Deflection d-Tensor Identities in the Relativistic Time Dependent Lagrange Geometry (original) (raw)
Local Bianchi identities in the relativistic time-dependent Lagrange geometry
V. Balan, M. Neagu: "Jet Single-Time Lagrange Geometry and Its Applications", John Wiley & Sons, Inc., Hoboken, New Jersey, 2011
The aim of this paper is to describe the local Bianchi identities for an h-normal \Gamma-linear connection of Cartan type ∇\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 ∇\Gamma and we give the general local expressions of Bianchi identities which connect these d-torsions and d-curvatures.
Torsion, Curvature and Deflection d-Tensors on J^1(T,M)
Balkan Journal of Geometry and Its Applications, 2001
The paper introduces the notion of \Gamma-linear connection \nabla on the -jet fibre bundle J^1(T,M), and presents its local components. We also describe the local Ricci and Bianchi identities of nabla\nablanabla.
Local Bianchi Identities in the Relativistic Non-Autonomous Lagrange Geometry
The aim of this paper is to describe the local Bianchi identities for an h-normal Γ-linear connection of Cartan type ∇Γ 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 ∇Γ and we give the general local expressions of Bianchi identities which connect these d-torsions and d-curvatures. : 53C60, 53C43, 53C07.
In this paper we describe the local Ricci and Bianchi identities for an h-normal N-linear connection D\Gamma(N) on the dual 1-jet space J^{1*}(T,M). To reach this aim, we firstly give the expressions of the local distinguished (d-) adapted components of torsion and curvature tensors produced by D\Gamma(N), and then we analyze their attached local Ricci identities. The derived deflection d-tensor identities are also presented. Finally, we expose the local expressions of the Bianchi identities (in the particular case of an h-normal N-linear connection of Cartan type), which geometrically connect the local torsion and curvature d-tensors of the linear connection D\Gamma(N).
Distinguished Torsion, Curvature and Deflection Tensors in the Multi-Time Hamilton Geometry
The purpose of this paper is to present the main geometrical objects on the dual 1-jet vector bundle J 1 * (T , M ) that characterize our approach of multi-time Hamilton geometry. In this direction, we firstly introduce the geometrical concept of a nonlinear connection N on the dual 1-jet space J 1 * (T , M ). Then, starting with a given N -linear connection D on J 1 * (T , M ), we describe the adapted components of the torsion, curvature and deflection distinguished tensors attached to the N -linear connection D.
The local description of the Ricci and Bianchi
2011
In this paper we describe the local Ricci and Bianchi identities for an hnormal N-linear connection DΓ(N) on the dual 1-jet space J 1 * (T , M). To reach this aim, we firstly give the expressions of the local distinguished (d-) adapted components of torsion and curvature tensors produced by DΓ(N), and then we analyze their attached local Ricci identities. The derived deflection d-tensor identities are also presented. Finally, we expose the local expressions of the Bianchi identities (in the particular case of an hnormal N-linear connection of Cartan type), which geometrically connect the local torsion and curvature d-tensors of the linear connection DΓ(N).
A Berwald Linear Connection in the Riemann-Lagrange Geometry of 1Jet Spaces
2007
This paper introduces the notions of a nonlinear connection Γ and of a Γ-linear connection ∇Γ on the 1-jet space J1(T,M). A particular nonlinear connection Γ0 and a Berwald Γ0 -linear connection BΓ0 are produced by a pair of semi-Riemannian metrics. The adapted components of the torsion and curvature d-tensors of our Berwald connection are described.
The Geometry of Relativistic Rheonomic Lagrange Spaces
In this paper we shall present a geometrization of time-dependent Lagrangians. The reader is invited to compare this geometrization with that contained in the book of Miron and Anastasiei . In order to develope the subsequent Relativistic Rheonomic Lagrange Geometry, Section 1 describes the main geometrical aspects of the 1-jet space J 1 (R, M ), in the sense of d-tensors, d-connections, d-torsions and d-curvatures. Section 2 introduces the notion of Relativistic Rheonomic Lagrange Space, which naturally generalizes that of Classical Rheonomic Lagrange Space [11], and constructs its canonical nonlinear connection Γ as well as its Cartan canonical Γ-linear connection. We point out that our geometry gives a model for both gravitational and electromagnetic field. From this point of view, Section 4 presents the Maxwell equations of the relativistic rheonomic Lagrangian electromagnetism. Section 5 describes the Einstein's gravitational field equations of a relativistic rheonomic Lagrange space. : 53C60, 53C80, 83C22
Torsions and Curvatures on Jet Fibre Bundle J^1(T,M)
The aim of this paper is twofold. On the one hand, to study the local representations of d-connections, d-torsions and d-curvatures with respect to adapted bases produced by a nonlinear connection Γ on the jet fibre bundle of order one J 1 (T, M ) → T × M . On the other hand, to open the problem of prolongations of tensors and connections from a product of two manifolds to 1-jet fibre bundle associated to these manifolds.
Ricci and Bianchi identities for h-normal Γ-linear connections on J1(T,M)
International Journal of Mathematics and Mathematical Sciences, 2003
The aim of this paper is to describe the local Ricci and Bianchi identities of an hnormal Γ -linear connection on the first-order jet fibre bundle J 1 (T , M). We present the physical and geometrical motives that determined our study and introduce the h-normal Γ -linear connections on J 1 (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 .