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 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 then 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 , 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 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
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 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 of H (since this is the Lie algebra of Gσ), and the −1 eigenspace will be denoted . Since σ is an automorphism of , this gives a direct sum decomposition
with
The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer is a Lie subalgebra of . The second condition means that is an -invariant complement to in . Thus any symmetric space is a reductive homogeneous space, but there are many reductive homogeneous spaces which are not symmetric spaces. The key feature of symmetric spaces is the third condition that brackets into .
Conversely, given any Lie algebra with a direct sum decomposition satisfying these three conditions, the linear map σ, equal to the identity on and minus the identity on , is an involutive automorphism.
Riemannian symmetric spaces satisfy the Lie-theoretic characterization If M is a Riemannian symmetric space, the identity component G of the isometry group of M is a Lie group acting transitively on M (that is, M is Riemannian homogeneous). Therefore, if we fix some point p of M, M is diffeomorphic to the quotient G/K, where K denotes the isotropy group of the action of G on M at p. By differentiating the action at p we obtain an isometric action of K on TpM. This action is faithful (e.g., by a theorem of Kostant, any isometry in the identity component is determined by its 1-jet at any point) and so K is a subgroup of the orthogonal group of TpM, hence compact. Moreover, if we denote by sp: M → M the geodesic symmetry of M at p, the map
is an involutive Lie group automorphism such that the isotropy group K is contained between the fixed point group and its identity component (hence an open subgroup) see the definition and following proposition on page 209, chapter IV, section 3 in Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces for further information.
To summarize, M is a symmetric space G/K with a compact isotropy group K. Conversely, symmetric spaces with compact isotropy group are Riemannian symmetric spaces, although not necessarily in a unique way. To obtain a Riemannian symmetric space structure we need to fix a K-invariant inner product on the tangent space to G/K at the identity coset eK: such an inner product always exists by averaging, since K is compact, and by acting with G, we obtain a G-invariant Riemannian metric g on G/K.
To show that G/K is Riemannian symmetric, consider any point (a coset of K, where h ∈ G) and define
where σ is the involution of G fixing K. Then one can check that sp is an isometry with (clearly) sp(p) = p and (by differentiating) dsp equal to minus the identity on TpM. Thus sp is a geodesic symmetry and, since p was arbitrary, M is a Riemannian symmetric space.
If one starts with a Riemannian symmetric space M, and then performs these two constructions in sequence, then the Riemannian symmetric space yielded is isometric to the original one. This shows that the "algebraic data" (G, K, σ, g) completely describe the structure of M.
Classification of Riemannian symmetric spaces
The algebraic description of Riemannian symmetric spaces enabled Élie Cartan to obtain a complete classification of them in 1926.
For a given Riemannian symmetric space M let (G, K, σ, g) be the algebraic data associated to it. To classify the possible isometry classes of M, first note that the universal cover of a Riemannian symmetric space is again Riemannian symmetric, and the covering map is described by dividing the connected isometry group G of the covering by a subgroup of its center. Therefore, we may suppose without loss of generality that M is simply connected. (This implies K is connected by the long exact sequence of a fibration, because G is connected by assumption.)
Classification scheme A simply connected Riemannian symmetric space is said to be irreducible if it is not the product of two or more Riemannian symmetric spaces. It can then be shown that any simply connected Riemannian symmetric space is a Riemannian product of irreducible ones. Therefore, we may further restrict ourselves to classifying the irreducible, simply connected Riemannian symmetric spaces.
The next step is to show that any irreducible, simply connected Riemannian symmetric space M is of one of the following three types: Euclidean type: M has vanishing curvature, and is therefore isometric to a Euclidean space. Compact type: M has nonnegative (but not identically zero) sectional curvature. Non-compact type: M has nonpositive (but not identically zero) sectional curvature.
A more refined invariant is the rank, which is the maximum dimension of a subspace of the tangent space (to any point) on which the curvature is identically zero. The rank is always at least one, with equality if the sectional curvature is positive or negative. If the curvature is positive, the space is of compact type, and if negative, it is of noncompact type. The spaces of Euclidean type have rank equal to their dimension and are isometric to a Euclidean space of that dimension. Therefore, it remains to classify the irreducible, simply connected Riemannian symmetric spaces of compact and non-compact type. In both cases there are two classes.
A. G is a (real) simple Lie group;
B. G is either the product of a compact simple Lie group with itself (compact type), or a complexification of such a Lie group (non-compact type).
The examples in class B are completely described by the classification of simple Lie groups. For compact type, M is a compact simply connected simple Lie group, G is M×M and K is the diagonal subgroup. For non-compact type, G is a simply connected complex simple Lie group and K is its maximal compact subgroup. In both cases, the rank is the rank of G.
The compact simply connected Lie groups are the universal covers of the classical Lie groups SO(n), SU(n), Sp(n) and the five exceptional Lie groups E6, E7, E8, F4, G2.
The examples of class A are completely described by the classification of noncompact simply connected real simple Lie groups. For non-compact type, G is such a group and K is its maximal compact subgroup. Each such example has a corresponding example of compact type, by considering a maximal compact subgroup of the complexification of G that contains K. More directly, the examples of compact type are classified by involutive automorphisms of compact simply connected simple Lie groups G (up to conjugation). Such involutions extend to involutions of the complexification of G, and these in turn classify non-compact real forms of G.
In both class A and class B there is thus a correspondence between symmetric spaces of compact type and non-compact type. This is known as duality for Riemannian symmetric spaces.
Classification result Specializing to the Riemannian symmetric spaces of class A and compact type, Cartan found that there are the following seven infinite series and twelve exceptional Riemannian symmetric spaces G/K. They are here given in terms of G and K, together with a geometric interpretation, if readily available. The labelling of these spaces is the one given by Cartan.