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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. [8]
The cosmological principle implies that the metric of the universe must be of the form = where ds 3 2 is a three-dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature. This metric is called the Friedmann–Lemaître–Robertson–Walker (FLRW ...
The local geometry of the universe is determined by whether the relative density Ω is less than, equal to or greater than 1. From top to bottom: a spherical universe with greater than critical density (Ω>1, k>0); a hyperbolic, underdense universe (Ω<1, k<0); and a flat universe with exactly the critical density (Ω=1, k=0). The spacetime of ...
The universe may have positive, negative, or zero spatial curvature depending on its total energy density. Curvature is negative if its density is less than the critical density; positive if greater; and zero at the critical density, in which case space is said to be flat. Observations indicate the universe is consistent with being flat. [139 ...
The index k is defined so that it can take only one of three values: 0, corresponding to flat Euclidean geometry; 1, corresponding to a space of positive curvature; or −1, corresponding to a space of positive or negative curvature. [153] The value of R as a function of time t depends upon k and the cosmological constant Λ. [151]
Friedmann's 1924 papers, including "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes" ("On the possibility of a world with constant negative curvature of space") published by the German physics journal Zeitschrift für Physik (Vol. 21, pp. 326–332), demonstrated that he had command of all three Friedmann models ...
Manifolds of constant curvature are most familiar in the case of two dimensions, where the elliptic plane or surface of a sphere is a surface of constant positive curvature, a flat (i.e., Euclidean) plane is a surface of constant zero curvature, and a hyperbolic plane is a surface of constant negative curvature. Einstein's general theory of ...
In history, there is a deep connection between the curvature of a manifold and its topology. The Bonnet–Myers theorem states that a complete Riemannian manifold that has Ricci curvature everywhere greater than a certain positive constant must be compact. The condition of positive Ricci curvature is most conveniently stated in the following ...