In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (M, g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space T p M {\displaystyle T_{p}M} as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about the endpoints). Both descriptions can also naturally be extended to the setting of pseudo-Riemannian manifolds. From the point of view of Lie theory, a symmetric space is the quotient G / H of a connected Lie group G by a closed subgroup H that is (a connected component of) the invariant group of an involution of G. This definition includes more than the Riemannian definition, and reduces to it when H is compact. Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. Their central role in the theory of holonomy was discovered by Marcel Berger. They are important objects of study in representation theory and harmonic analysis as well as in differential geometry.
Geometric definition Let M be a connected Riemannian manifold and p a point of M. A diffeomorphism f of a neighborhood of p is said to be a geodesic symmetry if it fixes the point p and reverses geodesics through that point, i.e. if γ is a geodesic with γ ( 0 ) = p {\displaystyle \gamma (0)=p} then f ( γ ( t ) ) = γ ( − t ) . {\displaystyle f(\gamma (t))=\gamma (-t).} It follows that the derivative of the map f at p is minus the identity map on the tangent space of p. On a general Riemannian manifold, f need not be isometric, nor can it be extended, in general, from a neighbourhood of p to all of M. M is said to be locally Riemannian symmetric if its geodesic symmetries are in fact isometric. This is equivalent to the vanishing of the covariant derivative of the curvature tensor. A locally symmetric space is said to be a (globally) symmetric space if in addition its geodesic symmetries can be extended to isometries on all of M.
Basic properties The Cartan–Ambrose–Hicks theorem implies that M is locally Riemannian symmetric if and only if its curvature tensor is covariantly constant, and furthermore that every simply connected, complete locally Riemannian symmetric space is actually Riemannian symmetric. Every Riemannian symmetric space M is complete and Riemannian homogeneous (meaning that the isometry group of M acts transitively on M). In fact, already the identity component of the isometry group acts transitively on M (because M is connected). Locally Riemannian symmetric spaces that are not Riemannian symmetric may be constructed as quotients of Riemannian symmetric spaces by discrete groups of isometries with no fixed points, and as open subsets of (locally) Riemannian symmetric spaces.
Examples Basic examples of Riemannian symmetric spaces are Euclidean space, spheres, projective spaces, and hyperbolic spaces, each with their standard Riemannian metrics. More examples are provided by compact, semi-simple Lie groups equipped with a bi-invariant Riemannian metric. Every compact Riemann surface of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric space. Every lens space is locally symmetric but not symmetric, with the exception of the real projective space L ( 2 ; 1 ) {\displaystyle L(2;1)} , which is symmetric. The lens spaces are quotients of the 3-sphere by a discrete isometry that has no fixed points. An example of a non-Riemannian symmetric space is anti-de Sitter space.
Algebraic definition Let G be a connected Lie group. Then a symmetric space for G is a homogeneous space G / H where the stabilizer H of a typical point is an open subgroup of the fixed point subgroup G σ {\displaystyle G^{\sigma }} of an involution σ in Aut(G). Thus σ is an automorphism of G with σ2 = idG and H is an open subgroup of the invariant subgroup
G σ = { g ∈ G : σ ( g ) = g } . {\displaystyle G^{\sigma }=\{g\in G:\sigma (g)=g\}.}
Because H is open, it is a union of components of Gσ (including, of course, the identity component). As an automorphism of G, σ fixes the identity element, and hence, by differentiating at the identity, it induces an automorphism of the Lie algebra g {\displaystyle {\mathfrak {g}}} of G, also denoted by σ, whose square is the identity. It follows that the eigenvalues of σ are ±1. The +1 eigenspace is the Lie algebra h {\displaystyle {\mathfrak {h}}} of H (since this is the Lie algebra of Gσ), and the −1 eigenspace will be denoted m {\displaystyle {\mathfrak {m}}} . Since σ is an automorphism of g {\displaystyle {\mathfrak {g}}} , this gives a direct sum decomposition
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