Non-Euclidean Geometry and Curved Space
What distinguishes the plane of Euclidean geometry from the surface of a sphere or a saddle surface is the curvature of each (see differential geometry); the plane has zero curvature, the surface of a sphere and other surfaces described by Riemann's geometry have positive curvature, and the saddle surface and other surfaces described by Lobachevsky's geometry have negative curvature. Similarly, in three dimensions the spaces corresponding to these three types of geometry also have zero, positive, or negative curvature, respectively.
As to which of these systems is a valid description of our own three-dimensional space (or four-dimensional space-time), the choice can be made only on the basis of measurements made over very large, cosmological distances of a billion light-years or more; the differences between a Euclidean universe of zero curvature and a non-Euclidean universe of very small positive or negative curvature are too small to be detected from ordinary measurements. One interesting feature of a universe described by Riemann's geometry is that it is finite but unbounded; straight lines ultimately form closed curves, so that a ray of light could eventually return to its source.
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