The requirements of homogeneity and isotropy for physical space, as held by Helmholtz and Lie, are valid only for a matter-empty universe.

The contrast with Helmholtz and Lie is this: both of them require homogeneity and isotropy for physical space. From a general Riemannian standpoint, the latter characteristics are valid only for a matter-empty universe. Such a universe is flat and Euclidean, whereas a universe that contains matter is inhomogeneous, anisotropic and of variable curvature.