) = ddt g ( T, ξ α ) − g ( T, ∇ T ξ α ) . (3.2)From (2.4), we also have ∇ T ξ α = − ϕT. (3.3)Using equations (3.2) and (3.3), we find ddt g ( T, ξ α ) = 0 , that is, η α ( T ) = cos θ α = constant. Let us assume θ α = θ for all α = 1 , ..., s , i.e., γ is slant. So, we have η α ( T ) = cos θ. (3.4)Equations (2.6) and (3.1) give us ∇ T T = κ v = − qϕT. (3.5)Then we get κ = | q | k ϕT k = | q | p − s cos θ. (3.6)If we write (3.6) in (3.5), we find − qϕT = κ v = | q | p − s cos θv , which gives us ϕT = − | q | q p − s cos θv = − sgn ( q ) p − s cos θv . (3.7)If κ = 0, then the magnetic curve is a Frenet curve of osculating order r = 2.Since κ is a constant, γ is a circle. From (3.7), we have η α ( ϕT ) = 0 = − sgn ( q ) p − s cos θη α ( v ) ,that is, η α ( v ) = 0 . If we differentiate the last equation along the curve γ , we obtain ∇ T η α ( v ) = 0 = g ( ∇ T v , ξ α ) + g ( v , ∇ T ξ α ) . So, we calculate g ( − κ T, ξ α ) + g (cid:16) v , sgn ( q ) p − s cos θv (cid:17) = 0 . S¸ABAN G¨UVENC¸ AND C˙IHAN ¨OZG¨UR
Since r = 2, we find − κ cos θ + sgn ( q ) p − s cos θ = 0 . Using equation (3.6) in the last equation, it is easy to see that | q | p − s cos θ (cid:18) − cos θ + 1 q (cid:19) = 0 . Since γ is non-geodesic, we have cos θ = 1 q . Then equation (3.6) becomes κ = | q | p − s cos θ = p q − s, where | q | > √ s . So the proof of b) is complete.Let κ = 0. From (2.1) and (3.4), we find ϕ T = − T + cos θ s X α =1 ξ α . (3.8)Using (2.3) and (3.4), we have( ∇ T ϕ ) T = − s cos θT + s X α =1 ξ α , (3.9)which gives us ∇ T ϕT = ( ∇ T ϕ ) T + ϕ ∇ T T = − s cos θT + s X α =1 ξ α + ϕ ( − qϕT )= − s cos θT + s X α =1 ξ α − q − T + cos θ s X α =1 ξ α ! . (3.10)Differentiating (3.7), we also find ∇ T ϕT = − sqn ( q ) p − s cos θ ( − κ T + κ v ) . (3.11)By the use of (3.6), (3.10) and (3.11), after some calculations, we obtain(1 − q cos θ ) − s cos θT + s X α =1 ξ α ! = − sgn ( q ) p − s cos θκ v . (3.12)If we find the norm of both sides in (3.12), we get κ = √ s | − q cos θ | . (3.13)Let us denote ε = sgn (1 − q cos θ ). If we write (3.13) in (3.12), we obtain s X α =1 ξ α = s cos θT − εsgn ( q ) √ s p − s cos θv . (3.14)Applying ϕ to (3.14), we find ϕv = − ε √ s cos θv . N SLANT MAGNETIC CURVES IN S -MANIFOLDS 7 If we apply ϕ to (3.7) and then use equations (3.4) and (3.14) together, we have ϕv = sgn ( q ) p − s cos θT + ε cos θ √ sv . (3.15)Let us choose a β ∈ { , ..., s } . From (3.15), we calculate η β ( v ) = − εsgn ( q ) √ − s cos θ √ s . If we differentiate (3.14) along the curve γ , we get s X α =1 ∇ T ξ α = s cos θ ∇ T T − εsgn ( q ) √ s p − s cos θ ∇ T v , which gives us − s (1 − q cos θ ) ϕT = − εsgn ( q ) √ s p − s cos θ ( − κ v + κ v ) . Since ϕT k v , we find κ = 0. This proves d) of the theorem.Let us examine Legendre case separately, that is, θ = π . Then we have ε = 1, κ = | q | , κ = √ s , κ = 0 and equation (3.14) gives us v = − sgn ( q ) √ s s X α =1 ξ α . This completes the proof of c).Conversely, let γ satisfy one of a ), b ), c ) or d ). Using the Frenet frame field andFrenet equations, it is straightforward to show that ∇ T T = − qϕT , i.e., γ is a slantnormal magnetic curve. (cid:3) The above theorem is a generalization of Theorem 3.1 of [10] (by Simona LuizaDruta-Romaniuc et al.) for S -manifolds. If we choose s = 1, since an S -manifoldbecomes a Sasakian manifold, we find their results. Remark.
The order of a slant magnetic curve in an S -manifold is still r ≤ r ≤ . Theorem 2.
Let ( M m + s , ϕ, ξ α , η α , g ) be an S -manifold and consider the contactmagnetic field F q for q = 0 . If γ is a normal magnetic curve associated to F q in M m + s , then the osculating order r ≤ .Proof. Let γ be a normal magnetic curve. Then, the Lorentz equation (2.6) givesus η α ( T ) = cos θ α , α = 1 , ..., s. If we differentiate this equation along the curve, we have η α ( E ) = 0for all α = 1 , ..., s. From the Frenet equations (3.1), we obtain − qϕT = κ v . From the definition of framed ϕ -structure, we calculate g ( ϕT, ϕT ) = 1 − A, S¸ABAN G¨UVENC¸ AND C˙IHAN ¨OZG¨UR where we denote A = s X α =1 cos θ α . Then, we have k ϕT k = √ − A and κ = | q | √ − A. Thus, ϕT can be rewritten as ϕT = − sgn ( q ) √ − Av . (3.16)Again, from the definition of framed ϕ -structure, we have ϕ T = − T + V, where we denote V = s X α =1 cos θ α ξ α . After some calculations, we get ∇ T ϕT = ( q − B ) T + (1 − A ) s X α =1 ξ α +( − q + B ) V, which corresponds to equation (3.10). Here, we denote B = s X α =1 cos θ α . From equation (3.16), we also find ∇ T ϕT = − sqn ( q ) √ − A ( − κ T + κ v ) , which corresponds to equation (3.11). In this last equation, we can replace κ = | q | √ − A . Finally, we have − sqn ( q ) √ − Aκ v = (1 − A ) s X α =1 ξ α + ( − q + B ) V (3.17)+ ( qA − B ) T. So, if we denote the norm of the right hand side of equation (3.17) by C , we find C = p (1 − A )( Aq − As + B − Bq + s ) , which is a constant. Hence, we obtain κ = C √ − A = p Aq − As + B − Bq + s = constant . From equation (3.17), we also have v ∈ span { T, ξ , ..., ξ s } . The angles between v and T, ξ , ..., ξ s are all constants since all the coefficients in equation (3.17) areconstants. Then, we can write v = c T + c ξ + ... + c s ξ s (3.18)for some constants c , ..., c s . If we differentiate equation (3.18), we get − κ v + κ v = c κ v − c ϕT − ... − c s ϕT. N SLANT MAGNETIC CURVES IN S -MANIFOLDS 9 Since ϕT is parallel to v , if we take the inner product of the last equation with v ,we find κ = 0. This proves the theorem. (cid:3) In particular, if γ is slant, i.e. θ α = θ for all α = 1 , ..., s, then we obtain thefollowing corollary: Corollary 1. If θ α = θ, for all α = 1 , ..., s, then A = s cos θ, B = s cos θ, V = cos θ s X α =1 ξ α ,C = p (1 − s cos θ ) s (1 − q cos θ ) and κ = √ s | − q cos θ | . Now, let us state the following proposition:
Proposition 1.
Let γ be a slant ϕ -helix of order 3 in an S -manifold ( M m + s , ϕ, ξ α , η α , g ) with contact angle θ . Then s X α =1 ξ α = s cos θT + ρv , (3.19) where ρ = g (cid:18) v , s P α =1 ξ α (cid:19) = sη α ( v ) is a real constant such that ρ = s − s cos θ .Hence, γ has the Frenet frame field ( T, ± ϕT √ − s cos θ , ± √ s √ − s cos θ − s cos θT + s X α =1 ξ α !) . Proof.
From the assumption, the Frenet frame field { T, v , v } is ϕ -invariant and η α ( T ) = cos θ .Differentiating the last equation along the curve, it is easy to see that η α ( v ) = 0 . (3.20)If we differentiate once again, we have g ( ϕT, v ) = − κ cos θ + κ η α ( v ) , (3.21)which means the value of η α ( v ) does not depend on α. Firstly, let us assume that θ = π . Since the space spanned by the Frenet framefield is ϕ -invariant, then ϕ T is in the set. Using (3.8) and (3.20), we can write s X α =1 ξ α ∈ sp { T, v } , that is, s X α =1 ξ α = s cos θT + ρv . (3.22)If we take the norm of both sides, we find ρ = s − s cos θ . Since the value of η α ( v ) does not depend on α , we obtain ρ = g v , s X α =1 ξ α ! = sη α ( v ) . If we apply ϕT to (3.22), we get g ( ϕT, v ) = 0. Since ϕT ⊥ T , ϕT ⊥ v and sp { T, v , v } is ϕ -invariant, we have ϕT k v . As a result, we find v = ± ϕT k ϕT k = ± ϕT √ − s cos θ . Now let us consider the Legendre case, i.e., θ = π . From (3.21), we find ∇ T g ( ϕT, v ) = − κ g ( ϕT, v ) . (3.23)Using (2.1) and (2.3), we calculate ∇ T ϕT = ( ∇ T ϕ ) T + ϕ ∇ T T (3.24)= s X α =1 ξ α + κ ϕv . Using the last equation, we obtain ∇ T g ( ϕT, v ) = g ( ∇ T ϕT, v ) + g ( ϕT, ∇ T v ) (3.25)= κ g ( ϕT, v ) . Equations (3.23) and (3.25) give us g ( ϕT, v ) = 0, that is, ϕT k v . Thus, wehave ϕT = ± v . Consequently, the Frenet frame field becomes { T, ± ϕT, v } . Now,we must show that v is parallel to s P α =1 ξ α . Since the space spanned by the Frenetframe field is ϕ -invariant, from orthonormal expansion, we can write ϕv = g ( ϕv , T ) T ± g ( ϕv , ± ϕT ) ϕT + g ( ϕv , v ) v , which reduces to ϕv = g ( ϕv , T ) T. (3.26)If we apply ϕ to equation (3.26) and use (2.1), we find − v + s X α =1 η α ( v ) ξ α = g ( ϕv , T ) ϕT. (3.27)Applying ϕT to (3.27) and using the Frenet frame field, we have g ( ϕv , T ) = 0. Asa result, we get ϕv = 0 and equation (3.27) becomes − v + s X α =1 η α ( v ) ξ α = 0 . We have already shown that the value of η α ( v ) does not depend on α ; so, we canwrite v = η α ( v ) s X α =1 ξ α . (3.28)Since v and ξ α are unit for all α = 1 , ..., s, we find η α ( v ) = ± √ s . Finally, for θ = π , we have ρ = s and s P α =1 ξ α = ρv , which completes the proof. (cid:3) Corollary 2.
Let γ be a Legendre ϕ -helix of order 3 in an S -manifold ( M m + s , ϕ, ξ α , η α , g ) .Then κ = √ s , v = ± ϕT and v = ± √ s s P α =1 ξ α . N SLANT MAGNETIC CURVES IN S -MANIFOLDS 11 Proof.
From equation (3.28), we already have v = ± √ s s X α =1 ξ α . If we differentiate this equation and use (3.1), we obtain − κ v = ± √ s s X α =1 ∇ T ξ α . (3.29)Using equations (2.4) and (3.29), we find that κ = √ s and v = ± ϕT . (cid:3) Finally, we can give the following theorem:
Theorem 3.
Let γ be a slant ϕ -helix of order r ≤ on an S -manifold ( M m + s , ϕ, ξ α , η α , g ) .Let θ denote the contact angle of γ . Then we havei. If cos θ = ± √ s , then γ is an integral curve of ± √ s s P α =1 ξ α , hence it is a normalmagnetic curve for F q with an arbitrary q .ii. If cos θ = 0 and κ = 0 (i.e. γ is a non-geodesic Legendre curve), then γ isa magnetic curve for F ± κ .iii. If cos θ = ε √ κ + s , then γ is a magnetic curve for F ε √ κ + s , where ε = − sgn ( g ( ϕT, v )) . In this case, γ is a slant ϕ -circle.iv. If cos θ = ε √ s ± κ √ s q κ + ( ε √ s ± κ ) , then γ is a magnetic curve for F ε q κ + ( ε √ s ± κ ) ,where ε = − sgn ( g ( ϕT, v )) and the sign ± corresponds to the sign of η α ( v ) .v. Except the above cases, γ is not a magnetic curve for any F q .Proof. Let cos θ = ± √ s . Then we have T = ± √ s s X α =1 ξ α , which gives us ∇ T T = 0. We also have ϕT = 0. So γ satisfies ∇ T T = − qϕT forany q , which proves i .Now let cos θ = 0 and κ = 0. Using Corollary 2, we have ∇ T T = κ ( ± ϕT ) = − qϕT, which gives us q = ± κ . This completes the proof of ii .From Proposition 1, we have the Frenet frame field ( T, ± ϕT √ − s cos θ , ± √ s √ − s cos θ − s cos θT + s X α =1 ξ α !) when r = 3 and (cid:26) T, ± ϕT √ − s cos θ (cid:27) when r = 2. If we differentiate v along the curve, after some calculations, in bothcases, we find (cid:18) ± κ cos θ √ − s cos θ (cid:19) − s cos θT + s X α =1 ξ α ! = ± κ p − s cos θv , (3.30)(taking κ = 0, when r = 2). Next, let us assume cos θ = ε √ κ + s , where we denote ε = − sgn ( g ( ϕT, v )).Then the left side of equation (3.30) vanishes. Thus we get κ = 0. From theassumption, we also have κ = constant , that is, γ is a slant ϕ -circle. Using theFrenet frame field, we find ∇ T T = − qϕT = κ v , where q = ε p κ + s . So, wehave just completed the proof of iii .Finally, let us assume cos θ = ε √ s ± κ √ s q κ + ( ε √ s ± κ ) , where ε = − sgn ( g ( ϕT, v )) andthe sign ± corresponds to the sign of η α ( v ). In this case, let us take κ = 0, sincewe have already investigated order r = 2. Using the Frenet frame field, after somecalculations, we obtain ∇ T T = − qϕT = κ v , where q = ε q κ + ( ε √ s ± κ ) .Hence, the proof of iv is complete.Since we have considered all cases, we can state that there exist no other slantmagnetic ϕ -helices in M . (cid:3) From the proof of Theorem 1, we can give the following proposition:
Proposition 2.
Let ( M m + s , ϕ, ξ α , η α , g ) be an S -manifold. There exist no non-geodesic slant ϕ -circles as magnetic curves corresponding to F q for < | q | ≤ √ s . Theorem 3 and Proposition 2 generalize Theorem 3.2 and Proposition 3.2 in [10]to S -manifolds, respectively. Under the condition s = 1, we obtain their results.4. Construction of slant normal magnetic curves in R n + s ( − s )In this section, we find parametric equations of slant normal magnetic curves in R n + s ( − s ). As a start, we recall structures defined on this S -manifold. Let ustake M = R n + s with coordinate functions { x , ...x n , y , ..., y n , z , ..., z s } and define ξ α = 2 ∂∂z α , α = 1 , ..., s,η α = 12 dz α − n X i =1 y i dx i ! , α = 1 , ..., s,ϕX = n X i =1 Y i ∂∂x i − n X i =1 X i ∂∂y i + n X i =1 Y i y i ! s X α =1 ∂∂z α ! ,g = s X α =1 η α ⊗ η α + 14 n X i =1 ( dx i ⊗ dx i + dy i ⊗ dy i ) , where X = n X i =1 (cid:18) X i ∂∂x i + Y i ∂∂y i (cid:19) + s X α =1 (cid:18) Z α ∂∂z α (cid:19) ∈ χ ( M ) . It is well-known that (cid:0) R n + s , ϕ, ξ α , η α , g (cid:1) is an S -space form with constant ϕ -sectional curvature − s. Hence it is denoted by R n + s ( − s ) [13]. The followingvector fields X i = 2 ∂∂y i , X n + i = ϕX i = 2( ∂∂x i + y i s X α =1 ∂∂z α ) , ξ α = 2 ∂∂z α N SLANT MAGNETIC CURVES IN S -MANIFOLDS 13 form a g -orthonormal basis and the Levi-Civita connection is ∇ X i X j = ∇ X n + i X n + j = 0 , ∇ X i X n + j = δ ij s X α =1 ξ α , ∇ X n + i X j = − δ ij s X α =1 ξ α , ∇ X i ξ α = ∇ ξ α X i = − X n + i , ∇ X n + i ξ α = ∇ ξ α X n + i = X i . (see [13]). Let γ : I → R n + s ( − s ) be a unit-speed slant curve with contact angle θ . Let us denote γ ( t ) = ( γ ( t ) , ..., γ n ( t ) , γ n +1 ( t ) , ..., γ n ( t ) , γ n +1 ( t ) , ..., γ n + s ( t )) , where t is the arc-length parameter. Then γ has the tangent vector field T = γ ′ ∂∂x + ... + γ ′ n ∂∂x n + γ ′ n +1 ∂∂y + ... + γ ′ n ∂∂y n + γ ′ n +1 ∂∂z + ... + γ ′ n + s ∂∂z s , which can be written as T = 12 (cid:2) γ ′ n +1 X + ... + γ ′ n X n + γ ′ X n +1 + ... + γ ′ n X n + (cid:0) γ ′ n +1 − γ ′ γ n +1 − ... − γ ′ n γ n (cid:1) ξ + ... + (cid:0) γ ′ n + s − γ ′ γ n +1 − ... − γ ′ n γ n (cid:1) ξ s (cid:3) . Since γ is slant curve, we have η α ( T ) = 12 (cid:0) γ ′ n + α − γ ′ γ n +1 − ... − γ ′ n γ n (cid:1) = cos θ for all α = 1 , ..., s . So, we obtain γ ′ n +1 = ... = γ ′ n + s = 2 cos θ + γ ′ γ n +1 + ... + γ ′ n γ n . (4.1)Since γ is a unit-speed, we can write( γ ′ ) + ... + ( γ ′ n ) = 4 (cid:0) − s cos θ (cid:1) . (4.2)These equations were obtained in our paper [12].Now, our aim is to find parametric equations for slant normal magnetic curves.So, let us assume that γ : I → R n + s ( − s ) is a normal magnetic curvature. Fromthe Lorentz equation, we have ∇ T T = − qϕT , (4.3)where q = 0 is a constant. Using the Levi-Civita connection, we calculate ∇ T T = 12 (cid:8)(cid:0) γ ′′ n +1 + 2 s cos θγ ′ (cid:1) X + ... + ( γ ′′ n + 2 s cos θγ ′ n ) X n (4.4)+ (cid:0) γ ′′ − s cos θγ ′ n +1 (cid:1) X n +1 + ... + ( γ ′′ n − s cos θγ ′ n ) X n (cid:9) and ϕT = 12 (cid:8) − γ ′ X − ... − γ ′ n X n + γ ′ n +1 X n +1 + ... + γ ′ n X n (cid:9) . (4.5)From equations (4.3), (4.4) and (4.5), we have γ ′′ n +1 + 2 s cos θγ ′ − γ ′ = ... = γ ′′ n + 2 s cos θγ ′ n − γ ′ n = γ ′′ − s cos θγ ′ n +1 γ ′ n +1 = ... = γ ′′ n − s cos θγ ′ n γ ′ n = − q, which is equivalent to γ ′′ n +1 − γ ′ = ... = γ ′′ n − γ ′ n = γ ′′ γ ′ n +1 = ... = γ ′′ n γ ′ n = λ, (4.6)where we denote λ = − q + 2 s cos θ . Firstly, let us assume λ = 0. From equation(4.6), if we select pairs γ ′′ n +1 − γ ′ = γ ′′ γ ′ n +1 , ..., γ ′′ n − γ ′ n = γ ′′ n γ ′ n , solving ODEs, we have( γ ′ ) + (cid:0) γ ′ n +1 (cid:1) = c , ..., ( γ ′ n ) + ( γ ′ n ) = c n , where c , ..., c n are arbitrary constants. Thus, we can write γ ′ = c cos f , ..., γ ′ n = c n cos f n , (4.7) γ ′ n +1 = c sin f , ..., γ ′ n = c n sin f n , where f , ..., f n are differentiable functions on I . From (4.6) and (4.7), we find f ′ = ... = f ′ n = − λ, which gives us f i = − λt + a i , i = 1 , , ..., n where a , ..., a n are arbitrary constants. Now, if we integrate (4.7), we have γ = c − λ sin f + b , ..., γ n = c n − λ sin f n + b n ,γ n +1 = c λ cos f + d , ..., γ n = c n λ cos f n + d n , where b i and d i are arbitrary constants ( i = 1 , ..., n ). Thus, we get γ ′ γ n +1 + ... + γ ′ n γ n = n X i =1 (cid:18) c i λ cos f i + c i d i cos f i (cid:19) .Using the last equation with (4.1), we obtain γ ′ n + α = 2 cos θ + n X i =1 (cid:18) c i λ cos f i + c i d i cos f i (cid:19) , where α = 1 , ..., s . If we integrate this last equation, we find γ n + α = 2 t cos θ − n X i =1 (cid:26) c i λ [sin (2 f i ) + 2 f i ] + c i d i λ sin f i (cid:27) + h α , for α = 1 , ..., s and h , ..., h s are arbitrary constants. Moreover, from (4.2) and(4.7), we have c + ... + c n = 4 (cid:0) − s cos θ (cid:1) . (4.8)Thus, we have just finished the case λ = 0.Secondly, let λ = 0. In this case, we have γ ′′ = γ ′′ = ... = γ ′′ n = 0 , which gives us γ i = c i t + d i , N SLANT MAGNETIC CURVES IN S -MANIFOLDS 15 for i = 1 , ..., n, where c i and d i are arbitrary constants. Using the last equation,we calculate γ ′ γ n +1 + ... + γ ′ n γ n = n X i =1 c i ( c n + i t + d n + i ) . So, equation (4.1) becomes γ ′ n + α = 2 cos θ + n X i =1 c i ( c n + i t + d n + i ) , which gives us γ n + α = 2 t cos θ + n X i =1 c i (cid:16) c n + i t + d n + i t (cid:17) + h α , where h α are arbitrary constants for α = 1 , ..., s . Since γ is unit-speed, from (4.2),we have c + ... + c n = 4 (cid:0) − s cos θ (cid:1) . To sum up, we give the following Theorem:
Theorem 4.
The slant normal magnetic curves on R n + s ( − s ) satisfying theLorentz equation ∇ T T = − qϕT have the parametric equationsa) γ i ( t ) = c i − λ sin f i ( t ) + b i ,γ n + i ( t ) = c i λ cos f i ( t ) + d i ,γ n + α ( t ) = 2 t cos θ − n X i =1 (cid:26) c i λ [sin (2 f i ( t )) + 2 f i ( t )] + c i d i λ sin f i ( t ) (cid:27) + h α ,f i ( t ) = − λt + a i ,α = 1 , ..., s, i = 1 , , ..., n,λ = − q + 2 s cos θ = 0 where a i , b i , c i , d i and h α are arbitrary constants such that c i satisfies c + ... + c n = 4 (cid:0) − s cos θ (cid:1) ; or b) γ i ( t ) = c i t + d i ,γ n + α ( t ) = 2 t cos θ + n X i =1 c i (cid:16) c n + i t + d n + i t (cid:17) + h α ,α = 1 , ..., s, i = 1 , ..., n, where c i , d i and h α are arbitrary constants such that c i satisfies c + ... + c n = 4 (cid:0) − s cos θ (cid:1) . In both cases, q = 0 is a constant and θ denotes the constant contact angle satisfying | cos θ | ≤ √ s . In particular, if s = 1, we obtain Theorem 3.5 in [10]. Acknowledgements.
This work is financially supported by Balikesir ResearchGrant no. BAP 2018/016.
References [1] Adachi ,T.: Curvature bound and trajectories for magnetic fields on a Hadamard surface.Tsukuba J. Math. , 225–230, (1996).[2] Blair, D. E.: Riemannian geometry of contact and symplectic manifolds, Second edition.Progress in Mathematics, 203. Birkhauser Boston, Inc., Boston, MA, (2010).[3] Blair, D. E.: Geometry of manifolds with structural group U ( n ) × O ( s ). J. Differential Ge-ometry, , 155-167, (1970).[4] Barros M., Romero, A., Cabrerizo, J. L., Fern´andez, M.: The Gauss-Landau-Hall problem onRiemannian surfaces. J. Math. Phys. , no. 11, 112905, 15 pp, (2005).[5] Baikoussis, C., Blair, D. E.: On Legendre curves in contact 3-manifolds. Geom. Dedicata ,135–142 (1994).[6] Cabrerizo, J. L., Fernandez M. and Gomez, J. S.: On the existence of almost contact structureand the contact magnetic field. Acta Math. Hungar. , 191-199, (2009).[7] Calvaruso, G., Munteanu, M. I., Perrone, A.: Killing magnetic curves in three-dimensionalalmost paracontact manifolds. J. Math. Anal. Appl. , no. 1, 423–439, (2015).[8] Cho, J. T., Inoguchi, J., Lee, J.E.: On slant curves in Sasakian 3-manifolds. Bull. Austral.Math. Soc. , 359–367 (2006).[9] Comtet, A.: On the Landau levels on the hyperbolic plane. Ann. Physics , 185-209, (1987).[10] Drut¸˘a-Romaniuc, S. L., Inoguchi, J., Munteanu, M. I., Nistor, A. I.: Magnetic curves inSasakian manifolds. Journal of Nonlinear Mathematical Physics, , 428-447, (2015).[11] Drut¸˘a-Romaniuc, S. L., Inoguchi, J., Munteanu, M. I., Nistor, A. I.: Magnetic curves incosymplectic manifolds. Rep. Math. Phys. , 33-48, (2016).[12] G¨uven¸c, S¸., ¨Ozg¨ur, C.: On slant curves in S -manifolds. Commun. Korean Math. Soc. , No.1, pp. 293-303, (2018).[13] Hasegawa, I., Okuyama, Y., Abe, T.: On p -th Sasakian manifolds. J. Hokkaido Univ. Ed.Sect. II A, , no. 1, 1–16, (1986).[14] Inoguchi J., Munteanu, M. I.: Periodic magnetic curves in Berger spheres. Tohoku Math. J. , 113-128, (2017).[15] Jleli, M., Munteanu, M. I., Nistor, A. I.: Magnetic trajectories in an almost contact metricmanifold R N +1 . Results Math. , 125-134, (2015).[16] Jleli, M., Munteanu, M. I.: Magnetic curves on flat para-Kahler manifolds. Turkish J. Math. , 963-969, (2015).[17] Munteanu, M. I., Nistor, A. I.: The classification of Killing magnetic curves in S × R . J.Geom. Phys. , 170-182, (2012).[18] Munteanu, M. I., Nistor, A. I.: A note on magnetic curves on S n +1 . C. R. Math. Acad. Sci.Paris , 447-449, (2014).[19] Munteanu, M. I., Nistor, A. I.: On some closed magnetic curves on a 3-torus. Math. Phys.Anal. Geom. , no. 2, Art. 8, 13 pp, (2017).[20] Nakagawa, H.: On framed f -manifolds. Kodai Math. Sem. Rep. S -space forms. Turkish J. Math. , no. 3, 454–461 (2014).[22] ¨Ozg¨ur, C.: On magnetic curves in the 3-dimensional Heisenberg group. Proceedings of theInstitute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, ,2, 278-286, (2017).[23] ¨Ozdemir, Z., G¨ok, I., Yaylı, Y., Ekmekci, N.: Notes on magnetic curves in 3D semi-Riemannian manifolds. Turkish J. Math. , 412-426, (2015).[24] Vanzura, J.: Almost r -contact structures. Ann. Scuola Norm. Sup. Pisa (3) . Singapore.World Scientific Publishing Co. 1984.(S¸. G¨uven¸c and C. ¨Ozg¨ur) Department of Mathematics, Balikesir University, 10145,C¸ a˘gıs¸, Balıkesir, TURKEY
E-mail address , S¸. G¨uven¸c: [email protected]
E-mail address , C. ¨Ozg¨ur:, C. ¨Ozg¨ur: