, , . With this coordinate transformation . Without embedding The geometry of a n-dimensional space can also be described with Riemannian geometry. An isotropic and homogeneous space can be described by the metric: . This reduces to Euclidean space when . But a space can be said to be "flat" when the Weyl tensor has all zero components. In three dimensions this condition is met when the Ricci tensor () is equal to the metric times the Ricci scalar (, not to be confused with the R of the previous section). That is . Calculation of these components from the metric gives that where . This gives the metric: . where can be zero, positive, or negative and is not limited to ±1. Open, flat, closed An isotropic and homogeneous space can be described by the metric: . In the limit that the constant of curvature () becomes infinitely large, a flat, Euclidean space is returned. It is essentially the same as setting to zero. If is not zero the space is not Euclidean. When the space is said to be closed or elliptic. When the space is said to be open or hyperbolic. Triangles which lie on the surface of an open space will have a sum of angles which is less than 180°. Triangles which lie on the surface of a closed space will have a sum of angles which is greater than 180°. The volume, however, is not . See also CAT(k) space Non-positive curvature Further reading The Feynman Lectures on Physics Vol. II Ch. 42: Curved Space External links Curved Spaces, simulator for multiconnected universes developed by Jeffrey Weeks Riemannian geometry Physical cosmology Differential geometry General relativity