Editing Riemannian manifold (section)

== Constant curvature and space forms ==
A Riemannian manifold is said to have ''[[constant curvature]]'' {{mvar|&kappa;}} if every [[sectional curvature]] equals the number {{mvar|&kappa;}}. This is equivalent to the condition that, relative to any coordinate chart, the [[Riemann curvature tensor]] can be expressed in terms of the [[metric tensor]] as
:<math>R_{ijkl}=\kappa(g_{il}g_{jk}-g_{ik}g_{jl}).</math>
This implies that the [[Ricci curvature]] is given by {{math|''R''<sub>''jk''</sub> {{=}} (''n'' − 1)''&kappa;g''<sub>''jk''</sub>}} and the [[scalar curvature]] is {{math|''n''(''n'' − 1)''&kappa;''}}, where {{mvar|n}} is the dimension of the manifold. In particular, every Riemannian manifold of constant curvature is an [[Einstein manifold]], thereby having constant scalar curvature. As found by [[Bernhard Riemann]] in his 1854 lecture introducing Riemannian geometry, the locally-defined Riemannian metric
:<math>\frac{dx_1^2+\cdots+dx_n^2}{(1+\frac{\kappa}{4}(x_1^2+\cdots+x_n^2))^2}</math>
has constant curvature {{mvar|&kappa;}}. Any two Riemannian manifolds of the same constant curvature are [[local isometry|locally isometric]], and so it follows that any Riemannian manifold of constant curvature {{mvar|&kappa;}} can be covered by coordinate charts relative to which the metric has the above form.{{sfnm|1a1=Wolf|1y=2011|1loc=Chapter 2}}

A ''[[space form|Riemannian space form]]'' is a Riemannian manifold with constant curvature which is additionally [[connected space|connected]] and [[geodesically complete]]. A Riemannian space form is said to be a ''[[spherical space form]]'' if the curvature is positive, a ''Euclidean space form'' if the curvature is zero, and a ''hyperbolic space form'' or ''[[hyperbolic manifold]]'' if the curvature is negative. In any dimension, the sphere with its standard Riemannian metric, Euclidean space, and hyperbolic space are Riemannian space forms of constant curvature {{math|1}}, {{math|0}}, and {{math|−1}} respectively. Furthermore, the [[Killing–Hopf theorem]] says that any simply-connected spherical space form is homothetic to the sphere, any simply-connected Euclidean space form is homothetic to Euclidean space, and any simply-connected hyperbolic space form is homothetic to hyperbolic space.{{sfnm|1a1=Wolf|1y=2011|1loc=Chapter 2}}

Using the [[covering space|covering manifold]] construction, any Riemannian space form is isometric to the [[quotient manifold]] of a simply-connected Riemannian space form, modulo a certain group action of isometries. For example, the isometry group of the {{mvar|n}}-sphere is the [[orthogonal group]] {{math|O(''n'' + 1)}}. Given any finite [[subgroup]] {{mvar|G}} thereof in which only the [[identity matrix]] possesses {{math|1}} as an [[eigenvalue]], the natural group action of the orthogonal group on the {{mvar|n}}-sphere restricts to a group action of {{mvar|G}}, with the [[quotient manifold]] {{math|''S''<sup>''n''</sup> / ''G''}} inheriting a geodesically complete Riemannian metric of constant curvature {{math|1}}. Up to homothety, every spherical space form arises in this way; this largely reduces the study of spherical space forms to problems in [[group theory]]. For instance, this can be used to show directly that every even-dimensional spherical space form is homothetic to the standard metric on either the sphere or [[real projective space]]. There are many more odd-dimensional spherical space forms, although there are known algorithms for their classification. The list of three-dimensional spherical space forms is infinite but explicitly known, and includes the [[lens space]]s and the [[Poincaré dodecahedral space]].{{sfnm|1a1=Wolf|1y=2011|1loc=Chapters 2 and 7}}

The case of Euclidean and hyperbolic space forms can likewise be reduced to group theory, based on study of the isometry group of Euclidean space and hyperbolic space. For example, the class of two-dimensional Euclidean space forms includes Riemannian metrics on the [[Klein bottle]], the [[Möbius strip]], the [[torus]], the [[cylinder]] {{math|''S''<sup>1</sup> × ℝ}}, along with the Euclidean plane. Unlike the case of two-dimensional spherical space forms, in some cases two space form structures on the same manifold are not homothetic. The case of two-dimensional hyperbolic space forms is even more complicated, having to do with [[Teichmüller space]]. In three dimensions, the Euclidean space forms are known, while the geometry of hyperbolic space forms in three and higher dimensions remains an area of active research known as [[hyperbolic geometry]].{{sfnm|1a1=Wolf|1y=2011|1loc=Chapters 2 and 3}}