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.

Read more about this topic:  Riemann Curvature Tensor

Famous quotes containing the words special and/or cases:

    The universal social pressure upon women to be all alike, and do all the same things, and to be content with identical restrictions, has resulted not only in terrible suffering in the lives of exceptional women, but also in the loss of unmeasured feminine values in special gifts. The Drama of the Woman of Genius has too often been a tragedy of misshapen and perverted power.
    Anna Garlin Spencer (1851–1931)

    We noticed several other sandy tracts in our voyage; and the course of the Merrimack can be traced from the nearest mountain by its yellow sand-banks, though the river itself is for the most part invisible. Lawsuits, as we hear, have in some cases grown out of these causes. Railroads have been made through certain irritable districts, breaking their sod, and so have set the sand to blowing, till it has converted fertile farms into deserts, and the company has had to pay the damages.
    Henry David Thoreau (1817–1862)