The field equations of generalized conformally flat spaces of metric $g_{\mu v} \left( {x,\xi ,\overline \xi } \right) = e^{2\sigma \left( {x,\xi \overline \xi } \right)} \eta _{uv}$

Abstract

The differential geometry in spaces which metric tensor depends on the spinor variables has been studied in [3]. In this work the authors study the form of spin connection coefficients, spin-curvature tensors and the field equations for generalized conformally flat spaces GCFS (M, g _µv (x,ξ,ξ)=e ^{2σ(x, ξ, ξ)} η _µv = where η _µv represents the Lorentz metric tensor η _µv = diag{+, —, —, —) and ξ,ξ represent the internal variables of the space. The introduction of these variables modifies the Riemannian structure of space-time and provides it with torsion. The case of conformally related metrics of Riemannian and generalized Lagrange spaces have been extensively studied in [1], [2]. It is remarkable, that in the above mentioned spaces GCFS, some spin connections and spin-curvature tensors are vanishing.