Contact Magnetic Curves: Classification...

Contact Magnetic Curves: Classification Results

Ana Nistor

”Gh. Asachi” Technical University of Iasi, Romania

A.I. Nistor (TU Iasi) Valenciennes, France 28.03-04.04 2017 1 / 37

Outline

Overview of the talk

1 Preliminaries

Magnetic curvesSlant curvesQuasi-Sasakian manifolds3-dimensional case

2 Classification results for magnetic curves in:

Sasakian manifoldsCosymplectic manifoldsQuasi-Sasakian manifoldsQuasi-Sasakian 3-manifolds


Preliminaries Magnetic curves

Preliminaries - magnetic curves

• The starting point in the study of magnetic curves was the classicaltreatment of static magnetic fields (time-independent) in E3.• First results were obtained for magnetic fields on Riemannian manifolds.

We give now some general definitions.

• A closed 2-form F on a (complete) Riemannian manifold (M, g) is calleda magnetic field.• The Lorentz force of a magnetic background (M, g ,F ) is the skewsymmetric (1, 1)−type tensor field Φ on M satisfying

g(Φ(X ),Y ) = F (X ,Y ), ∀X ,Y ∈ χ(M). (1)

• A trajectory generated by the magnetic field F is defined as a smoothcurve γ on M fulfilling the Lorentz equation (or Newton equation):

∇γ γ = Φ(γ), (2)

∇ denotes the Levi Civita connection of g .A.I. Nistor (TU Iasi) Valenciennes, France 28.03-04.04 2017 3 / 37


In a physical interpretation: the magnetic curve γ describes the trajectoryof a charged particle under the action of F in the magnetic background(M, g ,F ).

• The skew symmetry of Lorentz force yields a basic property of magneticcurves, i.e. the following conservation law: particles evolve with constantspeed (and so with constant energy) along the magnetic trajectories.

• Property for magnetic curves: ddt g(γ′, γ′) = 0. In particular, a magnetic

curve is called normal if it has unit energy, i.e. ||γ′|| = 1.In the sequel we study only unit speed curves.



Magnetic curves and geodesics

• For trivial magnetic field F = 0 (the magnetic field is absent) magneticcurves correspond to geodesics of (M, g).

• Geodesics are characterized as critical points of an energy action and sothey represent the trajectories for free fall particles (moving only under theinfluence of the gravity). Magnetic curves of (M, g ,F ) can be also viewed(at least locally) as the solutions of a variational principle.

• The existence and uniqueness of geodesics remain true for magneticcurves.



Magnetic curves on surfaces

M -oriented surface,dσ-the area element: dσ(X ,Y ) = 1 for any p.o. orth. X ,Y on M.

Any magnetic field on the surface M is determined from a smoothfunction f (the strength) by F = fdσ.A parallel magnetic field F , i.e. a magnetic field with constant strengthf = q, is called uniform magnetic field.

Proposition (Barros et al. 2005)

Let F = qdσ be a uniform magnetic field with constant strength q, on aRiemannian surface (M, g). A curve γ, with constant velocity v , hence con-stant energy, is a magnetic curve of (M, g ,F ) if and only if it has constantcurvature κ = q/v .



Uniform magnetic curves on E2 and S2(r)

On the Euclidean plane: circles with radius 1|q| .

On the 2-sphere: (small) circles with radius r√e√

e+r2q2(< r)

(e is the energy)



Uniform magnetic curves on H2(−r), r > 0

The situation in hyperbolic plane is quite different:Consider the upper half-plane model for the hyperbolic plane.The description of the flowlines are due to Comtet, 1987.

(i) if |q|√r> 1: geodesic circles, and therefore they are closed

curves;

(ii) if |q|√r≤ 1: non-closed curves of the upper half-plane; in

particular they are tangent to the boundary and they arehorocycles when |q| =

√r .



Landau Hall problem on surfaces

Landau Hall problem on a surface of revolution:M. Barros, J.L. Cabrerizo, M. Fernandez, A. Romero,The Gauss-Landau-Hall problem on Riemannian surfaces,J. Math. Phys. 46 (2005) 11, art. 112905.

The Landau-Hall problem on canal surfaces:M.I. Munteanu,The Landau Hall problem on canal surfaces,J. Math. Analysis Appl., 414 (2014) 2, 725–733.



Killing magnetic curves on Riemannian 3−manifolds

In the case of a 3-dimensional Riemannian manifold (M, g), 2-formsand vector fields may be identified via the Hodge star operator ? and thevolume form dvg of the manifold.

Thus, magnetic fields mean divergence free vector fields.In particular, Killing vector fields define an important class of magneticfields, called Killing magnetic fields.

Killing vector field V :

g(∇YV ,Z ) + g(∇ZV ,Y ) = 0

Killing vector fields −→ divergence free vector fields.



Killing magnetic curves on Riemannian 3−manifolds

In the case of a 3-dimensional Riemannian manifold (M, g), 2-formsand vector fields may be identified via the Hodge star operator ? and thevolume form dvg of the manifold.

Thus, magnetic fields mean divergence free vector fields.In particular, Killing vector fields define an important class of magneticfields, called Killing magnetic fields.

Killing vector field V :

g(∇YV ,Z ) + g(∇ZV ,Y ) = 0

Killing vector fields −→ divergence free vector fields.



One can define on M the cross product of two vector fields

g(X × Y ,Z ) = dvg (X ,Y ,Z ).

V is a Killing vector field on M:FV = dvg (V ,−,−) the corresponding Killing magnetic field.

Then, the Lorentz force corresponding to FV is Φ(X ) = V × X .

Consequently, the Lorentz equation can be written as

∇γ′γ′ = V × γ′.



S.L. Druta Romaniuc and M.I. Munteanu,Magnetic curves corresponding to Killing magnetic fields in E3,J. Math. Phys. 52, 113506 (2011).

M.I. Munteanu and A.I. Nistor,The classification of Killing magnetic curves in S2 × R,J. Geom. Phys. 62, (2012) 170 – 182.

The classification of magnetic curves in H2 × R was done in:

A. I. Nistor, Motion of charged particles in a Killing magnetic field inH2 × R, Rend. Sem. Mat. Univ. Politec. Torino, 2016.



Preliminaries - Frenet curves

According to [Blair2002], a curve γ in an m−dimensional Riemannianmanifold (M, g) is a Frenet curve of osculating order r , r ≥ 1 if thereexists an orthonormal frame of dimension r along γ, namelyT = γ, ν1, . . . , νr−1, such that

∇TT = κ1ν1,∇Tν1 = −κ1T + κ2ν2,∇Tνj = −κjνj−1 + κj+1νj+1, j = 2, . . . , r − 2∇Tνr−1 = −κr−1νr−2,

(3)

where κ1, κ2, . . . , κr−1 are positive C∞ functions of s. Moreover, κj iscalled the j-th curvature of γ.

D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds,Progress in Math. 203, 2002, Birkhauser, Boston-Basel-Berlin.


Preliminaries Quasi-Sasakian manifolds

Preliminaries - Quasi-Sasakian manifolds

“Sasakian manifolds have often been considered the odd-dimensionalanalogues of Kaehler manifolds. However, if M2n is a Kaehler manifold,M2n × R can be considered an odd-dimensional analogue, but M2n × Rcarries a natural cosymplectic (quasi-Sasakian of rank 1) structure. Thus,in a certain sense, quasi-Sasakian manifolds are better analogues ofKaehler manifolds.”

[D.E. Blair, The Theory of Quasi-Sasakian Structures, PhD Thesis,University of Illinois, 1966]



• A differentiable manifold M2n+1 is said to have an almost contact metricstructure (ϕ, ξ, η, g) if it admits a field ϕ of endomorphisms of tangentspaces, a vector field ξ, a 1-form η s.t.

η(ξ) = 1, ϕ2 = −I + η ⊗ ξ, ϕξ = 0, η ϕ = 0,

and a compatible Riemannian metric g such that

g(ϕX , ϕY ) = g(X ,Y )− η(X )η(Y ),

∀X ,Y ∈ X(M2n+1).• The fundamental 2-form of the almost contact metric structure(ϕ, ξ, η, g) is defined as a 2-form Ω on (M2n+1, ϕ, ξ, η, g) by

Ω(X ,Y ) = g(ϕX ,Y ), (4)

for all X , Y ∈ X(M2n+1).



• An almost contact metric manifold M2n+1 is said to be normal if thenormality tensor

S = Nϕ(X ,Y ) + 2dη(X ,Y )ξ

vanishes, where Nϕ is the Nijenhuis tensor field of ϕ defined by

Nϕ(X ,Y ) = [ϕX , ϕY ] + ϕ2[X ,Y ]− ϕ[ϕX ,Y ]− ϕ[X , ϕY ],

for any X ,Y ∈ X(M2n+1).



A normal almost contact metric manifold with Ω closed is called aquasi-Sasakian manifold.

If Ω = dη, then (M2n+1, ϕ, ξ, η, g) is called a contact metric manifold.

A normal contact metric manifold is called a Sasakian manifold.

A cosymplectic manifold is defined as a normal almost contact metricmanifold with both η and Ω closed.

According to [BlairPhD], the types of quasi-Sasakian manifolds range fromthe case of cosymplectic manifolds, dη = 0 (rank = 1) to the case ofcontact manifolds, η ∧ (dη)n 6= 0 (rank = 2n + 1).Recall that the rank of the quasi-Sasakian structure is the rank of the1-form η, i.e. η has rank = 2p if (dη)p 6= 0 and η ∧ (dη)p = 0, and hasrank = 2p + 1 if η ∧ (dη)p 6= 0 and (dη)p+1 = 0, see [BlairPhD].



Recall now the definition of a Kaehler manifold.

An almost complex manifold (B2k , J, gB) endowed with the almostcomplex structure J and the Riemannian metric gB such that

gB(JX ,Y ) = −gB(X , JY )

is called an almost Hermitian manifold.

If the Nijenhuis tensor field of the structure J vanishes, then the manifoldis Hermitian.

A Kaehler manifold is a Hermitian manifold with closed fundamental2-form, defined as

ΩB(X ,Y ) = gB(JX ,Y ).



quasi-Sasakian manifolds as product manifolds

We consider the quasi-Sasakian manifolds which can be written locally asa product of a Sasakian and a Kaeher manifold.• (N2p+1, ϕ, ξ, η, gN) a Sasakian manifold• (B2k , J, gB) a Kaehler manifold• M = N2p+1 × B2k is the product manifold endowed with thequasi-Sasakian structure (ϕ, ξ, η, g) defined as:

ϕ = ϕ+ J, ξ = (ξ, 0), η = η, g = gN + gB . (4)

Let γ be a smooth curve parametrized by arc-length s in thequasi-Sasakian manifold (M, ϕ, ξ, η, g) given as

γ : I → M, γ(s) = (γN(s), γB(s)), (5)

such that γN is a smooth curve in the Sasakian manifold N2p+1 and γB isa smooth curve in the Kaehler manifold B2k .


Preliminaries Slant curves

Preliminaries - Slant curves

Let M = (M, ϕ, ξ, η, g) be an almost contact metric manifold and γ(s) asmooth curve in M parametrized by arclength.

• The contact angle of γ is defined as the angle θ(s) ∈ [0, π] made by γwith the trajectories of ξ, that is we have

cos θ(s) = g(γ′(s), ξ).

• The curve γ(s) in M is said to be a slant curve if the contact angle θ isconstant.

• Slant curves of contact angle π/2 are called (almost) Legendre curves oralmost contact curves.


Preliminaries 3-dimensional case

Quasi-Sasakian 3-manifolds

Let M be a quasi-Sasakian 3-manifold. The following statements hold true:

rankM = 1 if and only if M is cosymplectic.

There are no quasi-Sasakian 3-manifolds with rankM = 2 ([Blair67]).

rankM = 3 if and only if η is a contact form on M.

Typical examples of cosymplectic 3-manifolds are: the Euclidean 3-spaceE3 and product manifolds S2 × R and H2 × R.

D.E. Blair, The theory of quasi-Sasakian structures, J. Differ. Geom. 1(1967), 331–345.



All the eight model spaces of Thurston geometry admit homogeneousalmost contact structure naturally associated to the metric.

In particular, other than Sol3, the naturally associated almost contactstructures are normal.

space forms and product spaces are cosymplectic.

The unit 3-sphere S3, the Heisenberg group Nil3 and the universalcovering SL2R of the special linear group SL2R equipped with thecompatible normal contact metric structure are Sasakian space forms.In particular, Nil3 is identified with the Sasakian space form R3(−3).

The hyperbolic 3-space H3 equipped with the compatible normalcontact metric structure is a Kenmotsu manifold.

The space Sol3 equipped with naturally associated almost contactstructure is a non-Sasakian contact metric 3-manifold.

Thus the six model spaces E3, S3, S2 × R, H2 × R, Nil3, SL2R arequasi-Sasakian.



Z. Olszak, Normal almost contact metric manifolds of dimension three,Ann. Pol. Math. 47 (1986), 41–50.Olszak studied quasi-Sasakian 3-manifolds and obtained the followingfundamental facts.Proposition Olszak

Let M be an almost contact metric 3-manifold. Then M is quasi-Sasakianif and only if M satisfies

(∇Xϕ)Y = α(g(X ,Y )ξ − η(Y )X

)(6)

for some function α satisfying dα(ξ) = 0.

On a quasi-Sasakian 3-manifold, we have

∇X ξ = −αϕX .

Note that on a quasi-Sasakian manifold of arbitrary odd dimension, ξ is aKilling vector field, especially, ∇ξξ = 0.


Classification results Magnetic curves in Sasakian manifolds

Contact magnetic fields

(M2n+1, ϕ, ξ, η, g) : a contact metric manifoldWe can define a magnetic field on M2n+1 by

Fq(X ,Y ) = qΩ(X ,Y )

where q is a real constant.We call Fq the contact magnetic field with the strength q.We assume q 6= 0.The Lorentz force φq associated to the contact magnetic field Fq

φq = qϕ



Magnetic curves in Sasakian manifolds

By definition, Sasakian manifolds: quasi-Sasakian manifolds of rank 2n+ 1.

The magnetic curves associated to the contact magnetic field defined bythe fundamental 2-form of a Sasakian manifold have maximum order 3.




Theorem (S.L. Druta-Romaniuc, J. Inoguchi, M.I. Munteanu, A.I.N.)

Let (N2p+1, ϕ, ξ, η, gN) be a Sasakian manifold and consider FqN , qN 6= 0,the contact magnetic field on N2p+1. Then γ is a normal magnetic curveassociated to FqN in N2p+1 if and only if γ belongs to the following list:

a) geodesics, obtained as integral curves of ξ;

b) non-geodesic ϕ-circles of curvature κ1 =√q2 − 1, for |q| > 1, and of

constant contact angle θ = arccos 1q ;

c) Legendre ϕ-curves in M2n+1 with curvatures κ1 = |q| and κ2 = 1, i.e.1-dimensional integral submanifolds of the contact distribution;

d) ϕ-helices of order 3 with axis ξ, having curvatures κ1 = |q| sin θ andκ2 = |q cos θ − 1|, where θ 6= π

2 is the constant contact angle.















































Magnetic curves in Kaehler manifolds

Magnetic curves in Kaehler manifolds were intensively studied by manyauthors, e.g.

T. Adachi, Kahler Magnetic Flow for a Manifold of Constant HolomorphicSectional Curvature, Tokyo J. Math. 18 (1995) 2, 473–483.

A. Comtet, On the Landau levels on the hyperbolic plane, Ann. of Phys.173 (1987), 185–209.

D. Kalinin, Trajectories of charged particles in Kahler magnetic fields, Rep.Math. Phys., 39 (1997), 299–309.

and it was shown that they are circles, thus, they have maximum order 2.


Classification results Magnetic curves in cosymplectic manifolds

Magnetic curves in cosymplectic manifolds

Cosymplectic manifolds: quasi-Sasakian manifolds of rank 1


Let (M2n+1, ϕ, ξ, η, g) be a cosymplectic manifold. Then γ is a normalmagnetic curve corresponding to the contact magnetic field FqM , qM 6= 0,on M2n+1, if and only if γ is given by one of the following cases:


b) Legendre circles of curvature κ1 = |qM |;c) ϕ−helices of order 3, with curvatures κ1 = |qM | sin θM ,

κ2 = |qM cos θM |, where θM 6= π2 is the constant contact angle of γ.





























Classification results Magnetic curves in quasi-Sasakian manifolds

Magnetic curves in quasi-Sasakian manifolds

Theorem (M.I. Munteanu, A.I.N.)

The magnetic curves corresponding to the contact magnetic field definedby the fundamental 2-form of a quasi-Sasakian manifold of arbitrarydimension have maximum order 5.



Idea of the proof

• (M = N2p+1 × B2k , ϕ, ξ, η, g) - a quasi-Sasakian manifold• the fundamental 2-form Ω on M is closed and it defines a contactmagnetic field,

Fq(X ,Y ) = qΩ(X ,Y ), (7)

where X ,Y ∈ X(M) and q is a real constant called the strength of Fq.The Lorentz force φq associated to Fq has the expression

φq = qϕ.

Hence, the curve γ parametrized by arc-length in (5) is a normal magneticcurve corresponding to the magnetic field Fq if and only of it is a solutionof the Lorentz equation:

∇γ γ = qϕγ. (8)



Using the product structure of the quasi-Sasakian manifold,

∇γ γ = ∇Nγ′Nγ′N +∇B

γ′Bγ′B (9)

= m2N∇N

γNγN + m2

B∇BγBγB

= m2NqNϕγN + m2

BqBJ γB .

and combining with the right hand side:

qϕγ = q(ϕγ′N + Jγ′B) (10)

= mNqϕ ˙γN + mBqJ γB .

one gets that the magnetic curves γN and γB correspond to contactmagnetic fields of strengths qN = q

mNin the Sasakian manifold N2p+1 and

qB = qmB

in the Kaehler manifold B2k , respectively.

Moreover, γN has maximum order 3 on the Sasakian manifold N2p+1

γB of order 2 on the Kaehler manifold B2k .


Classification results Magnetic curves in quasi-Sasakian 3-manifolds

Magnetic curves in quasi-Sasakian 3-manifolds

Let γ be a normal magnetic trajectory in a quasi-Sasakian 3-manifold Mwith respect to the Lorentz force qϕ. Namely, γ satisfies

∇γ′γ′ = q ϕγ′.

The first fundamental result is the following one.

Proposition

Every contact magnetic curve on a quasi-Sasakian 3-manifold is a slantcurve.



Curvature and torsion

On an arbitrary oriented Riemannian 3-manifold one can canonically definea cross product × of two vector fields X ,Y ∈ X(M):

g(X × Y ,Z ) = dvg (X ,Y ,Z ), for any Z ∈ X(M),

where dvg denotes the volume form defined by g .• When M is an almost contact metric 3-manifold, the cross product isgiven by the formula

X × Y = g(ϕX ,Y )ξ − η(Y )ϕX + η(X )ϕY .

Note that for a unitary vector field X orthogonal to ξ, the basisX , ϕX , ξ is considered to be positively oriented. Then we have

ξ × γ′ = ϕγ′.



Curvature and torsion

Take the Frenet frame field (T ,N,B) along γ. By definition T = γ′.• Hence, the magnetic equation is written as

∇γ′γ′ = qξ × γ′ = κN.

Consequently, we get:

γ has constant curvature κ = |q| sin θ;

and the torsion of γ is given by τ = α + q cos θ.

Recall that a magnetic curve in a Sasakian (respectively a cosymplectic)manifold is a helix. Unlike these situations, a magnetic curve on aquasi-Sasakian 3-manifold is not, in general, a helix.

In order to sustain this remark, we give the following example:



Example

We consider the following quasi-Sasakian 3-manifold introduced byJ. We lyczko, On Legendre curves in 3-dimensional normal almost contactmetric manifolds, Soochow J. Math. 33(2007), no. 4, 929-937.

Let M = (x , y , z) ∈ R3 | x > 0 be a half space. We equip M with aRiemannian metric g defined by

g = x2(dx2 + dy2) + η ⊗ η, where η = dz + 2xdy .

Then we can take a global orthonormal frame field

e1 =1

x

∂

∂x, e2 =

1

x

∂

∂y− 2

∂

∂z, e3 =

∂

∂z.

The Lie brackets satisfy

[e1, e2] = − 1

x2e2 −

2

x2e3, [e2, e3] = [e3, e1] = 0.



We define an endomorphism field ϕ by

ϕe1 = e2, ϕe2 = −e1, ϕe3 = 0

and put ξ = e3. Then (ϕ, ξ, η, g) is an almost contact metric structure onM. One can check that (M, ϕ, ξ, η, g) satisfies

∇X ξ =1

x2ϕX ,

where ∇ is the Levi-Civita connection of g .Thus, M is a quasi-Sasakian manifold with α = −1/x2 < 0.



Let us classify magnetic curves in We lyczko’s space.

The magnetic equation of We lyczko’s space writes as a system of threesecond order differential equations:

x2

x− 5y2

x− 2y z

x2+ x = −qy

6x y

x+

2x z

x2+ y = qx

−10x y − 4x z

x+ x = −2qxx .

(11)

We denoted by dot (·) the derivative with respect to the arclengthparameter s.



Solving (11), we get the following coordinate functions of a contactmagnetic curve in M:

x(s)2 = c0 + 2 sin θ

∫ s

0cos u(t)dt, (12)

where c0 is a positive constant.

y(s) = y0 + sin θ

∫ s

0

sin u(t)

x(t)dt, y0 ∈ R. (13)

z(s) = z0 + s cos θ − 2 sin θ

∫ s

0sin u(t)dt, z0 ∈ R. (14)

The key point is to obtain u, which is a solution of the followingintegro-differential equation:

2 cos θ + sin θ sin u(s) +(− q + u(s)

)[c0 + 2 sin θ

∫ s

0cos u(t)dt

]= 0.

Thus in general, normal magnetic curves in We lyczko space are not helices.A.I. Nistor (TU Iasi) Valenciennes, France 28.03-04.04 2017 34 / 37


Particular case: u = u0(const.)

The coordinate functions of the contact magnetic curve are:x(s) =

√c0,

y(s) = y0 + ε sin θ√c0

s,

z(s) = z0 + (cos θ − 2ε sin θ)s.Thus, along this magnetic curve, α is constant.Hence this magnetic curve is a helix.For θ = π

2 , that is γ is a Legendre magnetic curve, and for ε = 1 we obtain

γ(s) =

(√c0, y0 +

s√c0, z0 − 2s

).

Its strength is q = 1c0

.This magnetic curve is a helix with κ = τ = 1/c0 > 0.



Some references

S.L. Druta Romaniuc, J. Inoguchi, M.I. Munteanu, A.I. Nistor,Magnetic curves in Sasakian manifolds,J. Nonlinear Math. Phys., 22 (2015) 3, 428–447.

S.L. Druta Romaniuc, J. Inoguchi, M.I. Munteanu, A.I. Nistor,Magnetic curves in cosymplectic manifolds,Reports on Math. Physics, 78 (2016) 1, 33–48.

J. Inoguchi, M.I. Munteanu, A.I. Nistor, Magnetic curves inquasi-Sasakian 3−manifolds, preprint.

M.I. Munteanu, A.I. Nistor, A note on magnetic curves on S2n+1,C.R. Acad. Sci. Paris, Ser. I 352 (2014) 447–449.

M.I. Munteanu, A.I. Nistor, On some closed magnetic curves on a3−torus, Math. Phys. Analysis Geom. 20 (2017)2, art.8.

A.I. Nistor, New examples of F-planar curves, preprint.A.I. Nistor (TU Iasi) Valenciennes, France 28.03-04.04 2017 36 / 37

The end

Thank you for attention !

Supported by: Research project PN-II-RU-TE-2014-4-0004,272/01.10.2015, CNCS - UEFISCDI, Director: Professor Dorel Fetcu


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