A property of spacetime which describes how the laws of geometry work. In a space of only two dimensions which is flat, Euclidean geometry applies so that the sum of the internal angles of a triangle on the sheet is 180°. However, if this sheet is curved, the sum of the internal angles of a triangle will be either larger (if it is curved like a sphere) or smaller (if curved like a saddle). The curvature of 3D space is much harder to visualize, but the general principles are the same.

In general relativity the geometry of spacetime is intimately connected with the distribution of matter. Massive objects cause spacetime curvature. In the 2D flat-sheet analogy, the object will cause the sheet to bend. Objects moving on the sheet will then traverse the shortest distance if they travel in curved paths. This is, in essence, what happens in general relativity, as demonstrated by the bending of light around massive objects.

The Universe as a whole may have positive, negative, or zero curvature. A universe with positive curvature would curve back on itself like the surface of a sphere so that one could in principle travel out into space and eventually end back at the same place. Such a universe is a closed universe, having finite size. A universe with negative curvature, however, would be an open universe, infinite in size.

The curvature of space can be measured with geometric tests. Observations of the cosmic background radiation provide such a geometric test, with the result that the Universe has zero curvature (see cosmology). In other words, it is spatially flat (Euclidean) and infinite in both space and time. One possible explanation is provided by the theory of the inflationary universe, which proposes that the Universe expanded extremely quickly shortly after the Big Bang. If inflationary theory is true, our entire observable Universe, over 10 billion light years in size, may be only a speck within the greater Universe—and this may explain why we fail to see any curvature. Returning to the analogy of the surface of a sphere, if the triangle on the sphere is small compared with the size of the sphere, the sum of the angles in a triangle will not differ significantly from 180°. If the theory of inflation is correct, the Universe beyond our horizon may actually have positive or negative curvature, but we see too small a part of it to detect the curvature.