Editing Riemannian manifold (section)

=== 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.