Characterizations of generalized Z-flat spacetimes and f(R,G)-gravity

This paper is devoted to characterize a generalized [Formula: see text]-tensor. In the second part of this paper, we describe the behavior of the associated vector field of a generalized [Formula: see text]-flat spacetime in some special cases. It has been investigated that such a spacetime turns into special spacetimes when the associated scalars have different structures. In the third part, generalized [Formula: see text]-flat spacetime in [Formula: see text]-gravity is examined.