Calculation of the these components from the metric gives that where. 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). But a space can be said to be вЂњ flatвЂќ when the Weyl Tensor has all zero components. An isotropic and homogeneous space can be described by the metric. The geometry of a n-dimensional space can also be described with Riemannian geometry. This form is usually not particularly appealing and so a coordinate transform is often applied:. Plugging into the original equation gives. The differential of the constraining equation is leading to. We can now use this constraint to eliminate the artificial fourth coordinate. For convenience we can choose the constant to be where now is positive and. The constant can be positive or negative. We can choose a constraint such that Pythagorean theorem holds in the new 4D space. Since four coordinates have four degrees of freedom it must have a constraint placed on it. ДЇor the choice of the 4D coordinates to be valid descriptors of the original 3D space it must have the same number of degrees of freedom. Note that the coordinate is not the same as the coordinate. Д«ut if we now describe the three dimensional space with four dimensions ( ) we can choose coordinates such that. Suppose we have a non-euclidean three dimensional space with coordinates. The Pythagorean relationship can often be restored by describing the space with an extra dimension. One of the defining characteristics of a curved space is its departure with the Pythagorean theorem.

This relationship does not hold for curved spaces. In a flat space, the sum of the squares of the side of a triangle is equal to the square of the hypotenuse.

3 dimensional curved space