{'fraction_non_alphanumeric': 0.050489826676714394, 'fraction_numerical': 0.0066745613090752505, 'mean_word_length': 4.471142520612485, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Distinguished Riemann-Hamilton geometry in the polymomentum electrodynamics

In this paper we develop the distinguished (d-) Riemannian differential geometry (in the sense of d-connections, d-torsions, d-curvatures and some geometrical Maxwell-like and Einstein-like equations) for the polymomentum Hamiltonian which governs the multi-time electrodynamics.


Introduction
Let M n be a smooth real manifold of dimension n, whose local coordinates are x = (x i ) i=1,n , having the physical meaning of "space of events". In order to justify the "electrodynamics" terminology used in this paper, we recall that, in the study of classical electrodynamics, the Lagrangian function L : T M → R that governs the movement law of a particule of mass m = 0 and electric charge e, placed concomitantly into a gravitational field and an electromagnetic one, is given by where the semi-Riemannian metric ϕ ij (x) represents the gravitational potentials of the space M , A i (x) are the components of an 1-form on M representing the electromagnetic potential, P(x) is a smooth potential function on M and c is the velocity of light in vacuum. The Lagrange space L n = (M, L(x, y)), where L is given by (1), is known in the literature of specialty as the autonomous Lagrange space of electrodynamics. A deep geometrical study of the Lagrange space L n is now completely done by Miron and Anastasiei in the book [6]. More general, we point out that, in the study of classical rheonomic (time-dependent ) electrodynamics, a central role is played by the autonomous time-dependent Lagrangian function of electrodynamics expressed by where L : R × T M → R. Note that the non-dynamical character (i.e., the independence on the temporal coordinate t) of the spatial semi-Riemannian metric ϕ ij (x) determines the usage of the term "autonomous" in the preceding definition. Let (T m , h ab (t)) be a "multi-time" smooth Riemannian manifold of dimension m (please do not confuse with the mass m = 0), having the local coordinates t = (t c ) c=1,m , and let J 1 (T , M ) be the 1-jet space produced by the manifolds T and M . By a natural extension of the preceding examples of electrodynamics Lagrangian functions, we can consider the jet multi-time Lagrangian function where A (a) (i) (t, x) is a d-tensor on J 1 (T , M ) and P(t, x) is a smooth function on the product manifold T × M .
Remark 1 Throughout this paper, the indices a, b, c, ... run from 1 to m, while the indices i, j, k, ... run from 1 to n. The Einstein convention of summation is also adopted all over this work.
The pair EDM L n m = (J 1 (T , M ), L), where L is given by (3), is called the autonomous multi-time Lagrange space of electrodynamics. The distinguished Riemannian geometrization of the multi-time Lagrange space EDM L n m is now completely developed in the Neagu's works [8] and [9].
Via the classical Legendre transformation, the jet multi-time Lagrangian function of electrodynamics (3) leads us to the Hamiltonian function of polymomenta where H : J 1 * (T , M ) → R, and But, using as a pattern the Miron's geometrical ideas from [7], the distinguished Riemannian geometry for quadratic Hamiltonians of polymomenta (geometry in the sense of d-connections, d-torsions, d-curvatures and geometrical Maxwell-like and Einstein-like equations) is constructed on dual 1-jet spaces in the Oanȃ-Neagu's paper [11]. Consequently, in what follows, we apply the general geometrical result from [11] for the particular Hamiltonian function of polymomenta (4), which governs the multi-time electrodynamics.
Let χ a bc (t) (respectively γ k ij (x)) be the Christoffel symbols of the metric h ab (t) (respectively ϕ ij (x)). Obviously, if * χ a bc are the Christoffel symbols of the Riemannian metric h * ab (t), then we have * χ a bc = χ a bc . Using a general result from the geometrical theory of multi-time Hamilton spaces (see [2] and [11]), by direct computations, we find represents a nonlinear connection on E * . This nonlinear connection is called the canonical nonlinear connection of the multi-time Hamilton space of electrodynamics EDM H n m . Now, let be the adapted bases produced by the nonlinear connection N ED , where Working with these adapted bases, by direct computations, we can determine the adapted components of the generalized Cartan canonical connection of the space EDM H n m , together with its local d-torsions and d-curvatures (for details, see the general formulas from [11]).
(2) The torsion T of the generalized Cartan canonical linear connection of the space EDM H n m is determined by three effective adapted components: where χ c dab (t) (respectively R k rij (x)) are the classical local curvature tensors of the Riemannian metric h ab (t) (respectively semi-Riemannian metric ϕ ij (x)), and " ;a " and " :k " represent the following generalized Levi-Civita covariant derivatives: • the T -generalized Levi-Civita covariant derivative: 3 Electromagnetic-like model on the multi-time Hamilton space of electrodynamics EDMH n m In order to describe our geometrical electromagnetic-like theory (depending on polymomenta) on the multi-time Hamilton space of electrodynamics EDM H n m , we underline that, by a simple direct calculation, we obtain (see [11]) Proposition 5 The metrical deflection d-tensors of the space EDM H n m are expressed by the formulas: where " /b ", " |j " and "| (b) (j) " are the local covariant derivatives induced by the generalized Cartan canonical connection CΓ (N ) (see [10] and [11]).
Moreover, taking into account some general formulas from [11], we introduce where is called the polymomentum electromagnetic field attached to the multitime Hamilton space of electrodynamics EDM H n m . Now, particularizing the generalized Maxwell-like equations of the polymomentum electromagnetic field that govern a general multi-time Hamilton space M H n m , we obtain the main result of the polymomentum electromagnetism on the space EDM H n m (for more details, see [11]): Theorem 7 The polymomentum electromagnetic components (10) of the autonomous multi-time Hamilton space of electrodynamics EDM H n m are governed by the following geometrical Maxwell-like equations: where A {i,j} represents an alternate sum and {i,j,k} represents a cyclic sum.
4 Gravitational-like geometrical model on the multi-time Hamilton space of electrodynamics To expose our geometrical Hamiltonian polymomentum gravitational theory on the autonomous multi-time Hamilton space of electrodynamics EDM H n m , we recall that the fundamental vertical metrical d-tensor and the canonical nonlinear connection of the multi-time Hamilton space EDM H n m produce a polymomentum gravitational h * -potential G on E * = J 1 * (T , M ), locally expressed by We postulate that the geometrical Einstein-like equations, which govern the multi-time gravitational h * -potential G of the multi-time Hamilton space of electrodynamics EDM H n m , are the abstract geometrical Einstein equations attached to the Cartan canonical connection CΓ(N ) and to the adapted metric G on E * , namely where Ric(CΓ) represents the Ricci tensor of the Cartan connection, Sc(CΓ) is the scalar curvature, K is the Einstein constant and T is an intrinsic d-tensor of matter, which is called the stress-energy d-tensor of polymomenta.
In order to describe the local geometrical Einstein-like equations (together with their generalized conservation laws) in the adapted basis let CΓ (N ) = (χ c ab , 0, γ i jk , 0) be the generalized Cartan canonical connection of the space EDM H n m . Taking into account the expressions of its adapted curvature d-tensors on the space EDM H n m , we immediately find (see [11]): The Ricci tensor Ric(CΓ) of the autonomous multi-time Hamilton space of electrodynamics EDM H n m is characterized by two effective local Ricci d-tensors: χ ab = χ f abf , R ij = R r ijr . These are exactly the classical Ricci tensors of the Riemannian temporal metric h ab (t) and the semi-Riemannian spatial metric ϕ ij (x).
Consequently, using the notations χ = h ab χ ab and R = ϕ ij R ij , we get Theorem 9 The scalar curvature Sc(CΓ) of the generalized Cartan connection CΓ of the space EDM H n m has the expression (for details, see [11]) where χ and R are the classical scalar curvatures of the semi-Riemannian metrics h ab (t) and ϕ ij (x).
Particularizing the generalized Einstein-like equations and the generalized conservation laws of an arbitrary multi-time Hamilton space M H n m , we can establish the main result of the generalized polymomentum gravitational theory on the autonomous multi-time Hamilton space of electrodynamics EDM H n m (for more details, see [11]): Theorem 10 (1) The local geometrical Einstein-like equations, that govern the polymomentum gravitational potential of the space EDM H n m , have the form where T AB , A, B ∈ a, i, (i) (a) , are the adapted components of the polymomentum stress-energy d-tensor of matter T.
(2) The polymomentum conservation laws of the geometrical Einsteinlike equations of the space EDM H n m are expressed by the formulas where χ f b = h f d χ db and R r j = ϕ rs R sj .
Open Problem. There exist real physical interpretations for previous geometrical polymomentum field-like theories, which to be relevant for the physical domain of electrodynamics?