Canonical Nonlinear Connections on Jet Bundles of First Order

The aim of this paper is to open the problem of construction of a nonlinear connection $\Gamma=(M^{(i)}_{(\alpha)\beta}, N^{(i)}_{(\alpha)j})$ on the jet bundle of first order $J^1(T,M)$, which to be canonically produced by a Kronecker product vertical metrical d-tensor $G^{(\alpha)(\beta)}_{(i)(j)}=h^{\alpha\beta}g_{ij}$, possibly provided by multi-time dependent quadratic Lagrangians coming from various branches of theoretical physics: bosonic strings theory, electrodynamics or elasticity.Comment: 7 pages, Versions of this paper were presented at: -Workshop on Differential Geometry, Global Analysis and Lie Algebras, Aristotle University of Thessaloniki, Greece, Aug. 28-Sept. 2, 2000; -Seminar "Gh. Vranceanu", University of Bucharest, Romania, November, 200

Metrical Multi-Time Lagrange Geometry of Physical Fields

The author exposes the metrical multi-time Lagrange geometry of physical fields which naturally generalizes the classical Lagrangian developped by Miron and Anastasiei. In other words, one constructs a natural theory of physical fields on the 1-jet fibre bundle, attached to a Kronecker h-regular multi-time Lagrangian with partial derivatives of order one.Comment: 19 pages, A version of this paper was presented at Workshop on Differential Geometry, Global Analysis and Lie Algebras, Aristotle University of Thessaloniki, Greece, Aug. 28-Sept. 2, 200

Jet Berwald-Riemann-Lagrange Geometrization for Affine Maps between Finsler Manifolds

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.Comment: 22 page

The m-th root Finsler geometry of the Bogoslovsky-Goenner metric

In this paper we present the m-th root Finsler geometries of the three and four dimensional Bogoslovsky-Goenner metrics (good Finslerian anisotropic models in Special Relativity), in the sense of their Cartan torsion and curvature distinguished tensors or vertical Einstein-like equations.Comment: 7 page

Jet Geometrical Objects Depending on a Relativistic Time

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.Comment: 19 pages; The author thanks the referee of Analele Stiintifice ale Universitatii "Al.I.Cuza" din Iasi. Matematica for its remarks and useful suggestion

Harmonic Maps between Generalized Lagrange Spaces

In this paper are studied the harmonic maps between two generalized Lagrange spaces. At the same time, it is proved that the solutions of $C^2$ class of certain ODEs or PDEs are harmonic maps between certain convenient generalized Lagrange spaces.Comment: 11 pages, Southeast Asian Bulletin of Mathematics, Springer-Verlag, 2000, in pres

Solutions of Inverse Problems for Variational Calculus

The paper studies the harmonic maps on a direction between a Riemannian space and a generalized Lagrange space. Also, it is proved there that the solutions of C^2 class of certain ODEs or PDEs are harmonic maps, in the sense of this paper.Comment: 6 page

The Geometry of Relativistic Rheonomic Lagrange Spaces

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.Comment: 26 page

The Geometry of Autonomous Metrical Multi-Time Lagrange Space of Electrodynamics

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.Comment: 10 page

A Relativistic Approach on 1-Jet Spaces of the Rheonomic Berwald-Moor Metric

The aim of this paper is to develop on the 1-jet space J^1(R,M^4) the Finsler-like geometry (in the sense of d-connection, d-torsions and d-curvatures) of the rheonomic Berwald-Moor metric. A natural geometrical gravitational field theory produced by the rheonomic Berwald-Moor metric is also constructed.Comment: 14 page