2.2 Adapted connections on a contact metric manifold

A. Manea, C. Ida

Abstract.In this paper we develop the gauge theory on a contact man- ifold. We consider a Lagrangian which is supposed to be invariant under a global action of a Lie group and we obtain the equation of motion and the conservation laws. In order to get a local gauge invariant Lagrangian, we introduced some gauge fields and determine what form have to take such an invariant Lagrangian.

M.S.C. 2010: 53C12, 53C60.

Key words: Contact metric manifold; adapted connection; gauge theory.

1 Introduction

This paper is about Lagrangians depending by r scalar fields on a contact metric manifold. It answers at the question: ”What have to be the form of a Lagrangian to be invariant at the local action of a Lie group, also called infinitesimal transformation?”.

The concept of a non-abelian gauge theory as a generalization of Maxwell’s theory of electromagnetism, was introduced by Yang and Mills. A brief survey of interac- tion between the work of physics community and the mathematicians about gauge theory and differential manifolds could be found in [9]. In monographies [7], [8] were given the basics facts and tehnics of gauge field theories. Some topological aspects of gauge theory on contact 3-manifolds were studied in [10], [14], [15]. The topic of gauge-invariant Lagrangians in complex geometry was discussed in [11], [12], [13].

The gauge-invariance of Lagrangians and the gauge fields for tangent bundle and for foliated manifolds are the subjects of [1], [5], respectively.

Following the general case of foliated manifolds from [5], for a Lagrangian invari- ant at coordinates transformation, in this paper we express the equation of motions and the conservation laws for the scalar fields using some adapted connections on a contact manifold. So, the first section of the paper is devoted to determine the adapted connections. In the second section we consider a Lagrangian invariant at the coordinate transformation and we study what form it has to take for being invariant at global action of a Lie group, in subsection 2.1, then to be invariant at local action, in subsection 2.2. Here we need to introduce some new fields, called gauge fields, to ensure the local invariance. The last subsection is devoted to study the behaviour of gauge fields at local action of the Lie group.

Balkan Journal of Geometry and Its Applications, Vol.24, No.1, 2019, pp. 26-44.∗

⃝c Balkan Society of Geometers, Geometry Balkan Press 2019.

2 Contact metric manifolds

2.1 An adapted frame field on a contact metric manifold

LetM be a (2n+ 1)-dimensional manifold and (φ, ξ, η) an almost contact structure onM. That is,φis a tensor field of type (1,1),ξa vector field, called theReeb vector fieldonM, andη a 1-form onM, such that

(2.1) φ²=−I+η⊗ξ, η(ξ) = 1.

Moreover, if the (2n+1)-formη∧(dη)ⁿdoesn’t vanishes everywhere onM then (M, η) is acontact manifold.

A Riemannian metric compatible with the almost contact structure (φ, ξ, η) is a Riemannian metricg onM such that

(2.2) g(φX, φY) =g(X, Y)−η(X)η(Y), ∀X, Y ∈Γ(T M).

A manifoldM endowed with an almost contact structure and a Riemannian metric compatible with it is called analmost contact metric manifold.

There are well-known the following properties which derive from the conditions (2.1) and (2.2):

(2.3)

(a) φξ= 0,(b) φ³=−φ ,(c) η◦φ= 0,(d) η(X) =g(X, ξ), (e) dη(ξ, X) = 0, for everyX ∈Γ(T M). Also, if the almost contact metric manifold is contact, then we have

(2.4) dη(X, Y) =ϕ(X, Y),∀X, Y ∈Γ(T M), whereϕis thefundamental (orSasaki) 2-form onM given by (2.5) ϕ(X, Y) =g(X, φY),∀X, Y ∈Γ(T M).

Moreover, the almost contact metric manifold is said to be: K–contactif it is contact andξis Killing; normalif [φ, φ] + 2dη⊗ξ= 0;Sasakianif it is contact and normal.

IfM is Sasakian manifold then it isK–contact [6].

Also, we consider thecontact distributionDdefined by the subspaces Dx={Xx∈TxM|ηx(Xx) = 0},

which is the transversal distribution to thecharacteristic foliationFξ (1-dimensional foliation determined by the Reeb vector fieldξ). Then, the structural distribution of characteristic foliationFξ isTFξ :=⟨ξ⟩={f ξ|f ∈C^∞(M)}.

According to the general theory of foliations, [17, 18, 19], we can choose a local coordinate system (U, x= (x⁰, xⁱ)),i ∈ {1, . . . ,2n}, adapted to foliationFξ, that is ξ=∂/∂x⁰ onU.

Then, byη(ξ) = 1, we deduce that (2.6) η=dx⁰+ηidxⁱ, ηi=η

( ∂

∂xⁱ )

,∀i∈ {1, . . . ,2n}.

This allows us to consider onU the local basis {

∂/∂x⁰, δ/δxⁱ}

, i= 1, . . . ,2n, called adapted toFξ, where

(2.7) δ

δxⁱ = ∂

∂xⁱ −η_i ∂

∂x⁰,∀i∈ {1, . . . ,2n}. Obviously, the set{

δ/δxⁱ}

, i∈ {1, . . . ,2n} is a local basis in Γ(D|U), and the dual basis of{

∂/∂x⁰, δ/δxⁱ} is

(2.8) {

η, dx¹, dx², . . . , dx²ⁿ} .

In the following we shall evaluate the Lie brackets for the vector fields from the adapted basis{

∂/∂x⁰, δ/δxⁱ}

. By relation (2.3)(e) and

2dη(X, Y) =X(η(Y))−Y(η(X))−η[X, Y],∀X, Y ∈Γ(T M),

it followsη[ξ, X] =−ξ(η(X)). So, for anyX ∈Γ(D), we have [ξ, X]∈Γ(D). That

means [

∂

∂x⁰, δ δxⁱ

]

∈Γ(D).

On the other hand, a direct computation give us [ δ

δxⁱ, ∂

∂x⁰ ]

= ∂ηi

∂x⁰

∂

∂x⁰. We obtain that

(2.9)

[ ∂

∂x⁰, δ δxⁱ

]

= 0, η_i=η_i(x¹, x², ..., x²ⁿ).

Then, forδ/δxⁱ from (2.7), we can compute [ δ

δxⁱ, δ δx^j

]

= (∂ηi

∂x^j −∂ηj

∂xⁱ ) ∂

∂x⁰.

Also, we have the relations (2.4), (2.5) and (2.6), which express locally that

(2.10) dη

( δ δx^j, δ

δxⁱ )

=1 2

(∂ηi

∂x^j −∂ηj

∂xⁱ )

=ϕji=gjkφ^k_i, where we put

(2.11) ϕ_ij =ϕ ( δ

δxⁱ, δ δx^j

) , φ

( δ δxⁱ

)

=φ^j_i δ

δxⁱ, g_ij=g ( δ

δxⁱ, δ δx^j

) . Obviously,ϕij=−ϕjiandgij =gji, for alli, j∈ {1, . . . ,2n}. Hence, we obtain (2.12)

[ δ δxⁱ, δ

δx^j ]

= 2ϕ_ji ∂

∂x⁰.

By second relation (2.9) and (2.10) we remark that functionϕ_ij doesn’t depends by x⁰, for every i, j ∈ {1, . . . ,2n}. In the end of this subsection, we notice that the metricgcan be expressed with respect to adapted cobasis{dxⁱ, η},i∈ {1, . . . ,2n}in the form

(2.13) g=g_ijdxⁱ⊗dx^j+η⊗η.

2.2 Adapted connections on a contact metric manifold

Let us consider a contact metric manifold (M φ, ξ, η, g) as in the previous subsection and the Reeb foliationFξ on it, generated byξ. According to the orthogonal decom- positionT M =D ⊕ ⟨ξ⟩, we consider the projection morphismsv andhof Γ(T M) on Γ(⟨ξ⟩) and Γ(D), respectively.

According to the general theory of adapted connections on semi-Riemannian fo- liations, see [5], an adapted connection for the foliation Fξ (that means a linear connection on M which induces linear connections on both distributions D, ⟨ξ⟩), is given by

(2.14) ∇XY =h∇eXhY +v∇eXvY +h(Q(X, hY)) +v(Q(X, vY),

for anyX, Y ∈Γ(T M), where ∇e is an arbitrary linear connection onM andQis an arbitrary tensor field of type (1,2) onM.

In order to find some adapted connection on the contact metric manifold M, we shall use relation (2.14) for∇e the Levi-Civita connection of the metricg. Firstly, we compute the local coefficients of∇ewith respect to adapted local frame{

∂/∂x⁰, δ/δxⁱ} , using the well-known Koszul formula

2g(∇eXY, Z) =X(g(Y, Z))+Y(g(Z, X))−Z(g(Y, X))+g([X, Y], Z)−g([Y, Z], X)+g([Z, X], Y), and we obtain the following local expression of∇e:

(2.15)

∇e ^δ

δxj

δ

δxⁱ =F_ij^k_δx^δk + (

ϕij−¹₂^∂g_∂x^ij0

) ∂

∂x⁰,

∇e ^δ

δxj

∂

∂x⁰ =∇e ^∂

∂x0

δ δx^j =

(1

2g^{kl ∂g}_∂x^lj0 −φ^k_j ) δ

δx^k,

∇e ^∂

∂x0

∂

∂x⁰ = 0, where

(2.16) F_ij^k = 1

2g^kh (δghi

δx^j +δghj

δxⁱ −δgij

δx^h )

, and(

g^ij)

2n×2n is the inverse matrix of (gij)_2n×2n given in (2.11).

For an adapted connection∇^α, we denote its local coefficients by

(2.17)

∇α ^δ

δxj

δ δxⁱ =

α

F_ij^k _δx^δk, ∇^{α ∂}

∂x0

δ δxⁱ =

α

D^k_{i δx}^δk,

∇α ^δ

δxi

∂

∂x⁰ =

α

Li ∂

∂x⁰, ∇^α_∂x^∂₀ ∂x^∂0 =C^α _∂x^∂0.

Following some idea from [5], we consider four adapted connections on the contact metric manifold as follows.

Thefirst adapted connectionon the contact metric manifoldM is defined by (2.18) ∇^1XY =h∇eXhY +v∇eXvY, ∀X, Y ∈Γ(T M).

We notice that every vector fieldY ∈ Γ(T M) admits a decomposition with respect to the adapted basis{

∂/∂x⁰, δ/δxⁱ}

in the form

(2.19) Y =Yⁱ δ

δxⁱ +Y⁰ ∂

∂x⁰,

where, from δ/δxⁱ ∈Γ(D) = Kerη, we haveY⁰ =η(Y), so the projections ofY on Γ(⟨ξ⟩) and Γ(D) respectively, are given by

(2.20) vY =η(Y) ∂

∂x⁰, hY =Yⁱ δ δxⁱ. It results the following local form for (2.18):

(2.21) ∇^1X Y =X(η(Y)) ∂

∂x⁰ +η(Y)v∇eX

∂

∂x⁰ +X(Yⁱ) δ

δxⁱ +Yⁱh∇eX

δ δxⁱ. Using (2.21), by direct computation, we obtain the local coefficients of the first adapted connection∇¹ as

(2.22)

1

F_ij^k=F_ij^k,

1

D^k_i= 1 2g^kl∂gli

∂x⁰ −φ^k_i,

1

Li=C= 0.¹ Thesecond adapted connectionis defined by

(2.23) ∇^2X Y =h∇eXhY −h∇ehYvX+v∇eXvY −v∇evYhX, ∀X, Y ∈Γ(T M), and, by direct computation, its local coefficients are

(2.24)

2

F_ij^k=F_ij^k,

2

D^k_i= 0,

2

L_i=C²= 0.

Thethird adapted connection is defined by

(2.25) ∇^3XY =∇^1X Y +hQ(vX, hY), ∀X, Y ∈Γ(T M), where the tensor fieldQis defined by

g(hQ(vX, hY), hZ) =g(v[hY, hZ], vX).

Denoting bya^k_i the local components of the projection on Γ(D) of the vector field Q(

∂/∂x⁰, δ/δxⁱ)

, the above condition give usa^k_i =φ^k_i, so we obtain hQ

( ∂

∂x⁰, δ δxⁱ

)

=φ^k_i δ δx^k.

By direct computation, the local coefficients of the third adapted connections are (2.26)

3

F_ij^k=F_ij^k,

3

D^k_i= 1 2g^kl∂g_li

∂x⁰ +φ^k_i,

3

L_i=C³= 0.

Finally, thefourth adapted connection on the contact metric manifoldM, is defined as the average connection between∇¹ and∇³, that is

(2.27) ∇^4XY = 1 2

(1

∇^X Y+∇^3XY )

, ∀X, Y ∈Γ(T M),

and it has the same local coefficients with∇¹, excepting

4

D_i^k=g^{kl ∂g}_∂x^li₀.

Remark 2.1. (i) The horizontal coefficients of all four adapted connections coin- cides, that is

α

F_ij^k=F_ij^k,α∈ {1,2,3,4}.

(ii) The first adapted connection∇¹ is just the Schouten-Van Kampen connection associated to the Reeb foliationFξ, see for instance [3], p.107.

(iii) The second adapted connection∇² is just theD-connection on a contact metric manifold (introduced in [2]). Also, it can be viewed as the Vr˘anceanu connection or Vaisman connection associated to the Reeb foliationFξ, see [3, 18].

(iv) IfM isK-contact, then∂gij/∂x⁰= 0, and then∇²=∇⁴.

In the next section the adapted connections∇^α will be used to express the Euler- Lagrange equation for a Lagrangian on a contact manifold.

We finish this subsection with some considerations about basic connections (with respect to Reeb foliation) on a contact manifold. Generally speaking, on the foliated manifold (M,F) there is an adapted atlas whose coordinate system on the open set U ⊂M is (

xⁱ)

= (x^a, x^u), where a∈ {1, . . . , q}, u∈ {q+ 1, . . . , m}, such that the points in the same leafL∩Uhave their firstqcoordinates equal, and are distinguished by their last (m−q) coordinates. Locally, the structural bundle F is spanned by {∂/∂x^u},u∈ {q+ 1, . . . , m}.

Also, if we consider the canonical exact sequence associated to the foliation given by the integrable subbundleF, namely

0−→F −→^iF T M ^π−→^QF QF −→0,

then we recall that a connection∇on the normal bundle QF is said to be basicif

(2.28) ∇XY =π_QF[X,Ye]

for anyX ∈ Γ(F), Ye ∈Γ(T M) such thatπ_QF(eY) = Y. Obviously, the right-hand side of (2.28) does not depend by choice of vector field ˜Y, because the integrability ofF.

Now, returning to the contact metric manifoldM endoweed with the characteristic foliationFξ, and taking into account thatQFξ ∼=D, a linear connection ∇ onM is basic if and only if

(2.29) ∇ ^∂

∂x0Y =h [ ∂

∂x⁰,Ye ]

,

whereYe ∈Γ(T M) such that h(Ye) =Y. Locally, letYe =Y⁰∂/∂x⁰+Yⁱδ/δxⁱ. Then h(Ye) =Yⁱδ/δxⁱ and relation (2.29) locally becomes

∂Yⁱ

∂x⁰ +Y^kD_kⁱ =∂Yⁱ

∂x⁰, whereDⁱ_kare transversal (horizontal) components of∇ ^∂

∂x0

δ

δxⁱ. But the above equality means

Proposition 2.1. The connection∇ is basic if and only if all horizontal components of∇ ^∂

∂x0

δ

δxⁱ vanish.

Concerning now to adapted connections∇¹,∇²,∇³ and∇⁴, respectively, their locally coefficients given in (2.22), (2.24) and (2.26) show that

Proposition 2.2. From the four defined above connections, only the second connec- tion,∇², is basic with respect to the characteristic foliationFξ determined by the Reeb vector fieldξ.

3 Invariance of Lagrangians on a contact metric manifold

In [5], the equation of motion for r scalar fields Q^A, A ∈ {1, . . . , r}, on a semi- Riemannian foliated manifold (M,F, g), are expressed using covariant derivative with respect to the Vr˘anceanu connection on that manifold. In this section we apply that idea for the case of the contact metric manifold (M φ, ξ, η, g), endowed with the characteristic foliationFξ.

We start with a Lagrangian depending by r scalar fields Q^A = Q^A(x), A ∈ {1, . . . , r}, on the contact metric manifold (M φ, ξ, η, g), that is

(3.1) L(x) =L

(

Q^A(x),δQ^A

δxⁱ (x),∂Q^A

∂x⁰ (x) )

, which is invariant under the coordinate transformations onM.

Let us consider the functionH, locally defined byH(x) = √

|det(gij(x))|, i, j∈ {1, . . . ,2n}. From direct computation, we have the following transformation law in the intersectionU∩Ue ̸= f of two domains of local chart ofM

He = det

(∂xⁱ

∂ex^j

)·H , i, j∈ {1, . . . ,2n}.

Then

(3.2) L0(x) =H(x)· L(x),

is a Lagrangian density onM. Thus, the functional

(3.3) I(Ω) =

∫

Ω

L0(x)dx¹∧. . .∧dx²ⁿ∧η,

where Ω is a compact domain ofM, does not depend of the coordinates onM. As usual, we assume that the equations of motion for the fieldsQ^A(x) follow from the variational principleδ(I(Ω)) = 0. Hence, the Euler-Lagrange equations for fields Q^A are

(3.4) ∂L0

∂Q^A − ∂

∂xⁱ



 ∂L0

∂ (∂Q^A

∂xⁱ

)



− ∂

∂x⁰



 ∂L0

∂ (∂Q^A

∂x⁰

)



= 0.

Taking into account the relation (2.7), we obtain (from (3.1))

(3.5) ∂L0

∂ (∂Q^A

∂x⁰

) = ∂L0

∂ (∂Q^A

∂x⁰

)|δQA

δxi =ct.−ηi

∂L0

∂ (δQ^A

δxⁱ

).

Then, the equations (3.4) become

(3.6) ∂L0

∂Q^A− δ δxⁱ



 ∂L0

∂ (δQ^A

δxⁱ

)



− ∂

∂x⁰



 ∂L0

∂ (∂Q^A

∂x⁰

)|δQA δxi=ct



= 0,

and, from (3.2), we get

(3.7)

H· {

∂L

∂Q^A−_δx^δi

(

∂L

∂ (δQA

δxi

)

)

−_∂x^∂0

(

∂L

∂ (∂QA

∂x0

)|δQA δxi =ct

)}

= ^δH_δxi · ^∂L

∂(_δQA

δxi

)+_∂x^∂H₀ · ^∂L

∂(_∂QA

∂x0

)|δQA δxi =ct. Now we denote

(3.8) Qⁱ_A:= ∂L

∂ (δQ^A

δxⁱ

), Q⁰_A:= ∂L

∂ (∂Q^A

∂x⁰

)|δQA

δxi=ct, i∈ {1, . . . ,2n}. From the changing rules for horizontal vector fields, that is

δ

δxⁱ =∂xe^j

∂xⁱ δ δex^j,

it follows thatQⁱ_Aare components of rhorizontal vector fields hQA=Qⁱ_A δ

δxⁱ.

Taking into account relations (2.17), the covariant derivatives of Qⁱ_A, Q⁰_A with respect to an adapted connection∇^α,α= 1,2,3,4, given locally in subsection 2.2 are

Qⁱ_A|

j = δQⁱ_A

δx^j +Q^k_AF_kjⁱ , Qⁱ_A|

0 = ∂Qⁱ_A

∂x⁰ +Q^k_A

α

D_kⁱ, Q⁰_A|

0 = ∂Q⁰_A

∂x⁰ , Q⁰_A|

i= δQ⁰_A δxⁱ . Then, the equation (3.7) could be rewritten in the form

(3.9) ∂L

∂Q^A −Qⁱ_A|i−Q⁰_A|0 = (1

H δH δxⁱ −F_ij^j

)

Qⁱ_A+ 1 H

∂H

∂x⁰ ·Q⁰_A.

But, by direct calculus, we have 1

H δH δxⁱ = 1

2g^jsδgjs

δxⁱ , 1 H

∂H

∂x⁰ = 1 2g^js∂gjs

∂x⁰. On the other hand, taking into account the relations (2.16) it follows

F_ij^j = 1 H

δH δxⁱ.

Hence, we obtain that the equation of motion for the scalar fieldsQ^Ahave the form

(3.10) ∂L

∂Q^A −Qⁱ_A|

i−Q⁰_A|

0 =1

2g^js∂gjs

∂x⁰Q⁰_A,

where the covariant derivatives ofQⁱ_A,Q⁰_Aare taken with respect to one of the adapted connections introduced in subsection 2.2.

Remark 3.1. IfM isK-contact then the equation of motion for the scalar fieldsQ^A simplify in the nice form

(3.11) ∂L

∂Q^A −Qⁱ_A|

i−Q⁰_A|

0 = 0.

3.1 Globally gauge invariance

In this section we study the invariance of the Lagrangian (3.1) under the action of an arbitrarym-dimensional Lie groupGon the physical fieldsQ^A(x). We also consider thatGadmits ar-dimensional representationρ.

A Lie groupGis essentially uniquely determnined by its Lie algebra, defined by the basis{Xa}, a∈ {1, . . . , m}. The representationρassigns to every vector fieldXa

ar×r-matrix ([Xa]^A_B)r×r,A, B∈ {1, . . . , r}. There are well known relations [X_a, X_b] =C_ab^c X_c, C_ab^c =−C_ba^c ,

where the structure constantsC_ab^c obey the Jacobi identity C_ab^d C_dc^e +C_bc^dC_da^e +C_ca^d C_db^e = 0.

Moreover, the matrices generators are satisfying

(3.12) [Xa]^A_B[Xb]^B_C−[Xb]^A_B[Xa]^B_C=C_ab^c [Xc]^A_C.

Now, according to [3, 5], the groupGbeing given, for any vector fieldX =ε^aX_a on G, a global gauge action of Gon the scalar physical fieldsQ^A(x), A∈ {1, . . . , r}, is given by the infinitesimal transformations

(3.13) Q^′A(x) =Q^A(x) +δ(Q^A(x)), δ(Q^A(x)) =ε^a[Xa]^A_BQ^B(x).

Applying the operatorsδ/δxⁱ and∂/∂x⁰ to (3.13), we obtain

(3.14)

δQ^′A

δxⁱ =^δQ_δx^Ai +δ (δQ^A

δxⁱ

) , δ

(δQ^A δxⁱ

)

=ε^a[X_a]^A_B^δQ_δx^Bi

∂Q^′A

∂x⁰ =^∂Q_∂x^A0 +δ (∂Q^A

∂x⁰

) , δ

(∂Q^A

∂x⁰

)

=ε^a[Xa]^A_B^∂Q_∂x^B0 .

Now, we suppose that the Lagrangian (3.1) isglobally gauge G-invariant, that is, L is invariant under the infinitesimal transformations (3.13) and (3.14) This means thatδL= 0, or equivalently,Ldoes not depend byε^a. It follows that

(3.15)



 ∂L

∂Q^AQ^B+ ∂L

∂ (δQ^A

δxⁱ

)δQ^B

δxⁱ + ∂L

∂ (∂Q^A

∂x⁰

)|δQA δxi =ct

∂Q^B

∂x⁰



[Xa]^A_B = 0,

or, equivalently (3.16)

[ ∂L

∂Q^AQ^B+Qⁱ_AδQ^B

δxⁱ +Q⁰_A∂Q^B

∂x⁰ ]

[X_a]^A_B= 0, with the notations (3.8).

From relation (3.10) it follows (3.17) ∂L

∂Q^A[Xa]^A_BQ^B=Qⁱ_A|

i[Xa]^A_BQ^B+Q⁰_A|

0[Xa]^A_BQ^B+1 2g^js∂gjs

∂x⁰Q⁰_A[Xa]^A_BQ^B. Then it is natural to consider the scalar fields

J_aⁱ =−Qⁱ_A[X_a]^A_BQ^B, J_a⁰=−Q⁰_A[X_a]^A_BQ^B,

which are components of m horizontal vector fields hJ_a = J_aⁱδ/δxⁱ, and m colin- ear vertical vector fieldsvJ_a =J_a⁰ξ, called horizontal currentsandvertical currents, respectively.

Taking into account relations (2.17), the covariant derivatives of J_aⁱ, J_a⁰ with re- spect to an adapted connection∇^α, α= 1,2,3,4, given locally in subsection 2.2, are given by

J_aⁱ_|

j =δJ_aⁱ

δx^j +J_a^kF_kjⁱ , J_a⁰_|

0= ∂J_a⁰

∂x⁰. But, we have

δJ_aⁱ

δx^j =−δQⁱ_A

δx^j [Xa]^A_BQ^B−Qⁱ_A[Xa]^A_BδQ^B δx^j ,

∂J_a⁰

∂x⁰ =−∂Q⁰_A

∂x⁰ [X_a]^A_BQ^B−Q⁰_A[X_a]^A_B∂Q^B

∂x⁰ ,

and replacing (3.17) in (3.16) and, using also the expressions of covariant derivatives of fieldsQⁱ_A,Q⁰_A with respect to the same connection∇^α, we obtain the conservation laws

(3.18) J_aⁱ_|i+J_a⁰_|0 =1

2g^js∂gjs

∂x⁰Q⁰_A[Xa]^A_BQ^B.

According to the terminology from [3, 5], the vector fieldshJa andvJa will be called called the horizontal and the vertical currents on the contact metric manifold M, respectively.

Remark 3.2. If M is K-contact then the above conservation laws reduce in the simple form

(3.19) J_aⁱ_|

i+J_a⁰_|

0 = 0.

3.2 Locally gauge invariance

A group of global transformations is characterized by the parameters ϵ^a being in- dependent by the coordinates (x⁰, xⁱ). In this subsection we suppose now that the parameters of the group are coordinates dependent, that means that the action ofG on fieldsQ^A(x) is local. In this situation, the scalar fieldsQ^A(x) transform according to

(3.20) Q^′A(x) =Q^A(x)+δ^∗(Q^A(x)),^∗δ(Q^A(x)) =ε^a(x)[Xa]^A_BQ^B(x).

Then, from above relations, we have

(3.21) δQ^′A

δxⁱ = δQ^A

δxⁱ +ε^a(x)[X_a]^A_BδQ^B δxⁱ +δε^a

δxⁱ[X_a]^A_BQ^B,

(3.22) ∂Q^′A

∂x⁰ = ∂Q^A

∂x⁰ +ε^a(x)[Xa]^A_B∂Q^B

∂x⁰ + ∂ε^a

∂x⁰[Xa]^A_BQ^B.

Now, we have to remark that a globally invariant Lagrangian may be not invariant under the local transformations (3.20). The variation of the Lagrangian is

∗δ(L) = ∂L

∂Q^A

δ∗(Q^A) + ∂L

∂ (δQ^A

δxⁱ

)δ^∗ (δQ^A

δxⁱ )

+ ∂L

∂ (∂Q^A

∂x⁰

)|δQA δxi =ct

δ∗

(∂Q^B

∂x⁰ )

= ε^a



 ∂L

∂Q^AQ^B+ ∂L

∂ (δQ^A

δxⁱ

)δQ^B

δxⁱ + ∂L

∂ (∂Q^A

∂x⁰

)|δQA δxi=ct

∂Q^B

∂x⁰



[Xa]^A_B

+



 ∂L

∂ (δQ^A

δxⁱ

)δε^a

δxⁱ + ∂L

∂ (∂Q^A

∂x⁰

)|δQA δxi=ct

∂ε^a

∂x⁰



Q^B[Xa]^A_B.

Taking into account that the Lagrangian satisfy relation (3.15) (the global invariance), we obtain the variation ofLby the form

δ∗(L) = [

Qⁱ_Aδε^a

δxⁱ +Q⁰_A∂ε^a

∂x⁰ ]

Q^B[Xa]^A_B.

Hence, we need to add some new fields, calledgauge fields, see [7, 8], to obtain a locally invariant Lagrangian.

More exactly, we consider the horizontal and vertical 1-forms (3.23) H^a =H_i^a(x)dxⁱ, i∈ {1, . . . ,2n}, a∈ {1, . . . , r} and

(3.24) ζ^a=σ^a(x)η , a∈ {1, . . . , r}, respectively, whereH_i^a, σ^a∈C^∞(M).

Sinceη is a global 1-form onM, the functionsσ^aare globally defined onM, while H_i^a are locally defined functions onM, and they have to transform as it follows (at the local coordinate changing onM)

(3.25) H_i^a= ∂ex^j

∂xⁱHe_j^a. Now, we ask thatL=L(

Q^A, δQ^A/δxⁱ, ∂Q^A/∂x⁰, H_i^a, σ^a)

is an invariant Lagrangian to the local action of G. According to [7], the gauge fields have to transform as it follows:

(3.26) ^∗δ(H_i^a) =ε^bC_bc^aH_i^c+δε^a

δxⁱ, δ^∗(σ^a) =ε^bC_bc^aσ^c+∂ε^a

∂x⁰, and

δ∗(L) = ∂L

∂Q^A

δ∗(Q^A) + ∂L

∂ (δQ^A

δxⁱ

) ^∗δ (δQ^A

δxⁱ )

+ ∂L

∂ (∂Q^A

∂x⁰

)|δQA δxi=ct

δ∗

(∂Q^A

∂x⁰ )

+ ∂L

∂H_i^a

δ∗(H_i^a) + ∂L

∂σ^a

∗δ(σ^a),

must vanishes. That is



 ∂L

∂Q^AQ^B+ ∂L

∂ (δQ^A

δxⁱ

)δQ^B

δxⁱ + ∂L

∂ (∂Q^A

∂x⁰

)|δQA δxi =ct

∂Q^B

∂x⁰



[Xa]^A_Bε^a

+ [ ∂L

∂H_i^aC_bc^aH_i^c+ ∂L

∂σ^aC_bc^aσ^c ]

ε^b+



 ∂L

∂ (δQ^A

δxⁱ

)[X_a]^A_BQ^B+ ∂L

∂H_i^a



δε^a δxⁱ

+



 ∂L

∂ (∂Q^A

∂x⁰

)|δQA

δxi =ct[X_a]^A_BQ^B+ ∂L

∂σ^a



∂ε^a

∂x⁰ = 0.

Taking into account that parameters functions ε^a(x) are arbitrary, we obtain the following equivalent conditions for the vanishing ofδ^∗(L):

(3.27)

∂L

∂Q^A[Xa]^A_BQ^B+Qⁱ_A[Xa]^A_BδQ^B

δxⁱ +Q⁰_A[Xa]^A_B∂Q^B

∂x⁰ + ∂L

∂H_i^bC_ac^b H_i^c+ ∂L

∂σ^bC_ac^b σ^c= 0,

(3.28) Qⁱ_AQ^B[Xa]^A_B+ ∂L

∂H_i^a = 0, Q⁰_AQ^B[Xa]^A_B+ ∂L

∂σ^a = 0.

In order to obtain identities (3.27) and (3.28), it is enough to add some additional fields enter into Lagrangian from some expressions like covariant derivatives, called thehorizontal and vertical gauge-covariant derivativesof physical fields:

(3.29) D_iQ^A= δQ^A

δxⁱ −H_i^a[X_a]^A_BQ^B, D₀Q^A= ∂Q^A

∂x⁰ −σ^a[X_a]^A_BQ^B.