The contact magnetic flow in 3D Sasakian manifolds

The Contact Magnetic flow in 3D Sasakian manifolds J. L. Cabrerizo1, M. Fernández1and J. S. Gómez1 1Department of Geometry and Topology, University of Seville, Ap. 1160, 41080-Seville, Spain E-mail: [email protected], [email protected] and jsalv[email protected] Abstract We first present a geometrical approach to magnetic fields in three-dimensional Riemannian manifolds, because this particular dimension allows one to easily tie vector fields and 2-forms. When the vector field is divergence free, it defines a magnetic field on the manifold whose Lorentz force equation presents a simple and useful form. In particular, for any three-dimensional Sasakian manifold the contact magnetic field is studied, and the normal magnetics trajectories are determined. As an application, we consider the three-dimensional unit sphere, where we prove the existence of closed magnetic trajectories of the contact magnetic field, and that this magnetic flow is quantized in the set of rational numbers. PACS numbers: 02.40.Ky, 02.40.Ma Mathematics Subject Classification: 53C25, 53C80 1 Introduction The motion of a physical system in a semi-Riemannian manifold, (Mn, g), is determined by the principle of the least action (also called Maupertuis principle), and can be found by minimizing a certain Lagrangian (the action functional). Thus, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler-Lagrange equations of motion. By using the Hamilton-Jacobi approach to the geodesic equation, this statement can be given a very intuitive meaning: geodesics describe the motions of particles that are not experiencing any forces. Geodesics can also be defined as extremal curves for the following action energy functional E(γ) = 1 2Zg(γ0(t), γ0(t))dt, 1 where g is a Riemannian (or pseudo-Riemannian) metric. The defining property of geodesics is not that they minimize global lengths, but rather just local. The geodesic equation can then be obtained as the Euler-Lagrange equations of motion for this action. In this paper we study the normal trajectories of a charged particle in presence of a time-independent magnetic field in 3D manifolds, so that in physical terminology our approach belongs to the classical magnetostatic theory [19]. In Section 2 we first introduce some definitions and examples and we recall that the problem of characterizing magnetic flowlines in 2D manifolds (surfaces) from a global variational principle for a magnetic field Fwas solved in [3] for a certain class of magnetic fields, namely, the so called Gaussian magnetic fields. The associated Lorentz force equation of these magnetic fields corresponds with the field equations of a variational problem that describe the massive relativistic bosons. On the other hand, for magnetic fields associated to a global potential ω∈Λ1(Mn), i.e., F=dω on (Mn, g),we exhibit the global action that characterize the magnetic trajectories of (Mn, g, F ).In particular, this would happen if the second De Rham co-homology group vanishes, H2(Mn)=0. But there are other interesting backgrounds where this class of magnetic fields arise. In fact, the magnetic field F=dω associated with a potential gauge ωon principal circle bundles P(Mn,S1),the Kähler (uniform) magnetic field F=kΩJ,on Kähler manifolds (Mn, J, g)[1, 10], or the contact magnetic field Fξon Sasakian manifolds (M2n+1, ϕ, ξ, g),are some of them. In Section 3, 3D Riemannian manifolds (M3, g)are considered, where the essential point is that the particular dimension enables one to study magnetic fields by using its associated vector field, which must be divergence free. Then, the Lorentz force equation takes a intuitive form. In Section 4 we particularize to the case of Killing vector fields (which are divergence free), and a conservation law arises between their trajectories and the associated magnetic flowlines. In Section 5 we consider Sasakian 3D manifolds, where the contact (or Reeb) vector field ξis a global unit Killing vector field, and consequently it defines a magnetic field Fξ.Then, we show that the Lorentz force φof Fξand the tensor field ϕ of the Sasakian structure satisfy φ≡ϕ. This fact makes easy to prove that the normal magnetic trajectories of Fξare helices with axis ξ. Finally, in Section 6, the existence of periodic magnetic flowlines is investigated. This belongs to a class of non trivial problems known classically as closed curve problem. The solution is obtained for the 3D unit sphere S3,which is a Sasakian model space with the usual induced structure from R4. 2 2 Preliminaries and generalities Let (Mn, g)be a nD Riemannian manifold. A closed 2-form Fon Mnis said to be a magnetic field. In such case we shall write (Mn, g, F).Examples of magnetic fields are the following. Example 2.1 The curl magnetic field. Let V∈X(Mn)be an arbitrary vector field, and ∇the Levi-Civita covariant derivative of (Mn, g).The associated curl 2-form curl(V)is defined (see [14], p. 95) by curl(V)(X, Y ) = g(∇XV, Y )−g(∇YV, X), for all X, Y ∈X(Mn), which is an exact 2-form. In fact, if V[denotes the g-equivalent 1-form of V, then curl(V) = d V [. Example 2.2 The magnetic field associated with a potential gauge. Magnetic fields are also related to connections on principal circle bundles P(Mn,S1).In fact, if ωdenotes a potential gauge (i.e., a connection 1-form) on P, the strength or curvature 2-form is defined, via the structure equation, by dω =Ω. For any local section σU:U⊂Mn→P, we define FU:= −iσ∗ U(Ω)∈Λ2(U), which clearly satisfies dFU= 0. However, FU=FVon the overlapping open subsets U, V ⊂Mn, and consequently we get a closed 2-form, F, on the whole Mn. In other words, there exists an associated magnetic field Fon Mn.Moreover, if ω0stands for another potential on the same circle bundle and F0is the associated magnetic field, then F−F0=dα, where α∈Λ1(Mn). Thus, each principal circle bundle P(Mn,S1)defines a De Rham cohomology class, [F]∈H2(Mn). In particular, if Mnis simply connected, then the correspondence between principal circle bundles on Mnand H2(Mn)is bijective, [11]. The Lorentz force of a magnetic field Fon (Mn, g)is defined to be the operator φgiven by g(φ(X), Y ) = F(X, Y ), for all X, Y ∈X(M). The magnetic trajectories of Fare curves γin Mnthat satisfy the Lorentz equation ∇γ0γ0=φ(γ0).(1) Since the Lorentz force is skew-symmetric we have d dtg(γ0, γ0) = 2g(∇γ0γ0, γ0) = 0, 3 that is, magnetic curves have constant speed v(t) =kγ0(t)k=v0.When the magnetic curve γ(t)is arc-length parametrized (v0= 1), then it is called a normal magnetic curve. A special class of magnetic fields on a Riemannian manifold is that defined by the family of parallel 2-forms (∇F= 0), which are called uniform magnetic fields, and they play an important role in the classical Landau-Hall problem [3]. Notice that they correspond with parallel Lorentz forces (∇φ= 0). The existence and uniqueness of geodesics, remain true when one considers magnetic curves associated with an arbitrary magnetic field. Thus, for each p∈Mnand v∈TpMnthere exists exactly one inextensible (i.e. maximal) magnetic curve, γ: (−², ²)→Mn, of (Mn, g, F), with γ(0) = pand γ0(0) = v (see [16], for instance). Nevertheless, the well known homogeneity result for geodesics works quite different in non trivial magnetic fields. Therefore, if γis the inextendible magnetic curve of (M, g, F)determined from the initial data (p, v), the curve β, defined by β(t) = γ(λt), λ ∈R\{0}, is a magnetic trajectory of (M, g, λF)and also, when λ > 0, of (M, (1/λ)g, F ), in both cases determined from initial data (p, λv). Furthermore, the whole families of magnetic curves of (M, g, F)and (M, λg, λF)coincides, for any constant λ > 0. Consequently [3], for a non trivial magnetic field there exists no affine connection on Mnwhose geodesics are the magnetic curves of (Mn, g, F). It was shown also in [3] that a magnetic curve, γ: (a, b)→Mn,a, b ∈R, a < b, of (Mn, g, F )can be extended to the whole R(and hence is magnetically complete) if (Mn, g)is geodesically complete. A very special case of magnetic field appears when it is globally defined by an exact 2-form F=dω. But in general, this potential always exists at least locally. Denote by Γpq the space of smooth curves that connect two fixed points p, q of Mn. Now, we consider the action L: Γpq →Rdefined by L(γ) = 1 2Zγ g(γ0, γ0)dt +Zγ ω(γ0)dt. (2) The tangent space of Γpq in γis made up of the smooth vector fields, Z, along γthat vanish at pand q. To compute the extremals of (2), we first observe that Z¡ω(γ0)¢=γ0(ω(Z)) −g(φ(γ0), Z). Now, a standard computation involving integration by parts allows one to compute the first variation of this action to be (δL)(γ)[V] = Zγ g¡∇γ0γ0−φ(γ0), V ¢dt + [ω(Z)]∂γ, 4 Since [ω(Z)]∂γ = 0,we get that (δL)(γ)[V] = 0 for any V∈TγΓpq,if and only if γis a solution of the Lorentz equation (1). Therefore, the Lorentz equation is indeed the Euler-Lagrange equation associated with the functional L. 3 Magnetic fields in dimension three Dimension three is an interesting case for studying magnetic fields. In fact, there are several important facts which allows us to make a little bit different approach to their treatment. Let (M3, g)be an oriented 3D Riemannian manifold with volume 3-form Ω3. First, 2-forms and vector fields may be considered as the same thing. In fact, if Fis a 2-form Fon (M3, g),the Hodge star operator ?provides a 1-form ? F and hence the g-equivalent vector field (? F)]∈X(M3). The converse trip starts with a vector field V∈X(M3), then consider its g-equivalent 1-form V[and then take its star, ? V [. In this way, one obtains a 2-form which can be also written, using the interior contraction, as ? V [=iVΩ3. This defines a one-to-one map between 2-forms and vector fields. On the other hand, notice that magnetic fields means divergence free vector fields. In fact, it is well known that the Lie derivative of the volume form Ω3 satisfies LVΩ3=d(iVΩ3) = div(V)Ω3.This means that the 2-form ? V [=iVΩ3 is closed if and only if div(V) = 0, i.e., the volume element is invariant by the local flows of V. This allow us to regard magnetic fields in dimension three as divergence free vector fields. In particular, uniform magnetic fields correspond to parallel vector fields: if Vis a vector field in M3which is parallel, i.e., ∇V= 0, then it obviously has divergence zero and F=iVΩ3is a closed 2-form defining a magnetic field on (M3, g). Furthermore, it is clear that ∇F= 0. Conversely, suppose that F is a uniform magnetic field on (M3, g)and take V∈X(M3)its corresponding vector field, i.e., that defined by F=iVΩ3. A direct computation shows that Ω3(∇XV, Y, Z) = (∇XF) (Y, Z) = 0, for any X, Y, Z ∈X(M3)which proves that ∇V= 0. The 3D case gives us the possibility to find out a suitable expression for the Lorentz force φ:T(M3)−→ T(M3),namely, that Lorentz force can be viewed in terms of the cross product as follows. Definition 3.1 The cross product of any two vector fields X, Y ∈X(M3)is the vector field X∧Y∈X(M3)defined by g(X∧Y, Z) = Ω3(X, Y, Z).(3) 5 Proposition 3.2 Let V∈X(M3)with div(V)=0.Then, the Lorentz force φ of the magnetic field FV=iVΩ3is given by φ(X) = V∧X. (4) Proof. It suffices to see that for any Y∈X(M3)Eq. (3) gives g(V∧X, Y ) = Ω3(V, X, Y ) = (iVΩ3)(X, Y ) = FV(X, Y ) = g(φ(X), Y ). ¤ Consequently, the Lorentz force equation (1) that provides the magnetic flow, can be written as ∇γ0γ0=V∧γ0.(5) In particular, this equation shows that any integral curve of the magnetic field Vis a magnetic trajectory if and only of it is a geodesic. It is clear that the cross product on a 3-dimensional oriented Riemannian manifold satisfies the following two identities X∧(Y∧Z) = g(X, Z)Y−g(X, Y )Z, g(X∧Y, X ∧Z) = g(X, X)g(Y, Z)−g(X, Y )g(X, Z), Remark 3.3 (i) A geometrical construction similar to that showed in Eq. (3) has been used to define special almost contact structures on 7-dimensional manifolds endowed with a 2-fold vector cross product [13]. On the other hand, the cross product defined by Eq. (4) is an example of the r-fold cross product on manifolds introduced by Brown and Gray (see Refs. [6] and [8]). (ii) The Hall effect is a classical phenomenon for uniform magnetic fields in Euclidean space, and so in a free of gravity environment. It explains the dynamics of an electric current flow, Xin R3, when exposed to a perpendicular uniform magnetic field, V. The basic physical principle underlying the Hall effect is the Lorentz force appearing in the Lorentz force equation. Therefore, Xexperiences a force, the Lorentz force, acting normal to both, Xand V, and it moves in response to this force and the force effected by its internal electric field. Now, after Eq. (4), we notice that the Hall effect also appears in a more general context. For example, it applies to any magnetic field in R3, not necessarily uniform. Moreover, it also works in any Riemannian 3D space, (M3, g), even with non trivial gravity. Therefore, we have that when an electric current flow, X, moves through a conductor (M3, g)and perpendicular to an applied magnetic field V, it 6 experiences a force, the Lorentz force given by (4), acting normal to both directions and it moves in response to this force and the force effected by its internal electric field. 4 Killing magnetic fields in 3D manifolds Killing vector fields on a Riemannian manifold, (Mn, g), are those generating local flows of isometries, that is, K∈X(Mn)is Killing if and only if LKg= 0,or equivalently, ∇Kis a skew-symmetric operator, g(∇XK, Y ) + g(X, ∇YK) = 0. It is clear that any Killing vector field on (Mn, g)is divergence free. Consequently, if n= 3, then every Killing vector field Kdefines a magnetic field FK=iKΩ3which will be called a Killing magnetic field. In particular, uniform magnetic fields, ∇K= 0, are obviously Killing. Therefore, the class of Killing magnetic fields constitutes an important family of magnetic fields. Besides the conservation law which asserts that the speed of any magnetic trajectory is constant, we prove now that the magnetic trajectories of Killing magnetic fields in dimension three have another additional conservation law. Lemma 4.1 A magnetic field FKin a 3D Riemannian manifold is Killing if and only if for any magnetic curve γof FKthe product g(K, γ 0)is a constant along γ. Proof. In fact, if Kis Killing and γis a magnetic trajectory of FKthen d dt g(K, γ 0) = g(∇γ0K, γ 0) + g(K, ∇γ0γ0) = 0. Conversely, for p∈M3and v∈TpM3,let γbe a magnetic trajectory of FK such that γ(0) = p, γ 0(0) = v. We have 0 = d dt g(K, γ 0) = g(∇γ0K, γ 0) + g(K, K ∧γ0) = g(∇γ0K, γ 0). Therefore, g(∇vK, v) = 0, which means that Kis Killing. ¤ The Killing vector fields Kwith constant length kKk=g(K, K)1/2are called infinitesimal translations and they are characterized by the following property: A Killing vector field Vis an infinitesimal translation if and only if their trajectories are geodesics. A well known obstruction to the existence of infinitesimal translations on a Riemannian manifold is obtained in terms of the Ricci curvature. In fact, for a Killing vector field of constant length one has 0 = 41 2kKk2=k∇Kk2−Ric(K, K). Therefore, a Riemannian manifold with negative definite Ricci tensor does not admit a non trivial infinitesimal 7 translation. Finally, obviously a non trivial infinitesimal translation has no zero. Therefore, if it exists on a compact manifold then its Euler number must be zero. A very simple example is the Euclidean 3D space (R3, gR3)and the magnetic field defined by the vector field K=∂z,which is an infinitesimal translation. It is well known that the trajectories of (R3, gR3, F∂z)are helices with axis ∂z, that is, γ(t) = (x0+acos t, y0+asin t, z0+bt),where (x0, y0, z0)∈R3and a, b ∈R. 5 The contact magnetic field In order to study a more interesting background, namely, the 3D unit sphere S3, we now briefly recall the notion of an important class of 3D manifolds having a distinguished Killing magnetic field: the Sasakian manifolds [5], which are a particular type of almost contact metric manifolds. Sasakian manifolds constitute in some sense an odd-dimensional analogue of Kähler manifolds. For a given odd-dimensional manifold M2n+1 an almost contact structure on M2n+1 is triple (ϕ, ξ, η)where ϕis a field of endomorphisms of the tangent spaces, a global vector field ξand 1-form ηsuch that η(ξ) = 1, ϕ2(X) = −X+η(X)ξ, (6) for any X∈X(M2n+1).As a consequence of equations (6) we have also ϕ(ξ) = 0 and η·ϕ= 0.Moreover, ϕhas rank 2n. A Riemannian metric gon the almost contact manifold (M2n+1, ϕ, ξ, η)is said to be adapted or compatible [5] if for all X, Y ∈X(M2n+1)the following equation is satisfied: g(ϕ(X), ϕ(Y)) = g(X, Y )−η(X)η(Y).(7) Then (M2n+1, ϕ, ξ, η, g)is called an almost contact metric manifold. An immediate consequence is that g(ξ, ξ)=1and ηis the covariant form of ξ, that is, η(X) = g(ξ, X).The fundamental 2-form Φof the structure is defined by Φ(X, Y ) = g(X, ϕ(Y)).When the fundamental 2-form satisfies Φ = dη, then (M2n+1, ϕ, ξ, η, g)is called a contact metric manifold. In a contact metric manifold the integral curves of ξare geodesics. A contact metric manifold such that the vector field ξis a Killing vector field with respect to gis called a K-contact manifold. The first basic property of a K-contact manifold is that ∇Xξ=−ϕ(X).(8) ASasakian manifold is a contact metric manifold (M2n+1, ϕ, ξ, η, g)such that (∇Xϕ)Y=g(X, Y )ξ−η(Y)X, (9) 8 for all X, Y ∈X(M2n+1).It is easy to see that any Sasakian manifold is K- contact. In dimension 3, the converse is true. Now we shall prove that in any 3D almost contact metric manifold (in particular, Sasakian manifolds) the tensor field ϕis completely determined. Proposition 5.1 Let (M3, ϕ, ξ, g)be a 3D almost contact metric manifold. Then, the structure (1,1)-tensor field ϕis given by ϕ(X) = ξ∧X, (10) for any X∈X(M3). Proof. Let Gbe a local coordinate neighborhood in M3and take a unit vector field Uon Gorthogonal to ξ. Then ϕ(U)is a unit vector field which is orthogonal to Uand ξand hence {ξ, U, ϕ(U)}is a local orthonormal frame which is called aϕ-basis [5]. Define the orientation on M3such that the volume element Ω3 satisfies Ω3(ξ, U, ϕ(U)) = 1.First, it is clear that 0 = ϕ(ξ) = ξ∧ξ. On the other hand, as ϕ(U)is a unit vector field orthogonal to ξand U, then ϕ(U) = ±ξ∧U. But g(ξ∧U, ϕ(U)) = Ω3(ξ, U, ϕ(U)) = 1,and hence ϕ(U) = ξ∧U. Finally, we have ϕ(ϕ(U)) = −U=−ϕ(U)∧ξ=ξ∧ϕ(U).Therefore ϕ(X) = ξ∧Xfor any X∈X(M3). ¤ Recall that in a Sasakian manifold (M3, ϕ, ξ, g)the contact (or Reeb) vector field ξis Killing. Therefore, we have naturally defined on M3the Killing magnetic field Fξ=iξΩ3,which will be called the contact magnetic field of the Sasakian 3D manifold. The Lorentz force φof this magnetic field and the tensor ϕof the Sasakian structure are tied together as follows. Corollary 5.2 Let (M3, ϕ, ξ, g)be a 3D Sasakian manifold. Then, the structure tensor ϕand the Lorentz force φof the Killing magnetic field Fξsatisfy ϕ≡φ. (11) Proof. It follows immediately from Eq. (3) and Proposition 5.1. ¤ Next we shall determine the normal magnetic trajectories of the contact magnetic field Fξ. Theorem 5.3 Let (M3, ϕ, ξ, g)be a Sasakian manifold. The normal flowlines γ(t)of the contact magnetic field Fξare the helices of axis ξwith constant curvature κ0= sin θ0>0and constant torsion τ0= 1 + cos θ0,where θ0is the (constant) angle between γ0(t)and ξγ(t). 9