Editing Riemannian manifold (section)

== Riemannian metrics on Lie groups ==

=== 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}}

=== Homogeneous spaces ===

A Riemannian manifold {{math|(''M'', ''g'')}} is said to be [[homogeneous space|''homogeneous'']] if for every pair of points {{mvar|x}} and {{mvar|y}} in {{mvar|M}}, there is some isometry {{mvar|f}} of the Riemannian manifold sending {{mvar|x}} to {{mvar|y}}. This can be rephrased in the language of [[group action]]s as the requirement that the natural action of the [[isometry group]] is transitive. Every homogeneous Riemannian manifold is geodesically complete and has constant scalar curvature.{{sfnm|1a1=Kobayashi|1a2=Nomizu|1y=1963|1loc=Theorem IV.4.5}}

Up to isometry, all homogeneous Riemannian manifolds arise by the following construction. Given a Lie group {{mvar|G}} with compact subgroup {{mvar|K}} which does not contain any nontrivial [[normal subgroup]] of {{mvar|G}}, fix any [[complemented subspace]] {{mvar|W}} of the [[Lie algebra]] of {{mvar|K}} within the Lie algebra of {{mvar|G}}. If this subspace is invariant under the linear map {{math|ad<sub>''G''</sub>(''k''): ''W'' → ''W''}} for any element {{mvar|k}} of {{mvar|K}}, then {{mvar|G}}-invariant Riemannian metrics on the [[coset space]] {{math|''G''/''K''}} are in one-to-one correspondence with those inner products on {{mvar|W}} which are invariant under {{math|ad<sub>''G''</sub>(''k''): ''W'' → ''W''}} for every element {{mvar|k}} of {{mvar|K}}.{{sfnm|1a1=Besse|1y=1987|1loc=Section 7C}} Each such Riemannian metric is homogeneous, with {{mvar|G}} naturally viewed as a subgroup of the full isometry group.

The above example of Lie groups with left-invariant Riemannian metrics arises as a very special case of this construction, namely when {{mvar|K}} is the trivial subgroup containing only the identity element. The calculations of the Levi-Civita connection and the curvature referenced there can be generalized to this context, where now the computations are formulated in terms of the inner product on {{mvar|W}}, the Lie algebra of {{mvar|G}}, and the direct sum decomposition of the Lie algebra of {{mvar|G}} into the Lie algebra of {{mvar|K}} and {{mvar|W}}.{{sfnm|1a1=Besse|1y=1987|1loc=Section 7C}} This reduces the study of the curvature of homogeneous Riemannian manifolds largely to algebraic problems. This reduction, together with the flexibility of the above construction, makes the class of homogeneous Riemannian manifolds very useful for constructing examples.

=== Symmetric spaces ===
{{Main|Symmetric space}}

A connected Riemannian manifold {{math|(''M'', ''g'')}} is said to be [[symmetric space|''symmetric'']] if for every point {{mvar|p}} of {{mvar|M}} there exists some isometry of the manifold with {{mvar|p}} as a [[Fixed point (mathematics)|fixed point]] and for which the negation of the [[differential (mathematics)|differential]] at {{mvar|p}} is the [[identity map]]. Every Riemannian symmetric space is homogeneous, and consequently is geodesically complete and has constant scalar curvature. However, Riemannian symmetric spaces also have a much stronger curvature property not possessed by most homogeneous Riemannian manifolds, namely that the [[Riemann curvature tensor]] and [[Ricci curvature]] are [[parallel transport|parallel]]. Riemannian manifolds with this curvature property, which could loosely be phrased as "constant Riemann curvature tensor" (not to be confused with [[constant curvature]]), are said to be ''locally symmetric''. This property nearly characterizes symmetric spaces; [[Élie Cartan]] proved in the 1920s that a locally symmetric Riemannian manifold which is geodesically complete and [[simply-connected]] must in fact be symmetric.{{sfnm|1a1=Petersen|1y=2016|1loc=Chapter 10}}

Many of the fundamental examples of Riemannian manifolds are symmetric. The most basic include the sphere and [[real projective space]]s with their standard metrics, along with hyperbolic space. The complex projective space, [[quaternionic projective space]], and [[Cayley plane]] are analogues of the real projective space which are also symmetric, as are [[complex hyperbolic space]], quaternionic hyperbolic space, and Cayley hyperbolic space, which are instead analogues of hyperbolic space. [[Grassmannian manifold]]s also carry natural Riemannian metrics making them into symmetric spaces. Among the Lie groups with left-invariant Riemannian metrics, those which are bi-invariant are symmetric.{{sfnm|1a1=Petersen|1y=2016|1loc=Chapter 10}}

Based on their algebraic formulation as special kinds of homogeneous spaces, Cartan achieved an explicit classification of symmetric spaces which are ''irreducible'', referring to those which cannot be locally decomposed as [[product space]]s. Every such space is an example of an [[Einstein manifold]]; among them only the one-dimensional manifolds have zero scalar curvature. These spaces are important from the perspective of [[Riemannian holonomy]]. As found in the 1950s by [[Marcel Berger]], any Riemannian manifold which is simply-connected and irreducible is either a symmetric space or has Riemannian holonomy belonging to a list of only seven possibilities. Six of the seven exceptions to symmetric spaces in Berger's classification fall into the fields of [[Kähler geometry]], [[quaternion-Kähler manifold|quaternion-Kähler geometry]], [[G2 manifold|G<sub>2</sub> geometry]], and [[Spin(7) manifold|Spin(7) geometry]], each of which study Riemannian manifolds equipped with certain extra structures and symmetries. The seventh exception is the study of 'generic' Riemannian manifolds with no particular symmetry, as reflected by the maximal possible holonomy group.{{sfnm|1a1=Petersen|1y=2016|1loc=Chapter 10}}