text: Curvature The curvature of the universe places constraints on the topology. If the spatial geometry is spherical, i.e., possess positive curvature, the topology is compact. For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite. Many textbooks erroneously state that a flat or hyperbolic universe implies an infinite universe; however, the correct statement is that a flat universe that is also simply connected implies an infinite universe. For example, Euclidean space is flat, simply connected, and infinite, but there are tori that are flat, multiply connected, finite, and compact (see flat torus).
  In general, local to global theorems in Riemannian geometry relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in Thurston geometries.
  The latest research shows that even the most powerful future experiments (like the SKA) will not be able to distinguish between a flat, open and closed universe if the true value of cosmological curvature parameter is smaller than 10−4. If the true value of the cosmological curvature parameter is larger than 10−3 we will be able to distinguish between these three models even now.
  Final results of the Planck mission, released in 2018, show the cosmological curvature parameter, , to be , consistent with a flat universe. (i.e. positive curvature: , , , negative curvature: , , , zero curvature: , , ).