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On the nonexistence of a Lobachevsky geometry model of an isotropic and homogeneous universe. (English) Zbl 1011.83046

Summary: 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 \(\mathbb R^4\) of a homogeneous and isotropic universe for a fixed time which would fit the cosmological principle. Hence, there does not exist any model in \(\mathbb R^4\) of an isotropic universe which would be represented by a three-dimensional hypersurface with the Lobachevsky geometry.

MSC:

83F05 Relativistic cosmology
53C80 Applications of global differential geometry to the sciences
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