Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 875The condition g ij|k = 0 is equivalent withδg ijδx k = g mj ¯L m ik + g im ¯Lm jk .Applying the Christoffel process to the indices {i, j, k}, we find¯L i jk = gim2(δgjmδx k+ δg kmδx j− δg )jkδx m .By analogy, using the relations C i(1)j(k) = Ci(1) k(j) and g ij| (1)(k)= 0, followinga Christoffel process applied to the indices {i, j, k}, we get(¯C i(1)j(k) = gim ∂gjm2 ∂v k + ∂g km∂v j − ∂g )jk∂v m .As a rule, the Cartan canonical connection of a relativistic rheonomicLagrangian space RL n verifies also the propertiesh 11/1 = h 11|k = h 11 | (1)(k) = 0 and g ij/1 = 0. (5.209)The torsion d−tensor T of the Cartan canonical connection of a relativisticrheonomic Lagrangian space is determined by only six local components,because the properties of the Cartan canonical connection imply therelations Tijm = 0 and S(i)(1)(1)(1)(j)(k)= 0. At the same time, we point out thatthe number of the curvature local d−tensors of the Cartan canonical connectionnot reduces. In conclusion, the curvature d−tensor R of the Cartancanonical connection is determined by five effective local d−tensors. Thetorsion and curvature d−tensors of the Cartan canonical connection of anRL n are called the torsion and curvature of RL n .All torsion d−tensors of an autonomous relativistic rheonomic Lagrangianspace of electrodynamics vanish, except⎡R (m)(1)1j = −h 11g mk⎣H 1411U (1)(k)j(k)j+ ∂U (1)∂t⎤⎦ ,R (m)(1)ij = rm ijkv k + h 11g mk []U (1)4(k)i|j + U (1)(k)j|i,where r m ijk are the curvature tensors of the semi–Riemannian metric g ij.