Field equations in spaces with gμν(x, ξ, \̄gx) metric. Generalized conformally flat spaces

The gravitational field equations are derived here in the framework of spaces whose metric tensor depends also on spinor variables ξ and \̄gx. The attempt is to describe gravity by a tetrad field and the Lorentz connection coefficients in a more generalized framework than the standard one (cf. eg. [11]). An interesting case with generalized conformally flat spaces GCFS with metric gμν(x, ξ, \̄gx) = e2σ(x,ξ,\̄gx)ημν is studied.