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Concept in differential geometry
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a differentiable manifold) and is a single number that determines its local geometry.[1] The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point. For example, a sphere is a surface of constant positive curvature.
The classifications here are based on the universal covering space. There may be more than space that has the same universal covering space.
The Riemannian manifolds of constant curvature can be classified into the following three classes:
The Lorentzian manifolds of constant curvature can be classified into the following three classes:
The de Sitter and anti-de Sitter spaces of dimension 2 are the same (the sign of the curvature depends on which direction is referenced as "space-like").
For every signature, dimension and curvature, a similar classification exists.
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