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In physical cosmology, the shape of the universe refers to both its local and global geometry. Local geometry is defined primarily by its curvature, while the global geometry is characterised by its topology (which itself is constrained by curvature). General relativity explains how spatial curvature (local geometry) is constrained by gravity. The global topology of the universe cannot be deduced from measurements of curvature inferred from observations within the family of homogeneous general relativistic models alone, due to the existence of locally indistinguishable spaces with varying global topological characteristics. For example; a multiply connected space like a 3 torus has everywhere zero curvature but is finite in extent, whereas a flat simply connected space is infinite in extent (euclidean space).
Current observational evidence (WMAP, BOOMERanG, and Planck for example) imply that the observable universe is flat to within a 0.4% margin of error of the curvature density parameter with an unknown global topology.[1][2] It is currently unknown if the universe is simply connected like euclidean space or multiply connected like a torus. To date, no compelling evidence has been found suggesting the universe has a non-trivial (i.e.; not simply connected) topology, though it has not been ruled out by astronomical observations.
