Abstract:
The aim of this paper is to create a large geometrical background for the study of important branch of physics: electrodynamics, bosonic strings theory, magneto-hydrodynamics, and so forth. The geometrical construction is realized on the 1-jet fibre bundle J1(T,M) and is produced by a given quadratic multi-time Lagrangian function L. The Riemann-Lagrange geometry of the space EDMLpn=(J1(T,M),L), in the sense of d-connections, torsion and curvature d-tensors, allows the construction of a natural generalized multi-time field theory on EDMLPn, in the sense of generalized Maxwell and Einstein equations.

Abstract:
The aim of this paper is to describe the local Ricci and Bianchi identities of an h-normal Γ-linear connection on the first-order jet fibre bundle J1(T,M). We present the physical and geometrical motives that determined our study and introduce the h-normal Γ-linear connections on J1(T,M), emphasizing their particular local features. We describe the expressions of the local components of torsion and curvature d-tensors produced by an h-normal Γ-linear connection ∇Γ, and analyze the local Ricci identities induced by ∇Γ, together with their derived local deflection d-tensors identities. Finally, we expose the local expressions of Bianchi identities which geometrically connect the local torsion and curvature d-tensors of connection ∇Γ.

Abstract:
The aim of this paper is to construct a Riemann-Lagrange geometry on 1-jet spaces, in the sense of d-connections, d-torsions, d-curvatures, electromagnetic d-field and geometric electromagnetic Yang-Mills energy, starting from a given linear ODEs system or a given superior order ODE. The case of a non-homogenous linear ODE of superior order is disscused.

Abstract:
The aim of this paper is to construct natural geometrical objects on the 1-jet space J^1(T,R^5), where $T/subset R$, like a non-linear connection, a generalized Cartan connection, together with its d-torsions and d-curvatures, a jet electromagnetic d-field and a jet Yang-Mills energy, starting from the given Lorenz atmospheric DEs system and the pair of Euclidian metrics $/Delta = (1,/delta_{ij})$ on $T/times R^5$.

Abstract:
The aim of this paper is to construct a natural Riemann-Lagrange differential geometry on 1-jet spaces, in the sense of nonlinear connections, generalized Cartan connections, d-torsions, d-curvatures, jet electromagnetic fields and jet Yang-Mills energies, starting from some given non-linear evolution DEs systems modelling economic phenomena, like the Kaldor model of the bussines cycle or the Tobin-Benhabib-Miyao model regarding the role of money on economic growth.

Abstract:
In this paper we study a collection of jet geometrical concepts, we refer to d-tensors, relativistic time dependent semisprays, harmonic curves and nonlinear connections on the 1-jet space J1(R;M), necessary to the construction of a Miron's-like geometrization for Lagrangians depending on a relativistic time. The geometrical relations between these jet geometrical objects are exposed.

Abstract:
The aim of this paper is to describe the local Bianchi identities for an $h$-normal $\Gamma$-linear connection of Cartan type $\nabla\Gamma$ on the first-order jet space $J^1(R,M)$. In this direction, we present the local expressions of the adapted components of the torsion and curvature d-tensors produced by $\nabla\Gamma$ and we give the general local expressions of Bianchi identities which connect these d-torsions and d-curvatures.

Abstract:
In this paper we introduce a natural definition for the affine maps between two Finsler manifolds $(M, F)$ and $(N,\tilde F)$ and we give some geometrical properties of these affine maps. Starting from the equations of the affine maps, we construct a natural Berwald-Riemann-Lagrange geometry on the 1-jet space $J^1(TM;N)$, in the sense of a Berwald nonlinear connection $\Gamma^b_jet$, a Berwald $\Gamma^b_jet$-linear d-connection $B\Gamma^b_jet$, together with its d-torsions and d-curvatures, which geometrically characterizes the initial affine maps between Finsler manifolds.

Abstract:
The paper contains a geometrization of a time dependent Lagrangian function defined on the 1-jet space J^1(R,M) which identifies with R\times TM. The reader is invited to compare this geometrization with that developped by Miron and Anastasiei.

Abstract:
The paper contains a geometrization of the autonomous multi-time Lagrangian function of electrodynamics. We point out that this multi-time Lagrangian function comes from electrodynamics and the theory of bosonic strings.