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A few years ago, Cornish, Spergel and Starkman ͑CSS͒ suggested that a multiply connected ''small'' universe could allow for classical chaotic mixing as a preinflationary homogenization process. The smaller the volume, the more important the process. Also, a smaller universe has a greater probability of being spontaneously created. Previously DeWitt, Hart and Isham ͑DHI͒ calculated the Casimir energy for static multiply connected flat space-times. Because of the interest in small volume hyperbolic universes ͑e.g., CSS͒, we generalize the DHI calculation by making a numerical investigation of the Casimir energy for a conformally coupled, massive scalar field in a static universe, whose spatial sections are the Weeks manifold, the smallest universe of negative curvature known. In spite of being a numerical calculation, our result is in fact exact. It is shown that there is spontaneous vacuum excitation of low multipolar components.

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We present a modified version of the cosmic crystallography method, especially useful for testing closed models of negative spatial curvature. The images of clusters of galaxies in simulated catalogs are "pulled back" to the fundamental domain before the set of distances is calculated.

This paper deals with two aspects of relativistic cosmologies with closed (compact and boundless) spatial sections. These spacetimes are based on the theory of General Relativity, and admit a foliation into space sections ) (t S , which are spacelike hypersurfaces satisfying the postulate of the closure of space: each ) (t S is a 3-dimensional closed Riemannian manifold. The discussed topics are:(1) A comparison, previously obtained, between Thurston's geometries and Bianchi-Kantowski-Sachs metrics for such 3-manifolds is here clarified and developed.(2) Some implications of global inhomogeneity for locally homogeneous 3-spaces of constant curvature are analyzed from an observational viewpoint.

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The smallest known three-dimensional closed manifold of curvature k= --1 was discovered a few years ago by Weeks. This kind of manifold is constructed from a hyperbolic polyhedron with faces pairwise identified. Here it is used as the comoving spatial section of a Friedmann cosmological model