Editing Riemannian manifold (section)

=== Left-invariant metrics on Lie groups ===
Let {{mvar|G}} be a [[Lie group]], such as the [[3D rotation group|group of rotations in three-dimensional space]]. Using the group structure, any inner product on the tangent space at the identity (or any other particular tangent space) can be transported to all other tangent spaces to define a Riemannian metric. Formally, given an inner product {{math|''g''<sub>''e''</sub>}} on the tangent space at the identity, the inner product on the tangent space at an arbitrary point {{mvar|p}} is defined by
:<math>g_p(u,v)=g_e(dL_{p^{-1}}(u),dL_{p^{-1}}(v)),</math>
where for arbitrary {{mvar|x}}, {{math|''L''<sub>''x''</sub>}} is the left multiplication map {{math|''G'' → ''G''}} sending a point {{mvar|y}} to {{math|''xy''}}. Riemannian metrics constructed this way are ''left-invariant''; right-invariant Riemannian metrics could be constructed likewise using the right multiplication map instead.

The Levi-Civita connection and curvature of a general left-invariant Riemannian metric can be computed explicitly in terms of {{math|''g''<sub>''e''</sub>}}, the [[adjoint representation]] of {{mvar|G}}, and the [[Lie algebra]] associated to {{mvar|G}}.{{sfnm|1a1=Cheeger|1a2=Ebin|1y=2008|1loc=Proposition 3.18}} These formulas simplify considerably in the special case of a Riemannian metric which is ''bi-invariant'' (that is, simultaneously left- and right-invariant).{{sfnm|1a1=Cheeger|1a2=Ebin|1y=2008|1loc=Corollary 3.19|2a1=Petersen|2y=2016|2loc=Section 4.4}} All left-invariant metrics have constant scalar curvature.

Left- and bi-invariant metrics on Lie groups are an important source of examples of Riemannian manifolds. [[Berger sphere]]s, constructed as left-invariant metrics on the [[special unitary group]] SU(2), are among the simplest examples of the [[collapsing manifold|collapsing]] phenomena, in which a simply-connected Riemannian manifold can have small volume without having large curvature.{{sfnm|1a1=Petersen|1y=2016|1loc=Section 4.4.3 and p. 399}} They also give an example of a Riemannian metric which has constant scalar curvature but which is not [[Einstein metric|Einstein]], or even of parallel Ricci curvature.{{sfnm|1a1=Petersen|1y=2016|1p=369}} Hyperbolic space can be given a Lie group structure relative to which the metric is left-invariant.<ref>In the upper half-space model of hyperbolic space, the Lie group structure is defined by <math>(x_1,\ldots,x_n)\cdot(y_1,\ldots,y_n)=(x_1+y_nx_1,\ldots,x_{n-1}+y_nx_{n-1},x_ny_n).</math></ref>{{sfnm|1a1=Lee|1y=2018|1loc=Example 3.16f}} Any bi-invariant Riemannian metric on a Lie group has nonnegative sectional curvature, giving a variety of such metrics: a Lie group can be given a bi-invariant Riemannian metric if and only if it is the product of a [[compact Lie group]] with an [[abelian Lie group]].{{sfnm|1a1=Lee|1y=2018|1p=72|2a1=Milnor|2y=1976}}