Abstract
The aim of this paper is to find higher order geometrical corrections to the Einstein-Hilbert action that can lead to only second order equations of motion. The metric formalism is used, and static spherically symmetric and Friedmann-Lemaître space-times are considered, in four dimensions. The FKWC-basis are introduced in order to consider all the possible invariant scalars, and both polynomial and non-polynomial gravities are investigated.
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