On the nonexistence of a Lobachevsky geometry model of an isotropic and homogeneous universe

Abstract

According to the Einstein cosmological principle, our universe is homogeneous and isotropic, i.e. its curvature is constant at any point and in any direction. On large scales, when all local irregularities are ignored, this assumption has been confirmed by astronomers. We show that there is no reasonable hyperbolic geometry model in R 4 of a homogeneous and isotropic universe for a fixed time which would fit the cosmological principle. Hence, there does not exist any model in R 4 of an isotropic universe which would be represented by a three-dimensional hypersurface with the Lobachevsky geometry.