Riemann Curvature Tensor - Special Cases

Special Cases

Surfaces

For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor can be expressed as

where is the metric tensor and is a function called the Gaussian curvature and a, b, c and d take values either 1 or 2. The Riemann tensor has only one functionally independent component. The Gaussian curvature coincides with the sectional curvature of the surface. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by

Space forms

A Riemannian manifold is a space form if its sectional curvature is equal to a constant K. The Riemann tensor of a space form is given by

Conversely, except in dimension 2, if the curvature of a Riemannian manifold has this form for some function K, then the Bianchi identities imply that K is constant and thus that the manifold is (locally) a space form.