Editing Curvature (section)

==Curvature of space{{anchor|Space}}==
{{further|Curvature of Riemannian manifolds|Curved space}}
{{distinguish-redirect|Curvature of space|Curvature of space-time}}

By extension of the former argument, a space of three or more dimensions can be intrinsically curved. The curvature is ''intrinsic'' in the sense that it is a property defined at every point in the space, rather than a property defined with respect to a larger space that contains it. In general, a curved space may or may not be conceived as being embedded in a higher-dimensional [[ambient space]]; if not then its curvature can only be defined intrinsically.

After the discovery of the intrinsic definition of curvature, which is closely connected with [[non-Euclidean geometry]], many mathematicians and scientists questioned whether ordinary physical space might be curved, although the success of Euclidean geometry up to that time meant that the radius of curvature must be astronomically large. In the theory of [[general relativity]], which describes [[gravity]] and [[physical cosmology|cosmology]], the idea is slightly generalised to the "curvature of [[spacetime]]"; in relativity theory spacetime is a [[pseudo-Riemannian manifold]]. Once a time coordinate is defined, the three-dimensional space corresponding to a particular time is generally a curved Riemannian manifold; but since the time coordinate choice is largely arbitrary, it is the underlying spacetime curvature that is physically significant.

Although an arbitrarily curved space is very complex to describe, the curvature of a space which is locally [[isotropic]] and [[Homogeneous space|homogeneous]] is described by a single Gaussian curvature, as for a surface; mathematically these are strong conditions, but they correspond to reasonable physical assumptions (all points and all directions are indistinguishable). A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere or [[hypersphere]]. An example of negatively curved space is [[hyperbolic geometry]] (see also: [[non-positive curvature]]). A space or space-time with zero curvature is called '''''flat'''''.{{anchor|Flat space}} For example, [[Euclidean space]] is an example of a flat space, and [[Minkowski space]] is an example of a flat spacetime. There are other examples of flat geometries in both settings, though. A [[torus]] or a [[Cylinder (geometry)|cylinder]] can both be given flat metrics, but differ in their [[topology]]. Other topologies are also possible for curved space {{xref|(see also: [[Shape of the universe]])}}.