The metric space-time one described, for an unspecified geometry, distances between the points expressed in a certain frame of reference and their evolution in time.
The complete resolution of the equations of Einstein for metric unspecified is in practice impossible to realize. Symmetries of the problem make it possible to force the form of metric and to simplify it.
In the case of cosmology, it is the isotropy and the homogeneity which will make it possible to simplify the equations.