Constant curvature

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See also: Space form

In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points. The circle has constant curvature, also, in a natural (but different) sense.

The standard surface geometries of constant curvature are elliptic geometry (or spherical geometry) which has positive curvature, Euclidean geometry which has zero curvature, and hyperbolic geometry (pseudosphere geometry) which has negative curvature. Since Riemann surfaces can be taken to have constant curvature, there is a large supply of other examples, for negative curvature.

For higher dimensional manifolds, constant curvature is usually taken to mean constant sectional curvature, and a complete such a manifold is called a space form. As in the case of surfaces, there are three types of geometries (elliptic, flat, or hyperbolic) according to whether the curvature is positive, zero, or negative. The universal cover of a manifold of constant sectional curvature is one of the model spaces (sphere, Euclidean space, hyperbolic space), and the study of space forms is thus generalized crystallography.

See also: Curvature of Riemannian manifolds
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