Constant curvature
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space and is a single number that determines its local geometry. 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.
Classification
The classifications here are based on the universal covering space. There may be more than one space that has the same universal covering space.The Riemannian manifolds of constant curvature can be classified into the following three classes:
- Elliptic geometry – constant positive sectional curvature
- Euclidean geometry – constant vanishing sectional curvature
- Hyperbolic geometry – constant negative sectional curvature.
- De Sitter space – constant positive sectional curvature
- Minkowski space – constant vanishing sectional curvature
- Anti-de Sitter space – constant negative sectional curvature.
For every signature, dimension and curvature, a similar classification exists.
Properties
- Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel.
- Every space of dimension of constant curvature is locally maximally symmetric, i.e. it has local isometries.
- Conversely, there exists a similar but stronger statement: every maximally symmetric space, i.e. a space that has isometries, has constant curvature.
- The universal cover of a Riemannian manifold of constant sectional curvature is one of the model spaces:
- * sphere
- * plane
- * hyperbolic manifold
- A space of constant curvature that is geodesically complete is called a space form. The study of space forms is intimately related to generalized crystallography.
- Two space forms are isomorphic if and only if they have the same dimension, their metrics possess the same signature and their sectional curvatures are equal.