GEOMETRY OF THE UNIVERSE
The geometries of interest in cosmology are those that are uniform everywhere and hence satisfy the cosmological principle. Locally they can resemble Euclidean geometry but on a global scale be quite different.
Euclidean geometry postulates that through any given point one and only one line can be drawn that will never intersect a given line (that is, a parallel line), which defines a flat space. Other assumptions are possible that also produce a uniform space: For example no line can be drawn through the external point that will not intersect the given line; this situation defines a spherical space. Or through any point not on a given line any number of lines can be drawn that will never intersect the given line, which is the definiton for hyperbolic space.
Spherical space possesses what is referred to as positive curvature, as in the two-dimensional example of the surface of a sphere, where "straight" lines are arcs of great circles. Spherical space is centerless and edgeless even though it is finite in extent. In spherical space a beam of light sent off in one direction will eventually return to you from the opposite direction. Of an opposite nature, hyperbolic space possesses negative curvature, as on a two-dimensional saddle-shaped surface, where "straight" lines are arcs of hyperbolae. Hyperbolic space is of infinite extent and a beam of light sent off in some direction will never return. Both spherical and hyperbolic space contrast globally with Euclidean space, in which straight lines possess no, or zero, curvature. Like hyperbolic space, Euclidean space is infinite in extent so that a light beam will never return if sent off in some direction.
