scholarly journals On Almost φ-Lagrange Spaces

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
P. N. Pandey ◽  
Suresh K. Shukla
Keyword(s):  
Differential Equation ◽  
Lagrange Equations ◽  
Metric Tensor ◽  
Nonlinear Connection

We initiate a study on the geometry of an almost φ-Lagrange space (APL-space in short). We obtain the expressions for the symmetric metric tensor, its inverse, semispray coefficients, solution curves of Euler-Lagrange equations, nonlinear connection, differential equation of autoparallel curves, coefficients of canonical metrical d-connection, and h- and v-deflection tensors in an APL-space. Corresponding expressions in a φ-Lagrange space and an almost Finsler Lagrange space (AFL-space in short) have also been deduced.

scholarly journals Lagrange Spaces with (γ,β)-Metric

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Suresh K. Shukla ◽  
P. N. Pandey
Keyword(s):  
Lagrange Equations ◽  
Metric Tensor ◽  
Nonlinear Connection

We study Lagrange spaces with (γ,β)-metric, where γ is a cubic metric and β is a 1-form. We obtain fundamental metric tensor, its inverse, Euler-Lagrange equations, semispray coefficients, and canonical nonlinear connection for a Lagrange space endowed with a (γ,β)-metric. Several other properties of such space are also discussed.

Vector-tensor field theories and the Einstein-Maxwell field equations

1974 ◽  
Vol 341 (1626) ◽  
pp. 285-297 ◽  
Keyword(s):  
Dimensional Space ◽  
Field Theories ◽  
Maxwell Field ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Lagrange Equations ◽  
Third Order ◽  
Metric Tensor ◽  
First Order ◽  
First Derivative

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of the metric tensor g ij and its first two derivatives together with the first derivative of a vector field ψ i are investigated. In general, the Euler-Lagrange equations obtained by variation of g ij are of fourth order in g ij and third order in ψ i . It is shown that in a four dimensional space the only Euler-Lagrange equations which are of second order in g ij and first order in ψ i are the Einstein field equations (with an energy-momentum term). Various conditions are obtained under which the Einstein-Maxwell field equations are then an inevitable consequence.

scholarly journals On an R-Randers mth-Root Space

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
P. N. Pandey ◽  
Shivalika Saxena
Keyword(s):  
Finsler Space ◽  
Riemannian Metric ◽  
Root Space ◽  
Metric Tensor ◽  
Nonlinear Connection

We consider an n-dimensional Finsler space Fn(n>2) with the metric L(x,y)=F(x,y)+α(x,y), where F is an mth-root metric and α is a Riemannian metric. We call such space as an R-Randers mth-root space. We obtain the expressions for the fundamental metric tensor, Cartan tensor, geodesic spray coefficients, and the coefficients of nonlinear connection in an R-Randers mth-root space. Some other properties of such space have also been discussed.

Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

1949 ◽  
Vol 1 (3) ◽  
pp. 242-256 ◽  
Author(s):  
S. Minakshisundaram ◽  
Å. Pleijel
Keyword(s):  
Differential Equation ◽  
Riemannian Manifold ◽  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Line Element ◽  
Metric Tensor ◽  
Local Coordinates ◽  
Image Position ◽  
The Laplace Operator ◽  
Beltrami Laplace Operator

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (xi), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … xN) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

scholarly journals AN AP-STRUCTURE WITH FINSLERIAN FLAVOR I: THE PRINCIPAL IDEA

2009 ◽  
Vol 24 (22) ◽  
pp. 1749-1762 ◽  
Author(s):  
M. I. WANAS
Keyword(s):  
Geometric Structure ◽  
Building Blocks ◽  
The Other ◽  
Metric Tensor ◽  
Nonlinear Connection ◽  
Absolute Parallelism ◽  
Cartan Type ◽  
Linear Connections

A geometric structure (FAP-structure), having both absolute parallelism and Finsler properties, is constructed. The building blocks of this structure are assumed to be functions of position and direction. A nonlinear connection emerges naturally and is defined in terms of the building blocks of the structure. Two linear connections, one of Berwald type and the other of the Cartan type, are defined using the nonlinear connection of the FAP. Both linear connections are nonsymmetric and consequently admit torsion. A metric tensor is defined in terms of the building blocks of the structure. The condition for this metric to be a Finslerian one is obtained. Also, the condition for an FAP-space to be an AP-one is given.

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