Ivancevic_Applied-Diff-Geom

322 Applied Differential Geometry: A Modern Introductioncomes from the metric conditionsg ij|k = δg ijδx k −Lm ikg mj −L m jkg mi = 0,g ij;k = ∂g ij∂y k −Cm ikg mj −C m jkg mi = 0,where “ |k ” and “ ;k ” are the local h− and v− covariant derivatives of CΓ.The importance of the Cartan connection comes from its main role playedin the generalized Finsler–Lagrangian theory of physical fields.Regarding the unified field g ij (x, y) of GL n , the authors of [Miron et. al.(1988); Miron and Anastasiei (1994)] constructed a Sasakian metric on T M,G = g ij dx i ⊗ dx j + g ij δy i ⊗ δy j .In this context, the Einstein equations for the gravitational potentialsg ij (x, y) of a generalized Lagrangian space GL n , (n > 2), are postulatedas being the Einstein equations attached to CΓ and G,R ij − 1 2 Rg ij = KT H ij ,S ij − 1 2 Sg ij = KT V ij ,′ P ij = KT 1 ij,′′ P ij = −KT 2 ij,where R ij = Rijm m , S ij = Sijm m , ′ P ij = Pijm m , ′′ P ij = Pimj m are the Riccitensors of CΓ, R = g ij R ij and S = g ij S ij are the scalar curvatures, Tij H,Tij V , T ij 1 , T ij 2 are the components of the energy–momentum tensor T, andK is the Einstein constant (equal to 0 for vacuum). Moreover, the energy–momentum tensors TijH and T ij V satisfy the conservation laws [Miron et. al.(1988); Miron and Anastasiei (1994)]KT H j|m m = −1 hm(Pjs Rhm s + 2R s2mjPs m ), KT V j|m m = 0.The generalized Lagrangian theory of electromagnetism relies on thecanonical Liouville vector–field C = y i ∂∂yand the Cartan connection CΓiof the generalized Lagrangian space GL n . In this context, we can introducethe electromagnetic two—form on T M [Miron and Anastasiei (1994)]F = F ij δy i ∧ dx j + f ij δy i ∧ δy j ,whereF ij = 1 2 [(g imy m ) |j − (g jm y m ) |i ], f ij = 1 2 [(g imy m ) ;j − (g jm y m ) ;i ].Using the Bianchi identities attached to the Cartan connection CΓ, theyconclude that the electromagnetic components F ij and f ij are governed by