Editing Riemannian manifold (section)

=== 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}}