en>ChrisGualtieri: /* Development */TypoScan Project / General Fixes, typos fixed: etc → etc. using AWB

2012-06-05T04:14:46Z

Development: TypoScan Project / General Fixes, typos fixed: etc → etc. using AWB

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{{Unreferenced|date=December 2009}}
{{See also|space form|curvature of Riemannian manifolds|sectional curvature}}

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 all points. For example, a [[sphere]] is a surface of constant positive curvature.

==Classification==
The geometries 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.

==Properties==
* every space of constant curvature is [[locally symmetric]], i.e. its [[Riemann curvature tensor|curvature tensor]] is [[parallel (geometry)|parallel]] <math>\nabla \mathrm{R}=0</math>
* every space of constant curvature is locally [[maximally symmetric]], i.e. it has <math>\frac{1}{2} n (n+1)</math> number of [[Killing vector|local isometries]], where n is its dimension.
* conversely, there exists a similar but stronger statement: every [[maximally symmetric]] space, i.e. a space which has <math>\frac{1}{2} n (n+1)</math> (global) [[isometries]], has constant curvature.
* the [[universal cover]] of a manifold of constant sectional curvature is one of the model spaces
** [[sphere]] (sectional curvature positive)
** [[flat manifold|plane]] (sectional curvature zero)
** [[hyperbolic space|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 [[metric signature|signature]] and their sectional curvatures are equal.

{{DEFAULTSORT:Constant Curvature}}
[[Category:Differential geometry of surfaces]]
[[Category:Riemannian geometry]]
[[Category:Curvature (mathematics)]]

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en>ChrisGualtieri: /* Development */TypoScan Project / General Fixes, typos fixed: etc → etc. using AWB