Riemann-Lagrange Geometrical Background for Multi-Time Physical Fields

Mircea Neagu

Abstract

The aim of this paper is to create a large geometrical background which to allow the including of famous equations of mathematical physics (Maxwell or Einstein equations) as particular cases. The geometrical construction is realized on the 1-jet fibre bundle J^{1}(T, M) → T × M, using only a Kronecker h-regular quadratic Lagrangian function

L = h_{αβ}(t)g^{ij}(t, x)x_{α}^{i}x_{β}^{j} + U_{(α)}^{(i)}(t, x)x_{α}^{i} + F (t, x).

Section 1 exposes the main reasons that determined our study. Also, it contains certain physical and geometrical aspects of the already classical Lagrangian geometry of physical fields from [12], whose ideas represent the start point in our generalized metrical multi-time Lagrangian approach of the theory of physical fields. Section 2 introduces the ML_{p}^{n} spaces that represent the natural houses for our generalized field theory. A characterization theorem for these spaces is given. At the same time, the main local features of geometrical objects produced on a ML_{p}^{n} space are described. We reffer to the canonical nonlinear connection Γ, the generalized Cartan Γ-linear connection CΓ, together with its torsion and curvature d-tensors. Section 3 presents the metrical multi-time Lagrange theory of electromagnetism and describes its generalized Maxwell equations. Section 4 presents the generalized Einstein equations which govern the metrical multitime Lagrange theory of gravitational field. The generalized conservation laws of the gravitational field are derived.

Mathematics Subject Classification (2000): 53B40, 53C60, 53C80 Key words: 1-jet fibre bundle, metrical multi-time Lagrange space ML_{p}^{n}, generalized Cartan connection, generalized Maxwell and Einstein equations.

1 Geometrical and physical aspects

On the one hand, it is well known that the 1-jet fibre bundle J^{1}(T, M) → T × M is a basic object in the study of classical and quantum field theories [21]. On the other hand, the construction of a new field theory, described in multi-time terms on J^{1}(T, M), was imposed of certain relativistic invariant equations involving many time variables (chiral fields, sine-Gordon etc.), and of KP-hierarchy of integrable systems, in which the arbitrary variables t^{α} and t^{β} are quite equal in rights and there is no reason to prefer one to another by choosing it as time [3].

Balkan Journal of Geometry and Its Applications, Vol.6, No.2, 2001, pp. 49-70 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Using the covariant Hamiltonian multisymplectic or polisymplectic formalism, many researchers were studied in this direction [5], [6]. Therefore, we consider that a contravariant Lagrangian development of a field theory on J^{1}(T, M), created by Riemannian geometrical methodes, promises unpublished new points of view in this topic. Moreover, we believe that a contravariant generalized multi-time Riemann-Lagrange formalism may have interesting conections with covariant Hamiltonian multisymplectic or polysymplectic formalism of great interest in relativity, numerical theory and geometrical integrators theory [2], [23], [28].

We should like to point out that the construction of a generalized multi-time Riemann-Lagrange field theory was begun in [13]. In order to have a good distinction between the theory from paper [13] and the Riemann-Lagrange theory developed in this paper, let us try to emphasize the main differences and similitudes of these multi-time field theories.

Firstly, the geometrical construction exposed in [13] is realized on the configuration bundle ⊕_{α=1}^{p} IT IM → M, where the coordinates of α-th copy of IT IM are denoted (x^{i}, x_{α}^{i}). The geometrical invariance group of this vector bundle stands out by the ignoring of multi-time reparametrizations:

\{ \begin{matrix}x̃^{i} = x̃^{i}(x^{j}) \\ x̃_{α}^{i} = \frac{∂x̃^{i}}{∂x^{j}} x_{α}^{j}. \end{matrix} (1.1)

This fact emphasizes the absolute character of the multi-time coordinates involved in theory, which are regarded as fixed coordinates on IR^{p}. Comparatively, our geometrical construction is realized yet on the more suitable physical configuration bundle J^{1}(T, M) → T × M, whose local coordinates are (t^{α}, x^{i}, x_{α}^{i}). The geometrical invariance group of the jet fibre bundle of order one J^{1}(T, M), induced by the transformation group of T × M, stands out by the relativistic character of the multi-time coordinates t^{α}:

\{ \begin{matrix}t̃^{α} = t̃^{α}(t^{β}) \\ x̃^{i} = x̃^{i}(x^{j}) \\ x̃_{α}^{i} = \frac{∂x̃^{i}}{∂x^{j}} \frac{∂t^{β}}{∂t̃^{α}} x_{β}^{j}, \end{matrix} (1.2)

where the meaning of x_{α}^{i} is that of partial velocities or, alternatively, of partial directions.

Secondly, the geometrical development from [13] is realized by a given semi-Riemannian metric h_{αβ} on IR^{p}, together with an ”a priori” fixed nonlinear connection N = (N_{(α)j}^{(i)}) and a given ”Lagrangian” L : ⊕_{α=1}^{p} IT IM → IR.

Remark 1.1 In order to have a clearer understanding of geometrical concepts used in [13], comparatively with the concepts from this paper, we point out that in the paper [13] a Lagrangian is viewed as a real smooth function on E = ⊕_{α=1}^{p} IT IM. In contrast, we use the following distinct notions on E = J^{1}(T, M):

i) multi-time dependent Lagrangian function − A smooth map L : J^{1}(T, M) → IR.

ii) multi-time Lagrangian (Olver’s terminology) − A local function L on J^{1}(T, M) which transform by the rule L̃ = L| det J|, where J is the Jacobian matrix of coordinate transformations t^{α} = t^{α}(t̃^{β}). If L is a Lagrangian function on 1-jet fibre bundle, then L = L\sqrt{|h|} represent a Lagrangian on J^{1}(T, M).

For that reason, we used the quotation marks for the above notion of Lagrangian.

The geometrical objects taken in study in the the paper [13] allow the construction of the Sasakian-like metric on ⊕_{α=1}^{p} IT IM, whose physical meaning is that of multi-time gravitational potential, namely

G = g_{ij}dx^{i} ⊗ dx^{j} + G_{(i)(j)}^{(α)(β)}δx_{α}^{i} ⊗ δx_{β}^{j}, (1.3)

where δx_{α}^{i} = dx_{α}^{i} + N_{(α)j}^{(i)}dx^{j},

G_{(i)(j)}^{(α)(β)}(x^{k}, x_{γ}^{k}) = \frac{1}{2} \frac{∂^{2}L}{∂x_{α}^{i}∂x_{β}^{j}} ,

g_{ij}(x^{k}, x_{γ}^{k}) = h_{αβ}G_{(i)(j)}^{(α)(β)}(x^{k}, x_{γ}^{k}).

Now, it is important to note that, after the local description of the generalized Cartan connection induced by G, together with its local d-torsions and d-curvatures, the geometrical theory developed in [13] is stoped. Consequently, the generalized multitime field theory, in the sense of generalized Maxwell and Einstein equations, is not described in [13]. We believe that the theory from [13] was stoped because of the very complicated computations that were involved in the local description of the Bianchi identities attached to the generalized Cartan connection. In our opinion the description of these local Bianchi identities should be decisive to create the subsequent generalized field theory because the multi-time geometrical approach from [13] try to extend the geometrical ideas from [12].

In this paper, using as a pattern the geometrical ideas from [12], we try to construct a generalized Riemann-Lagrange geometry of physical fields, but on the 1-jet fibre bundle J^{1}(T, M). On this bundle of configurations, our generalized field theory is geometrical created by a given vertical fundamental metrical d-tensor of particular form

G_{(i)(j)}^{(α)(β)} = h^{αβ}(t^{γ})g_{ij}(t^{γ}, x^{k}),

where h = (h_{αβ}) is temporal semi-Riemannian metric, which is produced by a Kronecker h-regular quadratic Lagrangian function L : J^{1}(T, M) → IR, that is,

G_{(i)(j)}^{(α)(β)}(x^{k}, x_{γ}^{k}) = \frac{1}{2} \frac{∂^{2}L}{∂x_{α}^{i}∂x_{β}^{j}} = h^{αβ}(t^{γ})g_{ij}(t^{γ}, x^{k}).

Using this geometrical object, we derive a nonlinear connection Γ = (M_{(α)β}^{(i)}, N_{(α)j}^{(i)}) and then we construct the Sasakian-like metric G on J^{1}(T, M),

G = h_{αβ}dt^{α} ⊗ dt^{β} + g_{ij}dx^{i} ⊗ dx^{j} + G_{(i)(j)}^{(α)(β)}δx_{α}^{i} ⊗ δx_{β}^{j}, (1.4)

where δx_{α}^{i} = dx_{α}^{i}+M_{(α)β}^{(i)}dt^{β} +N_{(α)j}^{(i)}dx^{j}. The physical meaning of G is that of a multitime gravitational h-potential. In this geometrical context, the canonical generalized Cartan connection CΓ produced by G, together with its thirty Bianchi identities, extremely used in the development of the generalized multi-time field theory, can be locally described [15]. Therefore, it naturally follows the geometrical construction of our generalized multi-time theory for electromagnetic and gravitational fields.

Remark 1.2 In our opinion, from a physical point of view, our generalized multitime theory of physical fields appears as a generalized unified field theory because the multi-time electromagnetic field F, that we will take in our study, will be directly derived from G. As a consequence, our generalized multi-time field theory can be included in the set of ”metrical” field theories. At the same time, taking into account the form of the geometrical invariance group (1.2) of this theory, we appreciate that the metrical multi-time Riemann-Lagrange theory developed by us, can be also included in the set of ”parametrized” field theories. For more details upon the clasification of field theories, see [6].

Remark 1.3 We consider that the key of succes in our generalized multi-time field theory, comparatively with the theory developed in [13], is provided by the Kronecker h-regularity of G_{(i)(j)}^{(α)(β)}, which allows the description of the geometrical objects or identities on J^{1}(T, M) by a reduced number of local adapted components. Altough the Kronecker h-regularity condition imposed to G_{(i)(j)}^{(α)(β)} may be interpreted as a too restrictive one, we appreciate that this fact is not true. In this direction, let us analyse the form of vertical metrical d-tensors G_{(i)(j)}^{(α)(β)} = \frac{1}{2} \frac{∂^{2}L}{∂x_{α}^{i}∂x_{β}^{j}} produced by

several Lagrangian functions which govern various famous physical domains (bosonic string theory, electrodynamics, elasticity, Kaluza-Klein dynamics of ideal fluids without viscosity, hydrodinamics etc.) studied in [6], [7], [16] and [22]. The Kronecker decomposition of the vertical metrical d-tensors derived from these important physical Lagrangian functions show the naturalness of the Kronecker h-regularity condition.

In the sequel, we try to expose the main geometrical and physical aspects of the Lagrangian theory of physical fields from [11], [12]. From our point of view, all these aspects may be easily extended to our multi-time geometrical and physical backgrounds. In this sense, we recall that a Lagrange space L^{n} = (M, L(x, y)) ( this concept represents the geometrical background in [11] and [12]) is defined as a pair which consists of a real, smooth, n-dimensional manifold M, whose coordinates are (x^{i})_{i=\overline{1,n}}, and a regular ”Lagrangian” L : IT IM → IR, not necessarily homogenous with respect to the direction (y^{i})_{i=\overline{1,n}}. Let us consider

g_{ij}(x^{k}, y^{k}) = \frac{1}{2} \frac{∂^{2}L}{∂y^{i}∂y^{j}} , (1.5)

the fundamental metrical d-tensor attached to the ”Lagrangian” L. From physical point of view, this d-tensor has the physical meaning of an ”unified” gravitational field on IT IM, which consists of one ”external” (x)-gravitational field spanned by points {x}, and the other ”internal” (y)-gravitational field spanned by directions {y}. It should be emphasized that y is endowed with some microscopic character of the space-time structure. Moreover, since y is a vertical vector field on IT IM, the y-dependence has combined with the concept of anisotropy.

The field theory developed on a Lagrange space L^{n} relies on a nonlinear connection Γ = (N_{j}^{i}(x, y)) attached naturally to the given ”Lagrangian” L. This plays the role of mapping operator of internal (y)-field on the external (x)-field, and prescribes the ”interaction” between (x)- and (y)- fields. From geometrical point of view, the nonlinear connection allows the construction of the adapted bases

\begin{matrix}\{\frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − N_{i}^{j} \frac{∂}{∂y^{j}} , \frac{∂}{∂y^{i}} \} ⊂ X (IT IM), \\ \{dx^{i}, δy^{i} = dy^{i} + N_{j}^{i}dx^{j}\} ⊂ X^{∗}(IT IM). \end{matrix} (1.6)

Concerning the ”unified” field g_{ij}(x, y) of L^{n}, the authors construct a Sasakian-like metric on IT IM, G = g_{ij}dx^{i} ⊗ dx^{j} + g_{ij}δy^{i} ⊗ δy^{j}. (1.7)

As to the spatial structure, the most important thing is to determine the Cartan canonical connection CΓ = (L_{jk}^{i}, C_{jk}^{i}) with respect to g_{ij}, which comes from the metrical conditions

\{ \begin{matrix}g_{ij|k} = \frac{δg_{ij}}{δx^{k}} − L_{ik}^{m}g_{mj} − L_{jk}^{m}g_{mi} = 0 \\ g_{ij}|_{k} = \frac{∂g_{ij}}{∂y^{k}} − C_{ik}^{m}g_{mj} − C_{jk}^{m}g_{mi} = 0, \end{matrix} (1.8)

where ”_{|k}” and ”|_{k}” are the local h- and v- covariant derivatives of CΓ. The importance to the Cartan canonical connection comes from its main role played in the Lagrangian theory of physical fields.

In this context, the Einstein-Miron-Anastasiei equations of the gravitational potentials g_{ij}(x, y) of a Lagrange space L^{n}, n > 2, are postulated as being the abstract geometrical Einstein equations attached to CΓ and G, namely [12]

\{ \begin{matrix}R_{ij} − \frac{1}{2}Rg_{ij} = KT_{ij}^{H}, & ′P_{ij} = KT_{ij}^{1}, \\ S_{ij} − \frac{1}{2}Sg_{ij} = KT_{ij}^{V} , & ′′P_{ij} = −KT_{ij}^{2}, \end{matrix} (1.9)

where R_{ij} = R_{ijm}^{m}, S_{ij} = S_{ijm}^{m},′P_{ij} = P_{ijm}^{m},′′P_{ij} = P_{imj}^{m} are the Ricci d-tensors of CΓ, R = g^{ij}R_{ij}, S = g^{ij}S_{ij} are the scalar curvatures, T_{ij}^{H}, T_{ij}^{V}, T_{ij}^{1}, T_{ij}^{2} are the components of the stress-energy tensor T (equal to 0 for vacuum), and K is the Einstein constant. Moreover, the stress-energy d-tensors T_{ij}^{H} and T_{ij}^{V} satisfy the conservation laws

KT _{j|m}^{H m} = −\frac{1}{2}(P_{js}^{hm}R_{hm}^{s} + 2R_{mj}^{s}P_{s}^{m}), KT _{j|m}^{V m} = 0, (1.10)

where all notations are described in [12].

The Lagrangian theory of electromagnetism relies on the canonical Liouville vector field C=y^{i}(∂/∂y^{i}) and the Cartan canonical connection CΓ of the Lagrange space L^{n}. In this context, the authors introduce the electromagnetic 2-form on IT IM,

F = F_{ij}δy^{i} ∧ dx^{j} + f_{ij}δy^{i} ∧ δy^{j}, (1.11)

where

\begin{matrix}F_{ij} = \frac{1}{2}[(g_{im}y^{m})_{|j} − (g_{jm}y^{m})_{|i}], \\ f_{ij} = \frac{1}{2}[(g_{im}y^{m})|_{j} − (g_{jm}y^{m})|_{i}]. \end{matrix} (1.12)

Using certain geometrical identities, they deduce that the vertical electromagnetic components f_{ij} vanish always. At the same time, using the Bianchi identities attached to the Cartan canonical connection CΓ, they conclude that the horizontal electromagnetic components F_{ij} are governed by the following Maxwell-Miron-Anastasiei equations: ,

\{ \begin{matrix}F_{ij|k} + F_{jk|i} + F_{ki|j} = − ∑_{\{i,j,k\}} C_{imr}R_{jk}^{r}y^{m} \\ F_{ij}|_{k} + F_{jk}|_{i} + F_{ki}|_{j} = 0. \end{matrix} (1.13)

Finally, we point out that physical aspects of the Lagrangian electromagnetism are studied by Ikeda in [8].

2 The geometry of ML_{p}^{n} spaces

Let us consider T (resp. M) a ”temporal” (resp. ”spatial”) manifold of dimension p (resp. n), whose coordinates are (t^{α})_{α=\overline{1,p}} (resp. (x^{i})_{i=\overline{1,n}}). Let

E = J^{1}(T, M) → T × M

be the jet fibre bundle of order one associated to these manifolds. We recall that the bundle of configuration J^{1}(T, M) has the local coordinates (t^{α}, x^{i}, x_{α}^{i}), where α = \overline{1, p} and i = \overline{1, n}. We underline that, throughout this paper, the indices α, β, γ, … run from 1 to p, and the indices i, j, k, … run from 1 to n.

Remark 2.1 In the particular case T = IR (i. e., the temporal manifold T is the usual time axis represented by the set of real numbers), the coordinates (t^{1}, x^{i}, x_{1}^{i}) of the 1-jet space J^{1}(IR, M) ≡ IR × IT IM are denoted (t, x^{i}, y^{i}). Note that the spaces J^{1}(IR, M) and IR× IT IM identify only punctually, the geometrical invariance groups of these spaces being different. Thus, the geometrical transformation group of J^{1}(IR, M) is given by

\{ \begin{matrix}t̃= t̃(t) \\ x̃^{i} = x̃^{i}(x^{j}) \\ ỹ^{i} = \frac{∂x̃^{i}}{∂x^{j}} \frac{dt}{dt̃}y^{j}. \end{matrix} (2.1)

Comparatively, the transformation coordinates on IR × IT IM are

\{ \begin{matrix}t̃= t \\ x̃^{i} = x̃^{i}(x^{j}) \\ ỹ^{i} = \frac{∂x̃^{i}}{∂x^{j}} y^{j}. \end{matrix} (2.2)

We start our study considering a smooth multi-time dependent Lagrangian function L : E → IR , which is locally expressed by

E ∋ (t^{α}, x^{i}, x_{α}^{i}) → L(t^{α}, x^{i}, x_{α}^{i}) ∈ IR.

Definition 2.1 The distinguished tensor field on J^{1}(T, M),

G_{(i)(j)}^{(α)(β)} = \frac{1}{2} \frac{∂^{2}L}{∂x_{α}^{i}∂x_{β}^{j}} , (2.3)

is called the vertical fundamental metrical d-tensor attached to L.

Now, let h = (h_{αβ}(t^{γ})) be a fixed semi-Riemannian metric on the temporal manifold T and g_{ij}(t^{γ}, x^{k}, x_{γ}^{k}) be a ”spatial” metrical d-tensor on E, symmetric, of rank n, and having a constant signature.

Definition 2.2 A multi-time dependent Lagrangian function L : E → IR , whose vertical fundamental metrical d-tensor (2.3) is of the form

G_{(i)(j)}^{(α)(β)}(t^{γ}, x^{k}, x_{γ}^{k}) = h^{αβ}(t^{γ})g_{ij}(t^{γ}, x^{k}, x_{γ}^{k}), (2.4)

is called a Kronecker h-regular multi-time dependent Lagrangian function with respect to the temporal semi-Riemannian metric h = (h_{αβ}).

In this context, we can introduce the following

Definition 2.3 A pair ML_{p}^{n} = (J^{1}(T, M), L), where p = dim T and n = dim M, which consists of the 1-jet fibre bundle and a Kronecker h-regular multi-time dependent Lagrangian function L : J^{1}(T, M) → IR is called a metrical multi-time Lagrange space or a ML_{p}^{n} space.

Remarks 2.2 i) In the particular case (T, h) = (IR, δ), a metrical multi-time Lagrange space is called a relativistic rheonomic Lagrange space and is denoted RRL^{n} = (J^{1}(IR, M), L). The Riemann-Lagrange geometry of RRL^{n} spaces is now completely developed in [17]. From our point of view, the geometrical framework exposed in [17] establishes a generalized geometric foundation for relativistic timedependent Lagrangian mechanics of first order variational problems because the geometrical invariance group is given by (2.1). This transformation group emphasizes the relativistic character of the time t. We invite the reader to compare the geometrization from [17] with that realized in [12]. The geometrical background constructed in [12] is created in order to geometrize the absolute time-dependent Lagrangian mechanics. The absolute character of the time t involved in study comes from the form (2.2) of the geometrical invariance group that governs the theory from [12].

ii) If the temporal manifold T is 1-dimensional, then, via a temporal reparametrization, we have J^{1}(T, M) ≡ J^{1}(IR, M). In other words, a metrical multi-time Lagrangian space having dim T = 1 can be regarded as a reparametrized relativistic rheonomic Lagrange space.

Example 2.1 Suppose that the spatial manifold M is also endowed with a semi-Riemannian metric g = (g_{ij}(x)). Then, the multi-time dependent Lagrangian function representing a basic object in the physical theory of bosonic strings, L_{1} : J^{1}(T, M) → IR, L_{1} = h^{αβ}(t)g_{ij}(x)x_{α}^{i}x_{β}^{j} (2.5)

is a Kronecker h-regular multi-time dependent Lagrangian function. Consequently, the pair

BSML_{p}^{n} = (J^{1}(T, M), L_{1})

is a ML_{p}^{n} space that is called the metrical multi-time Lagrange space of bosonic strings. We underline that the multi-time Lagrangian L_{1} = L_{1}\sqrt{|h|} is exactly the energy multi-time Lagrangian whose extremals are the harmonic maps between the semi-Riemannian manifolds (T, h) and (M, g). For more details, see [4].

Example 2.2 In above notations, taking U_{(i)}^{(α)}(t, x) as a d-tensor field on E and F : T × M → IR a smooth map, the more general multi-time dependent Lagrangian function

L_{2} : E → IR, L_{2} = h^{αβ}(t)g_{ij}(x)x_{α}^{i}x_{β}^{j} + U_{(i)}^{(α)}(t, x)x_{α}^{i} + F (t, x) (2.6)

is also a Kronecker h-regular multi-time dependent Lagrangian function. The metrical multi-time Lagrange space

EDML_{p}^{n} = (J^{1}(T, M), L_{2})

is called the autonomous metrical multi-time Lagrange space of electrodynamics because, in the particular case (T, h) = (IR, δ), we discover a natural relativistic generalization of the classical absolute rheonomic Lagrangian space of electrodynamics [12] which governs the movement law of a particle placed concomitantly into a gravitational field and an electromagnetic one. For that reason, from physical point of view, the semi-Riemannian metric h_{αβ}(t) (resp. g_{ij}(x)) represents the gravitational potentials of the space T (resp. M), the d-tensor U_{(i)}^{(α)}(t, x) stands for the electromagnetic potentials and F is a function which is called potential function. The non-dynamical character of spatial gravitational potentials g_{ij}(x) motivates us to use the terminology ”autonomous”. We point out that the main geometrical and physical aspects of EDML_{p}^{n} spaces are deeply studied [16]. The Riemann-Lagrange geometry of these spaces stands out by the multi-time and partial directions dependence of geometrical or physical objects that involves.

Example 2.3 More general, if we consider g_{ij}(t, x) a d-tensor field on E, symmetric, of rank n and having constant signature on E, we can define the Kronecker h-regular multi-time dependent Lagrangian function

L_{3} : E → IR, L_{3} = h^{αβ}(t)g_{ij}(t, x)x_{α}^{i}x_{β}^{j} + U_{(i)}^{(α)}(t, x)x_{α}^{i} + F (t, x). (2.7)

In this context, the pair

NEDML_{p}^{n} = (J^{1}(T, M), L_{3})

is a metrical multi-time Lagrange space which is called the non-autonomous metrical multi-time Lagrange space of electrodynamics. Physically, we remark that the gravitational potentials g_{ij}(t, x) of the spatial manifold M are dependent of the temporal coordinates t^{γ}, emphasizing their dynamic character.

An important role and, at the same time, an obstruction in the subsequent development of the metrical multi-time Lagrangian geometry, is played by the following theorem proved in [19]:

Theorem 2.1 (characterization of metrical multi-time Lagrange spaces)

If we have p = dim T ≥ 2, then the following statements are equivalent:

i) L is a Kronecker h-regular multi-time dependent Lagrangian function on J^{1}(T, M).

ii) The multi-time dependent Lagrangian function L reduces to a non-autonomous electrodynamics multi-time dependent Lagrangian function, that is,

L = h^{αβ}(t)g_{ij}(t, x)x_{α}^{i}x_{β}^{j} + U_{(i)}^{(α)}(t, x)x_{α}^{i} + F (t, x).

In other words, in the case p ≥ 2, any ML_{p}^{n} space is equivalent with a NEDML_{p}^{n} space.

A direct consequence of the previous characterization theorem is

Corollary 2.2 The fundamental vertical metrical d-tensor of an ML_{p}^{n} space has the Kronecker form

G_{(i)(j)}^{(α)(β)} = \frac{1}{2} \frac{∂^{2}L}{∂x_{α}^{i}∂x_{β}^{j}} = \{ \begin{matrix}h^{11}(t)g_{ij}(t, x^{k}, y^{k}), & p = 1 \\ h^{αβ}(t^{γ})g_{ij}(t^{γ}, x^{k}), & p ≥ 2. \end{matrix} (2.8)

Remarks 2.3 i) It is obvious that the Theorem 2.1 is an obstruction in the development of a fertile geometrical theory for ML_{p}^{n} spaces. This obstruction will be removed in a subsequent paper by the introduction of a more general notion, that of generalized metrical multi-time Lagrange space [14]. The generalized metrical multi-time Lagrange geometry and its derived theory of physical fields are constructed in [14] using a given h-regular fundamental vertical metrical d-tensor G_{(i)(j)}^{(α)(β)} on the 1-jet space J^{1}(T, M), which can be not provided by a multi-time dependent Lagrangian function, together with an ”a priori” fixed nonlinear connection Γ = (M_{(α)β}^{(i)}, N_{(α)j}^{(i)}).

ii) In the case p = dim T ≥ 2, the Theorem 2.1 obliges us to continue the study of the metrical multi-time Lagrangian space theory, channeling our attention upon the non-autonomous metrical multi-time Lagrange spaces of electrodynamics.

Following the Riemann-Lagrange geometrical development from the paper [19], the fundamental vertical metrical d-tensor G_{(i)(j)}^{(α)(β)} of the metrical multi-time Lagrange space ML_{p}^{n} = (J^{1}(T, M), L) naturally induces a canonical nonlinear connection, defined by the local components Γ = (M_{(α)β}^{(i)}, N_{(α)j}^{(i)}) on J^{1}(T, M).

Theorem 2.3 The canonical nonlinear connection Γ of the metrical multi-time Lagrange space ML_{p}^{n} = (J^{1}(T, M), L) is defined by the temporal adapted components

M_{(α)β}^{(i)} = \{ \begin{matrix}−H_{11}^{1}y^{i}, & p = 1 \\ −H_{αβ}^{γ}x_{γ}^{i}, & p ≥ 2 \end{matrix} (2.9)

and the spatial adapted components

N_{(α)j}^{(i)} = \{ \begin{matrix}h_{11} \frac{∂G^{i}}{∂y^{j}} , & p = 1 \\ Γ_{jk}^{i}x_{α}^{k} + \frac{g^{ik}}{2} \frac{∂g_{jk}}{∂t^{α}} + \frac{g^{ik}}{4} h_{αβ}U_{(k)j}^{(β)}, & p ≥ 2, \end{matrix} (2.10)

where

\begin{matrix}G^{i} = \frac{g^{ik}}{4} (\frac{∂^{2}L}{∂x^{j}∂y^{k}} y^{j} − \frac{∂L}{∂x^{k}} + \frac{∂^{2}L}{∂t∂y^{k}} + \frac{∂L}{∂x^{k}} H_{11}^{1} + 2h^{11}H_{11}^{1}g_{kl}y^{l}), \\ H_{αβ}^{γ} = \frac{h^{γη}}{2} (\frac{∂h_{ηα}}{∂t^{β}} + \frac{∂h_{ηβ}}{∂t^{α}} − \frac{∂h_{αβ}}{∂t^{η}} ), \\ Γ_{jk}^{i} = \frac{g^{im}}{2} (\frac{∂g_{mj}}{∂x^{k}} + \frac{∂g_{mk}}{∂x^{j}} − \frac{∂g_{jk}}{∂x^{m}} ), \\ U_{(k)j}^{(β)} = \frac{∂U_{(k)}^{(β)}}{∂x^{j}} − \frac{∂U_{(j)}^{(β)}}{∂x^{k}} . \end{matrix} (2.11)

Remarks 2.4 i) Considering the particular case (T, h) = (IR, δ), we emphasize that the canonical nonlinear connection Γ = (0, N_{(1)j}^{(i)}) of the relativistic rheonomic Lagrange space RRL^{n} = (J^{1}(IR, M), L) represents a natural generalization of the canonical nonlinear connection used in the classical Lagrangian geometry from [12].

ii) The canonical nonlinear connection Γ = (M_{(α)β}^{(i)}, N_{(α)j}^{(i)}) of a ML_{p}^{n} space allows the construction of adapted bases

\begin{matrix}\{\frac{δ}{δt^{α}} , \frac{δ}{δx^{i}} , \frac{∂}{∂x_{α}^{i}} \} ⊂ X (E), \\ \{dt^{α}, dx^{i}, δx_{α}^{i}\} ⊂ X^{∗}(E), \end{matrix} (2.12)

where

\begin{matrix}\frac{δ}{δt^{α}} = \frac{∂}{∂t^{α}} − M_{(β)α}^{(j)} \frac{∂}{∂x_{β}^{j}} \\ \frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − N_{(β)i}^{(j)} \frac{∂}{∂x_{β}^{j}} \\ δx_{α}^{i} = dx_{α}^{i} + M_{(α)β}^{(i)}dt^{β} + N_{(α)j}^{(i)}dx^{j}. \end{matrix} (2.13)

The simple tensorial transformation rules of the elements of above adapted bases determine us to study the geometrical and physical objects on ML_{p}^{n} spaces, at level of local adapted components.

The naturalness of the construction of the nonlinear connection of a metrical multitime Lagrange space ML_{p}^{n} = (J^{1}(T, M), L) comes from the following result, proved also in [19]:

Theorem 2.4 The Euler-Lagrange equations of the energy action functional

E(f) = ∫_{T} L(t^{γ}, x^{k}, x_{γ}^{k})\sqrt{|h|}dt,

where f = (x^{i}(t^{γ})), are equivalent with the harmonic maps equations attached to the nonlinear connection Γ = (M_{(α)β}^{(i)}, N_{(α)j}^{(i)}). In other words, the extremals of E are equivalent with the solutions of the harmonic map equations [18]

h^{αβ}{x_{αβ}^{i} + M_{(α)β}^{(i)} + N_{(α)j}^{(i)}x_{β}^{j}} = 0,

where x_{αβ}^{i} represent the second derivatives of x^{i}(t^{γ}).

The main result of the metrical multi-time Lagrange geometry is the theorem of existence of the Cartan canonical h-normal Γ-linear connection CΓ which allows the subsequent development of the metrical multi-time Lagrangian theory of physical fields.

Theorem 2.5 (of existence and uniqueness of Cartan canonical connection)

On the metrical multi-time Lagrange space ML_{p}^{n} = (J^{1}(T, M), L), endowed with its canonical nonlinear connection Γ, there is a unique h-normal Γ-linear connection [15], defined by adapted components

CΓ = (H_{αβ}^{γ}, G_{jγ}^{k}, L_{jk}^{i}, C_{j(k)}^{i(γ)}),

having the metrical properties:

i) g_{ij|k} = 0, g_{ij}|_{(k)}^{(γ)} = 0,

ii) G_{jγ}^{k} = \frac{g^{ki}}{2} \frac{δg_{ij}}{δt^{γ}} , L_{ij}^{k} = L_{ji}^{k}, C_{j(k)}^{i(γ)} = C_{k(j)}^{i(γ)}.

Moreover, the components L_{jk}^{i} and C_{j(k)}^{i(γ)} of the Cartan canonical connection have the expressions

\begin{matrix}L_{jk}^{i} = \frac{g^{im}}{2} (\frac{δg_{mj}}{δx^{k}} + \frac{δg_{mk}}{δx^{j}} − \frac{δg_{jk}}{δx^{m}} ) , \\ C_{j(k)}^{i(γ)} = \frac{g^{im}}{2} (\frac{∂g_{mj}}{∂x_{γ}^{k}} + \frac{∂g_{mk}}{∂x_{γ}^{j}} − \frac{∂g_{jk}}{∂x_{γ}^{m}} ) . \end{matrix} (2.14)

Remarks 2.5 i) A proof of the Theorem 2.5 can be found in the paper [14], in the more general context of generalized metrical multi-time Lagrange spaces.

ii) In the particular case (T, h) = (IR , δ), the Cartan canonical δ-normal Γ-linear connection of the relativistic rheonomic Lagrange space RRL^{n} = (J^{1}(IR, M), L) naturally generalizes the canonical Cartan connection used in the classical rheonomic Lagrange geometry from [12].

iii) As a rule, the Cartan canonical connection of a metrical multi-time Lagrange space ML_{p}^{n} verifies also the metrical properties

h_{αβ/γ} = h_{αβ|k} = h_{αβ}|_{(k)}^{(γ)} = 0, g_{ij/γ} = 0.

iv) In the case p = dim T ≥ 2, the components of the Cartan connection of a metrical multi-time Lagrange space reduce to those of a NEDML_{p}^{n} space, namely

Ḡ_{αβ}^{γ} = H_{αβ}^{γ}, G_{jγ}^{k} = \frac{g^{ki}}{2} \frac{∂g_{ij}}{∂t^{γ}} , L_{jk}^{i} = Γ_{jk}^{i}, C_{j(k)}^{i(γ)} = 0.

The general theorems from [15], which characterize the d-torsion and d-curvature of a h-normal Γ-linear connection, applied to the Cartan connection of a ML_{p}^{n} space, imply the following important results:

Theorem 2.6 The torsion d-tensor T of the Cartan canonical connection of a metrical multi-time Lagrange space ML_{p}^{n} is determined by the local adapted components:

\begin{aligned}& h_{T} & & h_{M} & & & v \\ & p = 1 & p ≥ 2 & p = 1 & p ≥ 2 & p = 1 & & p ≥ 2 \\ h_{T} h_{T} & 0 & 0 & 0 & 0 & 0 & & R_{(μ)αβ}^{(m)} \\ h_{M}h_{T} & 0 & 0 & T_{1j}^{m} & T_{αj}^{m} & R_{(1)1j}^{(m)} & & R_{(μ)αj}^{(m)} \\ h_{M}h_{M} & 0 & 0 & 0 & 0 & R_{(1)ij}^{(m)} & & R_{(μ)ij}^{(m)} \\ vh_{T} & 0 & 0 & 0 & 0 & P_{(1)1(j)}^{(m) (1)} & & P_{(μ)α(j)}^{(m) (β)} \\ vh_{M} & 0 & 0 & P_{i(j)}^{m(1)} & 0 & P_{(1)i(j)}^{(m) (1)} & & 0 \\ vv & 0 & 0 & 0 & 0 & 0 & & 0 \end{aligned} (2.15)

where,

i) for p = dim T = 1, we have

T_{1j}^{m} = −G_{j1}^{m}, P_{i(j)}^{m(1)} = C_{i(j)}^{m(1)}, P_{(1)1(j)}^{(m) (1)} = −G_{j1}^{m},

P_{(1)i(j)}^{(m) (1)} = \frac{∂N_{(1)i}^{(m)}}{∂y^{j}} − L_{ji}^{m}, \frac{δN_{(1)i}^{(m)}}{δx^{j}} − \frac{δN_{(1)j}^{(m)}}{δx^{i}},

R_{(1)1j}^{(m)} = −\frac{∂N_{(1)j}^{(m)}}{∂t} + H_{11}^{1} [N_{(1)j}^{(m)} − \frac{∂N_{(1)j}^{(m)}}{∂y^{k}} y^{k}];

ii) for p = dim T ≥ 2, denoting

F_{i(μ)}^{m} = \frac{g^{mp}}{2} [\frac{∂g_{pi}}{∂t^{μ}} + \frac{1}{2}h_{μβ}U_{(p)i}^{(β)}],

H_{μαβ}^{γ} = \frac{∂H_{μα}^{γ}}{∂t^{β}} − \frac{∂H_{μβ}^{γ}}{∂t^{α}} + H_{μα}^{η}H_{ηβ}^{γ} − H_{μβ}^{η}H_{ηα}^{γ},

r_{pij}^{m} = \frac{∂Γ_{pi}^{m}}{∂x^{j}} − \frac{∂Γ_{pj}^{m}}{∂x^{i}} + Γ_{pi}^{k}Γ_{kj}^{m} − Γ_{pj}^{k}Γ_{ki}^{m},

we have T_{αj}^{m} = −G_{jα}^{m}, P_{(μ)α(j)}^{m (β)} = −δ_{γ}^{β}G_{jα}^{m}, R_{(μ)α(j)}^{(m)} = −H_{μαβ}^{γ}x_{γ}^{m},

R_{(μ)αj}^{(m)} = −\frac{∂N_{(μ)j}^{(m)}}{∂t^{α}} + \frac{g^{mk}}{2} H_{μα}^{β} [\frac{∂g_{jk}}{∂t^{β}} + \frac{h_{βγ}}{2} U_{(k)j}^{(γ)}],

R_{(μ)ij}^{(m)} = r_{ijk}^{m}x_{μ}^{k} + [F_{i(μ)|j}^{m} − F_{j(μ)|i}^{m}]; Theorem 2.7 The curvature d-tensor R of the Cartan canonical connection of a ML_{p}^{n} space is determined by the local adapted components:

h_{T}  h_{M}  v
p = 1 p ≥ 2 p = 1 p ≥ 2  p = 1  p ≥ 2
h_{T} h_{T}  0  H_{ηβγ}^{α}  0  R_{iβγ}^{l}  0  R_{(η)(i)βγ}^{(l)(α)}
h_{M}h_{T}  0  0  R_{i1k}^{l}  R_{iβk}^{l}  R_{(1)(i)1k}^{(l)(1)} = R_{i1k}^{l}  R_{(η)(i)βk}^{(l)(α)}
h_{M}h_{M}  0  0  R_{ijk}^{l}  R_{ijk}^{l}  R_{(1)(i)jk}^{(l)(1)} = R_{ijk}^{l}  R_{(η)(i)jk}^{(l)(α)}
vh_{T}  0  0  P_{i1(k)}^{(l) (1)}  0  P_{(1)(i)1(k)}^{(l)(1) (1)} = P_{i1(k)}^{(l) (1)}  0
vh_{M}  0  0  P_{ij(k)}^{l (1)}  0  P_{(1)(i)j(k)}^{(l)(1) (1)} = P_{ij(k)}^{l (1)}  0
vv  0  0  S_{i(j)(k)}^{l(1)(1)}  0  S_{(1)(i)(j)(k)}^{(l)(1)(1)(1)} = S_{i(j)(k)}^{l(1)(1)}  0

where R_{(η)(i)βγ}^{(l)(α)} = δ_{η}^{α}R_{iβγ}^{l} + δ_{i}^{l}H_{ηβγ}^{α}, R_{(η)(i)βk}^{(l)(α)} = δ_{η}^{α}R_{iβk}^{l}, R_{(η)(i)jk}^{(l)(α)} = δ_{η}^{α}R_{ijk}^{l} and i) for p = dim T = 1, we have

R_{i1k}^{l} = \frac{δG_{i1}^{l}}{δx^{k}} − \frac{δL_{ik}^{l}}{δt} + G_{i1}^{m}L_{mk}^{l} − L_{ik}^{m}G_{m1}^{l} + C_{i(m)}^{l(1)}R_{(1)1k}^{(m)},

R_{ijk}^{l} = \frac{δL_{ij}^{l}}{δx^{k}} − \frac{δL_{ik}^{l}}{δx^{j}} + L_{ij}^{m}L_{mk}^{l} − L_{ik}^{m}L_{mj}^{l} + C_{i(m)}^{l(1)}R_{(1)jk}^{(m)},

P_{i1(k)}^{l (1)} = \frac{∂G_{i1}^{l}}{∂y^{k}} − C_{i(k)/1}^{l(1)} + C_{i(m)}^{l(1)}P_{(1)1(k)}^{(m) (1)},

P_{ij(k)}^{l (1)} = \frac{∂L_{ij}^{l}}{∂y^{k}} − C_{i(k)|j}^{l(1)} + C_{i(m)}^{l(1)}P_{(1)j(k)}^{(m) (1)},

S_{i(j)(k)}^{l(1)(1)} = \frac{∂C_{i(j)}^{l(1)}}{∂y^{k}} − \frac{∂C_{i(k)}^{l(1)}}{∂y^{j}} + C_{i(j)}^{m(1)}C_{m(k)}^{l(1)} − C_{i(k)}^{m(1)}C_{m(j)}^{l(1)};

ii) for p = dim T ≥ 2, we have

H_{ηβγ}^{α} = \frac{∂H_{ηβ}^{α}}{∂t^{γ}} − \frac{∂H_{ηγ}^{α}}{∂t^{β}} + H_{ηβ}^{μ}H_{μγ}^{α} − H_{ηγ}^{μ}H_{μβ}^{α},

R_{iβγ}^{l} = \frac{δG_{iβ}^{l}}{δt^{γ}} − \frac{δG_{iγ}^{l}}{δt^{β}} + G_{iβ}^{m}G_{mγ}^{l} − G_{iγ}^{m}G_{mβ}^{l},

R_{iβk}^{l} = \frac{δG_{iβ}^{l}}{δx^{k}} − \frac{δΓ_{ik}^{l}}{δt^{β}} + G_{iβ}^{m}Γ_{mk}^{l} − Γ_{ik}^{m}G_{mβ}^{l},

R_{ijk}^{l} = r_{ijk}^{l} = \frac{∂Γ_{ij}^{l}}{∂x^{k}} − \frac{∂Γ_{ik}^{l}}{∂x^{j}} + Γ_{ij}^{m}Γ_{mk}^{l} − Γ_{ik}^{m}Γ_{mj}^{l}.

3 Generalized Maxwell equations for a multi-time electromagnetic field

Let ML_{p}^{n} = (J^{1}(T, M), L) be a metrical multi-time Lagrange space, together with Γ = (M_{(α)β}^{(i)}, N_{(α)j}^{(i)}) its canonical nonlinear connection. Let us consider the Cartan canonical connection of ML_{p}^{n}, locally expresed by CΓ = (H_{αβ}^{γ}, G_{iγ}^{k}, L_{ij}^{k}, C_{i(j)}^{k(γ)}).

Using the canonical Liouville d-tensor C= x_{α}^{i}(∂/∂x_{α}^{i}) and the fundamental vertical metrical d-tensor G_{(i)(k)}^{(α)(β)} of the metrical multi-time Lagrange space ML_{p}^{n}, we can construct the metrical deflection d-tensors

\begin{matrix}D̄_{(i)β}^{(α)} = G_{(i)(k)}^{(α)(γ)}D̄_{(γ)β}^{(k)} = x_{(i)/β}^{(α)}, \\ D_{(i)j}^{(α)} = G_{(i)(k)}^{(α)(γ)}D_{(γ)j}^{(k)} = x_{(i)|j}^{(α)}, \\ d_{(i)(j)}^{(α)(β)} = G_{(i)(k)}^{(α)(γ)}d_{(γ)(j)}^{(k)(β)} = x_{(i)}^{(α)}|_{(j)}^{(β)}, \end{matrix} (3.16)

where x_{(i)}^{(α)} = G_{(i)(k)}^{(α)(γ)}x_{γ}^{k} and ”_{/β}”, ”_{|j}” and ”|_{(j)}^{(β)}” are the local covariant derivatives induced by CΓ.

Taking into account the expressions of the local covariant derivatives of CΓ (see the papers [15], [20]), by a direct calculation, we obtain

Proposition 3.1 The metrical deflection d-tensors of a metrical multi-time Lagrange space ML_{p}^{n} have the expressions:

i) for p = 1,

\begin{matrix}D̄_{(i)1}^{(1)} = \frac{h^{11}}{2} \frac{δg_{im}}{δt} y^{m}, \\ D_{(i)j}^{(1)} = h^{11}g_{ik} [−N_{(1)j}^{(k)} + L_{jm}^{k}y^{m}] , \\ d_{(i)(j)}^{(1)(1)} = h^{11} [g_{ij} + g_{ik}C_{m(j)}^{k(1)}y^{m}]; \end{matrix} (3.17)

ii) for p ≥ 2,

\begin{matrix}D̄_{(i)β}^{(α)} = \frac{h^{αγ}}{2} \frac{∂g_{km}}{∂t^{β}} x_{γ}^{m}, \\ D_{(i)j}^{(α)} = −\frac{h^{αγ}}{2} \frac{∂g_{ij}}{∂t^{γ}} − \frac{1}{4}U_{(i)j}^{(α)}, \\ d_{(α)(j)}^{(i)(β)} = h^{αβ}g_{ij}. \end{matrix} (3.18)

In order to construct the metrical multi-time Lagrangian theory of electromagnetism, we introduce

Definition 3.1 The distinguished 2-form on J^{1}(T, M), locally defined by

F = F_{(i)j}^{(α)}δx_{α}^{i} ∧ dx^{i} + f_{(i)(j)}^{(α)(β)}δx_{α}^{i} ∧ δx_{β}^{j}, (3.19)

where

\begin{matrix}F_{(i)j}^{(α)} = \frac{1}{2} [D_{(i)j}^{(α)} − D_{(j)i}^{(α)}], \\ f_{(i)(j)}^{(α)(β)} = \frac{1}{2} [d_{(i)(j)}^{(α)(β)} − d_{(j)(i)}^{(α)(β)}], \end{matrix}

is called the multi-time electromagnetic field of the space ML_{p}^{n}.

Remark 3.1 The naturalness of the previous definition comes considering the particular case of a relativistic rheonomic Lagrange space (i. e., (T, h) = (IR, δ)). In this particular case, we find a natural relativistic generalization of the electromagnetic d-tensors used in [12].

Proposition 3.2 The components F_{(i)j}^{(α)} and f_{(i)(j)}^{(α)(β)} of the electromagnetic d-form F of the metrical multi-time Lagrange space ML_{p}^{n} are described by the formulas:

i) in the case p = 1,

F_{(i)j}^{(1)} = \frac{h^{11}}{2} [g_{jm}N_{(1)i}^{(m)} − g_{im}N_{(1)j}^{(m)} + (g_{ik}L_{jm}^{k} − g_{jk}L_{im}^{k})y^{m}] , f_{(i)(j)}^{(1)(1)} = 0;

ii) in the case p ≥ 2,

F_{(i)j}^{(α)} = \frac{1}{8} [U_{(j)i}^{(α)} − U_{(i)j}^{(α)}] , f_{(i)(j)}^{(α)(β)} = 0.

Because the vertical electromagnetic components f_{(i)(j)}^{(α)(β)} vanish, it follows that the main laws that govern the electromagnetic metrical multi-time Lagrangian theory must be intimately conected by the horizontal electromagnetic components F_{(i)j}^{(α)}.

Theorem 3.3 The electromagnetic components F_{(i)j}^{(α)} of the metrical multi-time Lagrange space ML_{p}^{n} are governed by the following generalized Maxwell equations:

i) for p = 1,

\{ \begin{matrix}F_{(i)k/1}^{(1)} = \frac{1}{2}A_{\{i,k\}} \{D̄_{(i)1|k}^{(1)} + D_{(i)m}^{(1)}T_{1k}^{m} + d_{(i)(m)}^{(1)(1)}R_{(1)1k}^{(m)} − [T_{1i|k}^{p} + C_{k(m)}^{p(1)}R_{(1)1i}^{(m)}] y_{(p)}\} \\ ∑_{\{i,j,k\}} F_{(i)j|k}^{(1)} = −\frac{1}{2} ∑_{\{i,j,k\}} C_{(i)(l)(m)}^{(1)(1)(1)}R_{(1)jk}^{(m)}y^{l} \\ ∑_{\{i,j,k\}} F_{(i)j}^{(1)}|_{(k)}^{(1)} = 0, \end{matrix}

ii) for p ≥ 2,

\{ \begin{matrix}F_{(i)k/β}^{(α)} = \frac{1}{2}A_{\{i,k\}} \{D̄_{(i)β|k}^{(α)} + D_{(i)m}^{(α)}T_{βk}^{m} + d_{(i)(m)}^{(α)(μ)}R_{(μ)βk}^{(m)} − [T_{βi|k}^{p} + C_{k(m)}^{p(μ)}R_{(μ)βi}^{(m)}] x_{(p)}^{(α)}\} \\ ∑_{\{i,j,k\}} F_{(i)j|k}^{(α)} = 0 \\ ∑_{\{i,j,k\}} F_{(i)j}^{(α)}|_{(k)}^{(γ)} = 0, \end{matrix}

where y_{(p)} = G_{(p)(q)}^{(1)(1)}y^{q}, C_{(i)(l)(m)}^{(1)(1)(1)} = G_{(l)(q)}^{(1)(1)}C_{i(m)}^{q(1)} = \frac{h^{11}}{2} \frac{∂^{3}L}{∂y^{i}∂y^{l}∂y^{m}}, x_{(p)}^{(α)} = G_{(p)(q)}^{(α)(β)}x_{β}^{q}.

Proof. Firstly, we point out that the Ricci identities [20] applied to the spatial metrical d-tensor g_{ij} imply that the following curvature d-tensor identities:

R_{miβk} + R_{imβk} = 0, R_{mijk} + R_{imjk} = 0, P_{mij(k)}^{(γ)} + P_{imj(k)}^{(γ)} = 0,

where R_{miβk} = g_{ip}R_{mβk}^{p}, R_{mijk} = g_{ip}R_{mjk}^{p} and P_{mij(k)}^{(γ)} = g_{ip}P_{mj(k)}^{p (γ)}, are true. Now, let us consider the following general deflection d-tensor identities [20]: d_{1}) D̄_{(ν)β|k}^{(p)} − D_{(ν)k/β}^{(p)} = x_{ν}^{m}R_{mβk}^{p} − D_{(ν)m}^{(p)}T_{βk}^{m} − d_{(ν)(m)}^{(p)(μ)}R_{(μ)βk}^{(m)},

d_{2}) D_{(ν)j|k}^{(p)} − D_{(ν)k|j}^{(p)} = x_{ν}^{m}R_{mjk}^{p} − d_{(ν)(m)}^{(p)(μ)}R_{(μ)jk}^{(m)},

d_{3}) D_{(ν)j}^{(p)}|_{(k)}^{(γ)} − d_{(ν)(k)|j}^{(p)(γ)} = x_{ν}^{m}P_{mj(k)}^{p (γ)} − D_{(ν)m}^{(p)}C_{j(k)}^{m(γ)} − d_{(ν)(m)}^{(p)(μ)}P_{(μ)j(k)}^{(m) (γ)},

where D̄_{(α)β}^{(i)} = x_{α/β}^{i}, D_{(α)j}^{(i)} = x_{α|j}^{i}, d_{(α)(j)}^{(i)(β)} = x_{α}^{i}|_{(j)}^{(β)}. Contracting these deflection

d-tensor identities by G_{(i)(p)}^{(α)(ν)} and using the above curvature d-tensor equalities, we obtain the metrical deflection d-tensors identities:

d′_{1}) D̄_{(i)β|k}^{(α)} − D_{(i)k/β}^{(α)} = −x_{(m)}^{(α)}R_{iβk}^{m} − D_{(i)m}^{(α)}T_{βk}^{m} − d_{(i)(m)}^{(α)(μ)}R_{(μ)βk}^{(m)},

d′_{2}) D_{(i)j|k}^{(α)} − D_{(i)k|j}^{(α)} = −x_{(m)}^{(α)}R_{ijk}^{m} − d_{(i)(m)}^{(α)(μ)}R_{(μ)jk}^{(m)},

d′_{3}) D_{(i)j}^{(α)}|_{(k)}^{(γ)} − d_{(i)(k)|j}^{(α)(γ)} = −x_{(m)}^{(α)}P_{ij(k)}^{m (γ)} − D_{(i)m}^{(α)}C_{j(k)}^{m(γ)} − d_{(i)(m)}^{(α)(μ)}P_{(μ)j(k)}^{(m) (γ)}.

At the same time, we recall that the following Bianchi identities hold good [15]:

b_{1}) A_{{j,k}} {R_{jαk}^{l} + T_{αj|k}^{l} + C_{k(m)}^{l(μ)}R_{(μ)αj}^{(m)}} = 0,

b_{2}) ∑_{{i,j,k}} {R_{ijk}^{l} − C_{k(m)}^{l(μ)}R_{(μ)ij}^{(m)}} = 0,

b_{3}) A_{{j,k}} {P_{jk(p)}^{l (ε)} + C_{j(p)|k}^{l(ε)} + C_{k(m)}^{l(μ)}P_{(μ)j(p)}^{(m) (ε)}} = 0,

where A_{{j,k}} means alternate sum and ∑_{{i,j,k}} means cyclic sum.

Now, in order to obtain the first Maxwell identity, we permute i and k in d′_{1} and we subtract the new identity from the initial one. Finally, using the Bianchi identity b_{1}, we obtain what we were looking for.

Doing a cyclic sum by the indices {i, j, k} in d′_{2} and using the Bianchi identity b_{2}, it follows the second Maxwell equation.

Applying a Christoffel process to the indices {i, j, k} in d′_{3} and combining with the Bianchi identity b_{3} and the relation P_{(μ)j(p)}^{(m) (ε)} = P_{(μ)p(j)}^{(m) (ε)}, we get a new identity. The cyclic sum by the indices {i, j, k} applied to this last identity implies the third Maxwell equation.

Remark 3.2 In the particular case (T, h) = (IR, δ), we discover a natural generalization of the Maxwell equations appeared in the absolute rheonomic Lagrangian electromagnetism from [12].

4 Multi-time gravitational h-potential on ML_{p}^{n}. Generalized Einstein equations and conservation laws

Let h = (h_{αβ}) be a fixed semi-Riemannian metric on the temporal manifold T and Γ = (M_{(α)β}^{(i)}, N_{(α)j}^{(i)}) a fixed nonlinear connection on the 1-jet space E = J^{1}(T, M). In order to develope the metrical multi-time Lagrange theory of gravitational field, we introduce the following abstract physical concept:

Definition 4.1 From physical point of view, an adapted metrical d-tensor G on E, locally expressed by

G = h_{αβ}dt^{α} ⊗ dt^{β} + g_{ij}dx^{i} ⊗ dx^{j} + h^{αβ}g_{ij}δx_{α}^{i} ⊗ δx_{β}^{j},

where g_{ij} = g_{ij}(t^{γ}, x^{k}, x_{γ}^{k}) is a d-tensor field on E, symmetric, of rank n = dim M and having a constant signature on E, is called a multi-time gravitational h-potential.

Remark 4.1 In the particular case (T, h) = (IR, δ), the gravitational δ-potentials naturally generalize the gravitational potentials used in [12].

Now, taking ML_{p}^{n} = (J^{1}(T, M), L) a metrical multi-time Lagrange space, via its fundamental vertical metrical d-tensor G_{(i)(j)}^{(α)(β)} (see (2.8)) and its canonical nonlin-

ear connection Γ = (M_{(α)β}^{(i)}, N_{(α)j}^{(i)}) (see (2.9), (2.10)), we can construct a multi-time gravitational h-potential, naturally attached to the space ML_{p}^{n}, setting

G = h_{αβ}dt^{α} ⊗ dt^{β} + g_{ij}dx^{i} ⊗ dx^{j} + h^{αβ}g_{ij}δx_{α}^{i} ⊗ δx_{β}^{j}.

Let us consider CΓ = (H_{αβ}^{γ}, G_{jγ}^{k}, L_{jk}^{i}, C_{j(k)}^{i(γ)}) the Cartan canonical connection of ML_{p}^{n}.

We postulate that the generalized Einstein equations, which govern the multi-time gravitational h-potential G of the metrical multi-time Lagrange space ML_{p}^{n}, are the abstract geometrical Einstein equations attached to the Cartan canonical connection CΓ of ML_{p}^{n} and the adapted metric G on E, that is,

Ric(CΓ) − \frac{Sc(CΓ)}{2}G = KT , (4.1)

where Ric(CΓ) represents the Ricci d-tensor of the Cartan connection, Sc(CΓ) is its scalar curvature, K is the Einstein constant and T is an intrinsec tensor of matter (equal to 0 for vacuum), which is called the stress-energy d-tensor.

In the adapted basis (X_{A}) = (\frac{δ}{δt^{α}} , \frac{δ}{δx^{i}} , \frac{∂}{∂x_{α}^{i}} ) attached to the canonical nonlinear

connection Γ of ML_{p}^{n}, the curvature d-tensor R of the canonical Cartan connection is locally expressed by R(X_{C}, X_{B})X_{A} = R_{ABC}^{D}X_{D}. It follows that we have R_{AB} = Ric(X_{A}, X_{B}) = R_{ABD}^{D} and Sc(CΓ) = G^{AB}R_{AB}, where

(4.2) G^{AB} = \{ \begin{matrix}h_{αβ}, & for A = α, B = β \\ g^{ij}, & for A = i, B = j \\ h_{αβ}g^{ij}, & for A = {(i) \atop (α)}, B = {(j) \atop (β)} \\ 0, & otherwise. \end{matrix}

Taking into account, on the one hand, the form of the fundamental vertical metrical d-tensor G_{(i)(j)}^{(α)(β)} of the metrical multi-time Lagrange space ML_{p}^{n}, and, on the other hand, the expressions of local curvature d-tensors attached to the Cartan canonical connection CΓ, by direct computations, we deduce

Theorem 4.1 The Ricci d-tensor Ric(CΓ) of the Cartan canonical connection CΓ of a metrical multi-time Lagrange space ML_{p}^{n} is determined by the following adapted components: i) for p = dim T = 1,

\begin{matrix}R_{11} =^{not} H_{11} = 0, R_{i1} = R_{i1m}^{m}, R_{ij} = R_{ijm}^{m}, R_{i(j)}^{(1)} =^{not} P_{i(j)}^{(1)} = −P_{im(j)}^{m (1)}, \\ R_{(i)j}^{(1)} =^{not} P_{(i)j}^{(1)} = P_{ij(m)}^{m (1)}, R_{(i)1}^{(1)} =^{not} P_{(i)1}^{(1)} = P_{i1(m)}^{m (1)}, R_{(i)(j)}^{(1)(1)} =^{not} S_{(i)(j)}^{(1)(1)} = S_{i(j)(m)}^{m(1)(1)}; \end{matrix}

ii) for p = dim T ≥ 2,

\begin{matrix}R_{(α)(β)} =^{not} H_{αβ} = H_{αβμ}^{μ}, R_{iα} = R_{iαm}^{m}, R_{ij} = R_{ijm}^{m}, R_{i(j)}^{(α)} =^{not} P_{i(j)}^{(α)} = 0, \\ R_{(i)j}^{(α)} =^{not} P_{(i)j}^{(α)} = 0, R_{(i)β}^{(α)} =^{not} P_{(i)β}^{(α)} = 0, R_{(i)(j)}^{(α)(β)} =^{not} S_{(i)(j)}^{(α)(β)} = 0. \end{matrix}

Moreover, denoting H = h^{αβ}H_{αβ}, R = g^{ij}R_{ij} and S = h_{αβ}g^{ij}S_{(i)(j)}^{(α)β)}, it follows

Corollary 4.2 The scalar curvature Sc(CΓ) of the Cartan canonical connection CΓ of a metrical multi-time Lagrange space ML_{p}^{n} is given by the formulas

i) for p = dim T = 1, Sc(CΓ) = R + S;

ii) for p = dim T ≥ 2, Sc(CΓ) = H + R.

The main result of the metrical multi-time Lagrange theory of gravitational field is given by the following theorem:

Theorem 4.3 The generalized Einstein equations, which govern the multi-time gravitational h-potential G induced by the Kronecker h-regular Lagrangian of a metrical multi-time Lagrange space ML_{p}^{n}, have the following adapted local form:

i) for p = dim T = 1,

(E_{1})  \{ \begin{matrix}−\frac{R + S}{2} h_{11} = KT_{11} \\ R_{ij} − \frac{R + S}{2} g_{ij} = KT_{ij} \\ S_{(i)(j)}^{(1)(1)} − \frac{R + S}{2} h^{11}g_{ij} = KT_{(i)(j)}^{(1)(1)}, \end{matrix}

(E_{2})  \{ \begin{matrix}0 = T_{1i}, & R_{i1} = KT_{i1}, & P_{(i)1}^{(1)} = KT_{(i)1}^{(1)} \\ 0 = T_{1(i)}^{(1)}, & P_{i(j)}^{(1)} = KT_{i(j)}^{(1)}, & P_{(i)j}^{(1)} = KT_{(i)j}^{(1)}, \end{matrix}

ii) for p = dim T ≥ 2,

(E_{1})  \{ \begin{matrix}H_{αβ} − \frac{H + R}{2}h_{αβ} = KT_{αβ} \\ R_{ij} − \frac{H + R}{2}g_{ij} = KT_{ij} \\ −\frac{H + R}{2}h^{αβ}g_{ij} = KT_{(i)(j)}^{(α)(β)}, \end{matrix}

(E_{2})  \{ \begin{matrix}0 = T_{αi}, & R_{iα} = KT_{iα}, & 0 = T_{(i)β}^{(α)} \\ 0 = T_{α(i)}^{(β)}, & 0 = T_{i(j)}^{(α)}, & 0 = T_{(i)j}^{(α)}, \end{matrix}

where T_{AB}, A, B ∈ \{α, i, {(α) \atop (i)}\} are the adapted local components of the stress-energy d-tensor T .

Remark 4.2 Asumming that p = dim T > 2 and n = dim M > 2, the set (E_{1}) of the generalized Einstein equations can be rewritten in the more natural form

(E′_{1})  \{ \begin{matrix}H_{αβ} − \frac{H}{2} h_{αβ} = KT̃_{αβ} \\ R_{ij} − \frac{R}{2}g_{ij} = KT̃_{ij}, \end{matrix}

where T̃_{AB}, A, B ∈ {α, i} are the adapted local components of a new stress-energy d-tensor T̃. This new form of the generalized Einstein equations is derived in the more general case of a generalized metrical multi-time Lagrange space [14].

It is obvious that, in order to have the compatibility of the generalized Einstein equations, it is necessary that certain adapted local components of the stress-energy d-tensor T to vanish ”a priori”. Moreover, from physical point of view, it is well known that a good stress-energy d-tensor T must verify the local conservation laws

T_{A|B}^{B} = 0, ∀ A ∈ \{α, i, {(α) \atop (i)}\},

where T_{A}^{B} = G^{BD}T_{DA}. Consequently, by a direct calculation, we find

Theorem 4.4 The conservation laws of the generalized Einstein equations of a metrical multi-time Lagrange space ML_{p}^{n} are given by the formulas:

i) for p = 1,

\{ \begin{matrix}[\frac{R + S}{2} ]_{/1} = R_{1|m}^{m} − P_{(1)1}^{(m)}|_{(m)}^{(1)} \\ [R_{j}^{m} − \frac{R + S}{2} δ_{j}^{m}]_{|m} = −P_{(1)j}^{(m)}|_{(m)}^{(1)} \\ [S_{(1)(j)}^{(m)(1)} − \frac{R + S}{2} δ_{j}^{m}] |_{(m)}^{(1)} = −P _{(j)|m}^{m(1)}, \end{matrix} (4.3)

where R_{1}^{i} = g^{im}R_{m1}, P_{(1)1}^{(i)} = h_{11}g^{im}P_{(m)1}^{(1)}, R_{j}^{i} = g^{im}R_{mj}, P_{(1)j}^{(i)} = h_{11}g^{im}P_{(m)j}^{(1)},

P _{(j)}^{i(1)} = g^{im}P_{m(j)}^{(1)} and S_{(1)(j)}^{(i)(1)} = h_{11}g^{im}S_{(m)(j)}^{(1)(1)};

i) for p ≥ 2,

\{ \begin{matrix}[H_{β}^{μ} − \frac{H + R}{2}δ_{β}^{μ}]_{/μ} = −R_{β|m}^{m} \\ [R_{j}^{m} − \frac{H + R}{2}δ_{j}^{m}]_{|m} = 0, \end{matrix} (4.4)

where H_{β}^{μ} = h^{μγ}H_{γμ}, R_{j}^{i} = g^{im}R_{mj} and R_{β}^{i} = g^{im}R_{mβ}.

Remarks 4.3 i) In the case p > 2, n > 2, taking into account the components T̃_{αβ} and T̃_{ij} of the new stress-energy d-tensor T̃ from (E′_{1}), we point out that the conservation laws modify in the following simple and natural new form (see the paper [14]):

T̃_{β/μ}^{μ} = 0, T̃_{j|m}^{m} = 0. (4.5)

ii) In the particular case (T, h) = (IR , δ), the generalized Einstein equations, together with their generalized conservation laws, naturally generalize the similar equations described in [12].

\underline{Open problems}.

i) What is the real physical interpretation of our abstract geometrical theory?

ii) The development of an analogous metrical multi-time Lagrangian geometry of physical fields on the jet space of order two J^{2}(T, M) is in our attention.

Acknowledgements. A version of this paper was presented at Workshop on Diff. Geom., Global Analysis, Lie Algebras, Aristotle University of Thessaloniki, Greece, Aug. 28-Sept. 2, 2000. The author thanks to all participants at this conference, especially to Prof. Dr. C. Udrişte, Prof. Dr. Gr. Tsagas, Prof. Dr. A. Janussis and Prof. Dr. N. Spyrou for their useful sugestions. Also, many thanks go to Conf. Dr. K. Teleman and Prof. Dr. P. Olver, who advised me to distroy previous versions of this paper.

References

[1] G. S. Asanov, Gauge-Covariant Stationary Curves on Finslerian and Jet Fibrations and Gauge Extension of Lorentz Force, Tensor N. S., 50, 122-137, (1991).

[2] T. J. Bridges, Multi-Symplectic Structures and Wave Propagation, Math. Proc. Camb. Phil. Soc., 121, 147-190, (1997).

[3] L. A. Dickey, Solitons Equations and Hamiltonian Systems, Advanced Series in Mathematical Physics, 12, World Scientific, Singapore, (1991); Ch. 17, ”Multi-Time Lagrangian and Hamiltonian Formalism”.

[4] J. Eells, L. Lemaire, A Report on Harmonic Maps, Bull. London Math. Soc., 10, 1-68, (1978).

[5] G. Giachetta, L. Mangiarotti, G. Sardanashvily, Covariant Hamiltonian Field Theory, http://xxx.lanl.gov/abs/hep-th/9904062, (1999).

[6] M. J. Gotay, J. Isenberg, J. E. Marsden, Momentum Maps and the Hamiltonian Structure of Classical Relativistic Fields, http://xxx.lanl.gov/abs/hepth/9801019, (1998).

[7] D. D. Holm, J. E. Marsden, T. S. Raţiu, The Euler-Poincaré Equations and Semidirect Products with Applications to Continuum Theories, http://xxx.lanl.gov/abs/chao-dyn/9801015, (1998).

[8] S. Ikeda, Some Remarks on the Lagrangian Theory of Electromagnetism, Tensor N. S., 49, 204-208, (1990).

[9] L. A. Cordero, C. T. J. Dodson, M. de Léon, Differential Geometry of Frame Bundles, Kluwer Academic Publishers, (1989).

[10] N. Kamran, P. J. Olver, Le Probléme d’Equivalence à une Divergence prés dans le Calcul des Variations des Intégrales Multiples, C. R. Acad. Sci. Paris, t. 308, Série I, p. 249-252, (1989).

[11] R. Miron, M. Anastasiei, Vector Bundles. Lagrange Spaces. Applications in Relativity Theory, Editura Academiei R. S. R, Bucureşti, (1987); in Romanian.

[12] R. Miron, M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Academic Publishers, (1994).

[13] R. Miron, M. S. Kirkovits, M. Anastasiei, A Geometrical Model for Variational Problems of Multiple Integrals, Proc. of Conf. of Diff. Geom. and Appl., Dubrovnik, Yugoslavia, June 26-July 3, 209-215, (1988).

[14] M. Neagu, Generalized Metrical Multi-Time Lagrange Geometry of Physical Fields, Forum Mathematicum, Journals from de Gruyter, Berlin, New York, (2001), in press; http://xxx.lanl.gov/abs/math.DG/0011003, (2000).

[15] M. Neagu, Upon h-Normal Γ-Linear Connections on J^{1}(T, M), (2000), http://xxx.lanl.gov/abs/math.DG/0009070.

[16] M. Neagu, The Geometry of Autonomous Metrical Multi-Time Lagrange Space of Electrodynamics, International Journal of Mathematics and Mathematical Sciences, Hindawi Publishing Corporation, (2001), in press; http://xxx.lanl.gov/abs/math.DG/0010091, (2000).

[17] M. Neagu, The Geometry of Relativistic Rheonomic Lagrange Spaces, Workshop on Diff. Geom., Global Analysis, Lie Algebras, Aristotle University of Thessaloniki, Greece, June 24-28, (1997); BSG Proceedings 5, 142-168, Editor: Prof. Dr. Grigorios Tsagas, Geometry Balkan Press, Bucharest, (2001); http://xxx.lanl.gov/abs/math.DG/0010090, (2000).

[18] M. Neagu, C. Udrişte, Mullti-Time Dependent Sprays and Harmonic Maps on J^{1}(T, M), Third Conference of Balkan Society of Geometers, Politehnica University of Bucharest, Romania, July 31-August 3, (2000); http://xxx.lanl.gov/abs/math.DG/0009049, (2000).

[19] M. Neagu, C. Udrişte, The Geometry of Metrical Multi-Time Lagrange Spaces, http://xxx.lanl.gov/abs/math.DG/0009071, (2000).

[20] M. Neagu, C. Udrişte, Torsion, Curvature and Deflection d-Tensors on J^{1}(T, M), Balkan Journal of Geometry and Its Applications, 6, 1 , 45-60, (2001), in press; http://xxx.lanl.gov/abs/math.DG/0009069, (2000).

[21] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, [22] P. J. Olver, Canonical Elastic Moduli, Journal of Elasticity, 19, 189-212, (1988).

[23] D. Opriş, C. Giulvezan, Geometrical Integrators and Applications, Editura Mirton, Timişoara, (1999); in Romanian.

[24] D. Saunders, The Geometry of Jet Bundles, Cambridge University Press, New York, London, (1989).

[25] C. Udrişte, Solutions of DEs and PDEs as Potential Maps using First Order Lagrangians, Centennial Vrânceanu, Romanian Academy, University of Bucharest, June 30-July 4, (2000); http://xxx.lanl.gov/abs/math.DS/0007061, (2000).

[26] C. Udrişte, M. Neagu, Geometrical Interpretation of Solutions of Certain PDEs, Balkan Journal of Geometry and Its Applications, 4, 1, 145-152, (1999).

[27] C. Udrişte, M. Neagu, From PDE Systems and Metrics to Generalized Field Theories, http://xxx.lanl.gov/abs/math.DG/0101207, (2001).

[28] C. Udrişte, M. Postolache, Least Squares Problem and Geometric Dynamics, Ital. J. Pure. Appl. Math., 15, (2000), in press.

University POLITEHNICA of Bucharest Department of Mathematics I Splaiul Independentei 313 77206 Bucharest, Romania e-mail: mircea@mathem.pub.ro