Generalized projective 𝒫-flat spacetimes and applications to f(ℛ,𝒢) gravity

We investigate spacetimes endowed with the generalized projective [Formula: see text]-curvature tensor and study geometric consequences of the [Formula: see text]-flat condition. First, algebraic restrictions under which the generalized projective tensor inherits the classical symmetries of the Riemann curvature tensor are obtained. Under suitable assumptions on the defining coefficients, several consequences of [Formula: see text]-flatness are derived, including Einstein-type conditions, conformal flatness, and constant sectional curvature. Applications to generalized Robertson–Walker and standard static spacetimes are then examined, yielding characterizations of the fiber and base manifolds under the vanishing of the generalized projective tensor. Finally, consequences for four-dimensional generalized Robertson–Walker spacetimes in [Formula: see text] gravity are investigated, including the form of the effective energy-momentum tensor, effective fluid interpretation, and associated energy conditions.