Curved Space - Without Embedding

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 the 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.