Consistency Conditions of f(R,G)-Gravity Field Equations for Bianchi-Type III Metric

In this paper, we study the -gravitation theory under the assumption that the standard matter-energy content of the universe is a perfect fluid with linear barotropic equation of state within the framework of Bianchi-Type III model from the class of homogeneous and anisotropic universe models. However, whether such a restriction lead to any contradictions or inconsistencies in the field equations will create an issue that needs to be examined. Under the effective fluid approach, we will be concerned mainly the field equations in an orthonormal tetrad framework with an equimolar and examined the situation of establishing the functional form of  together with the scale factors, which are their solutions. Unlike similar studies, which are very few in the literature, instead of assuming preliminary solutions, we determined the consistency conditions of the field equations by assuming the matter energy content of the universe as an isotropic perfect fluid for Bianchi-Type III.