Friedmann Cosmology and Almost Isotropy

Details Friedmann Cosmology and Almost Isotropy Friedmann Cosmology and Almost Isotropy

2003-02-19

arxiv.org/abs/math/0302244v3

GAFA, Geom. funct. anal. Vol. 14 (2004) 853-912

Mathematics

Differential Geometry

45 pages, 7 figures (eps), AIM Workshop on General Relativity 2002, added some details to the explanation, fixed typos/grammar

Scientific paper

In the Friedmann Model of the universe, cosmologists assume that spacelike slices of the universe are Riemannian manifolds of constant sectional curvature. This assumption is justified via Schur's Theorem by stating that the spacelike universe is locally isotropic. Here we define a Riemannian manifold as almost locally isotropic in a sense which allows both weak gravitational lensing in all directions and strong gravitational lensing in localized angular regions at most points. We then prove that such a manifold is Gromov Hausdorff close to a length space $Y$ which is a collection of space forms joined at discrete points. Within the paper we define a concept we call an "exponential length space" and prove that if such a space is locally isotropic then it is a space form.

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