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Concept
Constant curvature
Summary
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining 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.
The Riemannian manifolds of constant curvature can be classified into the following three cases:
elliptic geometry – constant positive sectional curvature
Euclidean geometry – constant vanishing sectional curvature
hyperbolic geometry – constant negative sectional curvature.
Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel .
Every space of constant curvature is locally maximally symmetric, i.e. it has number of local isometries, where is its dimension.
Conversely, there exists a similar but stronger statement: every maximally symmetric space, i.e. a space which has (global) isometries, has constant curvature.
(Killing–Hopf theorem) The universal cover of a manifold of constant sectional curvature is one of the model spaces:
sphere (sectional curvature positive)
plane (sectional curvature zero)
hyperbolic manifold (sectional curvature negative)
A space of constant curvature which is geodesically complete is called space form and the study of space forms is intimately related to generalized crystallography (see the article on space form for more details).
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.
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Hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane.
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.