aa r X i v : . [ m a t h . DG ] F e b MAGNETIC CURVES IN C -MANIFOLDS S¸ABAN G ¨UVENC¸
Abstract.
In this paper, we study normal magnetic curves in C -manifolds.We prove that magnetic trajectories with respect to the contact magnetic fieldsare indeed θ α -slant curves with certain curvature functions. Then, we give theparametrizations of normal magnetic curves in R n + s with its structures as a C -manifold. Introduction
Let (
M, g ) be a Riemannian manifold, F a closed 2-form and let us denote theLorentz force on M by Φ, which is a (1 , F is associated bythe relation(1.1) g (Φ X, Y ) = F ( X, Y ) , ∀ X, Y ∈ χ ( M ) , then it is called a magnetic field ([1], [2] and [5]). Let ∇ be the Riemannianconnection associated to the Riemannian metric g and γ : I → M a smooth curve.If γ satisfies the Lorentz equation(1.2) ∇ γ ′ ( t ) γ ′ ( t ) = Φ( γ ′ ( t )) , then it is called a magnetic curve or a trajectory for the magnetic field F . TheLorentz equation can be considered as a generalization of the equation for geodesics.Magnetic trajectories have constant speed. If the speed of the magnetic curve γ is equal to 1, then it is called a normal magnetic curve [6]. For fundamentals ofalmost contact metric manifolds, we refer to Blair’s book [4]. This paper is basedon a similar idea of Ozgur and the present author’s previous paper [7].2. Preliminaries
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(2.1) ϕ X = − X + s X α =1 η α ( X ) ξ α ,η α ( ξ β ) = δ αβ , ϕξ α = 0 , η α ( ϕX ) = 0 , η α ( X ) = g ( X, ξ α ) , (2.2) g ( ϕX, ϕY ) = g ( X, Y ) − s X α =1 η α ( X ) η α ( Y ) , Mathematics Subject Classification.
Primary 53C25; Secondary 53C40, 53A04.
Key words and phrases. C -manifold, magnetic curve, θ α -slant curve. 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. The fundamental 2-form and
Nijenhuis tensor is givenby: Ω(
X, Y ) = g ( X, ϕY ) ,N ϕ ( X, Y ) = − s X α =1 dη α ( X, Y ) ξ α . If d Ω = 0 and dη α = 0, M = ( M, ϕ, ξ α , η α , g ) is called a C -manifold. In a C -manifold, it is known that ( ∇ X ϕ ) Y = 0and ∇ X ξ α = 0 , (see [3] and [4]). 3. Magnetic Curves in C -manifolds Let γ : I → M be a unit-speed curve in an n -dimensional Riemannian manifold( M, g ). The curve γ is called a Frenet curve of osculating order r (1 ≤ r ≤ n ),if there exists orthonormal vector fields T, v , ..., v r along the curve validating theFrenet equations T = γ ′ , ∇ T T = κ v , ∇ T v = − κ v + κ v , (3.1) ... ∇ T v r = − κ r − v r − , where κ , ..., κ r − are positive functions called the curvatures of γ . If κ = 0 , then γ is called a geodesic . If κ is a non-zero positive constant and r = 2 , γ is calleda circle. If κ , ..., κ r − are non-zero positive constants, then γ is called a helix oforder r ( r ≥ . If r = 3, it is shortly called a helix .A submanifold of an C -manifold is said to be an integral submanifold if η α ( X ) =0 , α ∈ { , , ..., s } , where X is tangent to the submanifold. A Legendre curve isa 1-dimensional integral submanifold of an C -manifold ( M n + s , ϕ, ξ α , η α , g ) . Moreprecisely, a unit-speed curve γ : I → M is a Legendre curve if T is g-orthogonal toall ξ α ( α = 1 , , ...s ), where T = γ ′ . Definition 1.
Let γ be a unit-speed curve in a C -manifold ( M, ϕ, ξ α , η α , g ) . γ iscalled a θ α − slant curve if there exist constant contact angles such that η α ( T ) =cos θ α , α = 1 , , ..., s . If θ α = θ for all α = 1 , , ..., s , then γ is shortly called slant.Moreover, if θ α = π for all α = 1 , , ..., s , then γ is called a Legendre curve. For θ α − slant curves, we can give the following inequality for the constant contactangles: s X α =1 cos θ α ≤ . The equality case is only valid when γ is a geodesic as an integral curve of ± s P α =1 cos θ α ξ α . AGNETIC CURVES IN C -MANIFOLDS 3 Let γ be a unit-speed Legendre curve in a C -manifold ( M, ϕ, ξ α , η α , g ). If wedifferentiate η α ( T ) = 0, we obtain η α ( v ) = 0. We can continue this process untilwe find η α ( v r ) = 0. Thus, we can state the following proposition: Proposition 1. If γ is a unit-speed Legendre curve in a C -manifold ( M, ϕ, ξ α , η α , g ) ,then ξ α is g -orthogonal to sp { T, v , ..., v r } , for all α = 1 , , ..., s . If we consider equations (1.1), (1.2) and (3.1) together, for a normal magneticcurve of a magnetic field F with charge q , we find ∇ T T = Φ T,F ( X, Y ) = g (Φ X, Y ) ,F q ( X, Y ) = q Ω (
X, Y )= qg ( X, ϕY ) , which gives us Φ q = − qϕ. Here, T denotes the tangential vector field of the normal magnetic curve γ for themagnetic field F q in M . Then, we have the following equations:(3.2) ∇ T T = − qϕT, ∇ T ξ α = 0 , ∇ T ϕT = ( ∇ T ϕ ) T + ϕ ∇ T T = ϕ ( − qϕT )= − qϕ T = − q − T + s X α =1 η α ( T ) ξ α ! = qT − q s X α =1 η α ( T ) ξ α . If we take the inner product of equation (3.2) with ξ α , we obtain0 = g ( − qϕT, ξ α ) = g ( ∇ T T, ξ α )= ddt g ( T, ξ α ) . Integrating both sides, we get η α ( T ) = cos θ α = constant, for all α = 1 , , ..., s . Equations (3.1) and (3.2) give us(3.3) ∇ T T = κ v = − qϕT,g ( ϕT, ϕT ) = g ( T, T ) − s X α =1 ( η α ( T )) = 1 − s X α =1 cos θ α S¸ABAN G¨UVENC¸ and k ϕT k = vuut − s X α =1 cos θ α . From equation (3.3), we find(3.4) κ = | q | vuut − s X α =1 cos θ α = constant, − qϕT = κ v = | q | vuut − s X α =1 cos θ α v and(3.5) ϕT = − sgn ( q ) vuut − s X α =1 cos θ α v . If κ = 0 , then r = 2 and γ is a circle. If we apply η α to equation (3.5), we obtain η α ( v ) = 0 , which gives us ∇ T η α ( v ) = 0= g ( ∇ T v , ξ α ) + g ( T, ∇ T ξ α )= − κ cos θ α . As a result, we get cos θ α = 0 , for all α = 1 , , ..., s . Hence, γ is a Legendre circle, k ϕT k = 1 and κ = | q | . Let κ = 0. Using equations (2.1) and (3.1), we calculate ∇ T ϕT = ( ∇ T ϕ ) T + ϕ ∇ T T = ϕ ( − qϕT )(3.6) = − q − T + s X α =1 cos θ α ξ α ! . Differentiating equation (3.5), we also have(3.7) ∇ T ϕT = − sgn ( q ) vuut − s X α =1 cos θ α ( − κ T + κ v )In view of (3.4), (3.6) and (3.7), it is easy to see that(3.8) q " s X α =1 cos θ α ξ α − s X α =1 cos θ α ! T = sgn ( q ) vuut − s X α =1 cos θ α κ v (3.9) κ = | q | vuut s X α =1 cos θ α AGNETIC CURVES IN C -MANIFOLDS 5 If we write (3.9) in (3.8), we have(3.10) s X α =1 cos θ α ξ α = s X α =1 cos θ α ! T + vuut s X α =1 cos θ α vuut − s X α =1 cos θ α v If we differentiate (3.10), we find κ = 0. From equations (3.5) and (3.10), we canwrite(3.11) v = − sgn ( q ) s − s P α =1 cos θ α ϕT (3.12) v = 1 s s P α =1 cos θ α s − s P α =1 cos θ α s X α =1 cos θ α ξ α − s X α =1 cos θ α ! T ! Finally, if κ = 0, after some calculations, by (2.1) and (3.5), we obtain T = ± s P α =1 cos θ α ξ α , where s P α =1 cos θ α = 1. So, we can give the following theorem: Theorem 1.
Let γ : I → M = ( M, ϕ, ξ α , η α , g ) be a unit-speed curve in a C -manifold. Then γ is a normal magnetic curve for F q ( q = 0) in M if and onlyif i) γ is a geodesic θ α − slant curve as an integral curve of ± s P α =1 cos θ α ξ α , where s P α =1 cos θ α = 1 ; orii) γ is a Legendre circle with κ = | q | having the Frenet frame field { T, − sgn ( q ) ϕT } ; or iii) γ is a non-Legendre θ α − slant helix with κ = | q | vuut − s X α =1 cos θ α ,κ = | q | vuut s X α =1 cos θ α , having the Frenet frame field { T, v , v } , where s P α =1 cos θ α < , v and v are given in equations (3.11) and (3.12), respec-tively. Corollary 1. If γ is a unit-speed slant curve in M , then it is a normal magneticcurve if anf only ifi) it is a geodesic as an integral curve of ± √ s s P α =1 ξ α ; or S¸ABAN G¨UVENC¸ ii) γ is a Legendre circle with κ = | q | having the Frenet frame field { T, − sgn ( q ) ϕT } ; or iii) γ is a non-Legendre slant helix with κ = | q | √ − s cos θ, κ = | q | √ sε cos θ, having the Frenet frame field ( T, − sgn ( q ) √ − s cos θ ϕT, ε √ s √ − s cos θ s X α =1 ξ α − s cos θT !) , where θ = π is the contact angle satisfying | cos θ | < √ s and ε = sgn (cos θ ) . Proof.
Since θ α = θ for all α = 1 , , ..., s, if we use s X α =1 cos θ α = s cos θ and s X α =1 cos θ α ξ α = cos θ s X α =1 ξ α in Theorem 1, the proof is clear. (cid:3) Remark.
If we take s = 1, we have Proposition 1 in [8].Let M = ( M, ϕ, ξ α , η α , g ) be a C -manifold. A Frenet curve of order r = 2 iscalled a ϕ -curve in M if sp { T, v , ξ , ..., ξ s } is a ϕ − invariant space. A Frenet curveof order r ≥ ϕ -curve if sp { T, v , ..., v r } is ϕ − invariant. A ϕ − helix oforder r is a ϕ − curve with constant curvatures κ , ..., κ r − . A ϕ − helix of order 3 isshortly named a ϕ − helix. Proposition 2. If γ is a Legendre ϕ − helix in a C -manifold, then it is a Legendre ϕ − circle.Proof. Let γ be a Legendre ϕ − helix. Then the contact angles θ α = π for all α = 1 , , ..., s and the Frenet frame field { T, v , v } is ϕ − invariant. Thus, we canwrite(3.13) g ( ϕT, v ) = cos µ, (3.14) ϕT = cos µv ± sin µv , for some function µ = µ ( t ). If we differentiate equation (3.13), we find − µ ′ sin µ = κ g ( ϕT, v )(3.15) = ± κ sin µ. Firstly, let us assume that µ = 0, i.e. ϕT = v . Hence, we have ∇ T ϕT = − κ T = − κ T + κ v , which is equivalent to κ = 0. Likewise, if µ = π , we obtain κ = 0. Finally, let usassume that µ = 0 , π . In this case, since γ is a helix, using (3.15), we have κ = constant,κ = ∓ µ ′ = constant. If we differentiate (3.14), we calculate κ ϕv = − κ cos µT. AGNETIC CURVES IN C -MANIFOLDS 7 If we apply ϕ to both sides, we conclude ϕT = ± v , which gives κ = 0. Thiscompletes the proof. (cid:3) Remark.
For s = 1, we obtain Proposition 2 of [8]. Likewise, the followingtheorem generalizes Theorem 1 of [8] to C -manifolds: Theorem 2. If γ be ϕ − helix of order r ≤ in a C -manifold M = ( M, ϕ, ξ α , η α , g ) .Then, the following statements are valid:i) If cos θ α ( α = 1 , , ..., s ) are constants such that s P α =1 cos θ α = 1 , then γ is anintegral curve of ± s P α =1 cos θ α ξ α , hence it is a normal magnetic curve for arbitrary q . ii) If cos θ α = 0 for all α = 1 , , ..., s , i.e. γ is a Legendre ϕ − curve, then it is amagnetic circle generated by the magnetic field F ± κ .iii) If cos θ α ( α = 1 , , ..., s ) are constants such that s P α =1 cos θ α = κ κ + κ , then γ is a magnetic curve for F ± √ κ + κ .iv) Except above cases, γ cannot be a magnetic curve for any magnetic field F q .Proof. In view of Theorem 1 and Proposition 2, it is straightforward to show that ∇ T T = − qϕT for valid q . (cid:3) Magnetic Curves of R n + s with its structures as a C -manifold In this section, we consider parametrizations of normal magnetic curves in M = R n + s as a C -manifold. Let { x , ..., x n , y , ..., y n , z , ..., z s } be the coordinate func-tions and define X i = ∂∂x i , Y i = ∂∂y i , ξ α = ∂∂z α , for i = 1 , ..., n and α = 1 , , ..., s . { X i , Y i , ξ α } is an orthonormal basis of χ ( M ) withrespect to the usual metric g = n X i =1 h ( dx i ) + ( dy i ) i + s X α =1 ( dz α ) . Let us define a (1 , ϕ as ϕX i = − Y i , ϕY i = X i , ϕξ α = 0 . Finally, let η α = dz α for α = 1 , , ..., s . It is well-known that ( M, ϕ, ξ α , η α , g ) isa C -manifold, since dη α = 0 and d Ω = 0, where Ω (
X, Y ) = g ( X, ϕY ) for all
X, Y ∈ χ ( M ) (see [3] and [4]).Let us denote normal magnetic curve by γ = ( γ , ..., γ n , γ n +1 , ..., γ n , γ n +1 , ..., γ n + s ) . Then T = γ ′ = (cid:0) γ ′ , ..., γ ′ n , γ ′ n +1 , ..., γ ′ n , γ ′ n +1 , ..., γ ′ n + s (cid:1) , which gives us ∇ T T = (cid:0) γ ′′ , ..., γ ′′ n , γ ′′ n +1 , ..., γ ′′ n , γ ′′ n +1 , ..., γ ′′ n + s (cid:1) ,ϕT = (cid:0) γ ′ n +1 , ..., γ ′ n , − γ ′ , ..., − γ ′ n , , ..., (cid:1) . S¸ABAN G¨UVENC¸
Since ∇ T T = − qϕT, we have η α ( T ) = γ ′ n + α = cos θ α = constant and γ n + α = cos θ α t + h α . We also get γ ′′ i = − qγ ′ n + i ,γ ′′ n + i = qγ ′ i for i = 1 , ..., n . As a result, we obtain γ ′ i γ ′′ i + γ ′ n + i γ ′′ n + i = 0 , i.e. ( γ ′ i ) + (cid:0) γ ′ n + i (cid:1) = c i . If we consider differentiable functions f i : I → R , we can write γ ′ i = c i cos f i ,γ ′ n + i = c i sin f i . After calculations, we find f i ( t ) = qt + d i . Finally, we have γ i = c i q sin ( qt + d i ) + b i ,γ n + i = − c i q cos ( qt + d i ) + b n + i . Thus, we give the following theorem:
Theorem 3.
The normal magnetic curves on R n + s satisfying the Lorentz equation ∇ T T = − qϕT have the parametric equations γ i = c i q sin ( qt + d i ) + b i ,γ n + i = − c i q cos ( qt + d i ) + b n + i ,γ n + α = cos θ α t + h α , where i = 1 , ..., n, α = 1 , , ..., s, b i , h α are arbitrary constants and θ α are theconstant contact angles. AGNETIC CURVES IN C -MANIFOLDS 9 References [1] Adachi ,T.:
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Email address , S¸. G¨uven¸c:, S¸. G¨uven¸c: