Riemannian symmetric space
Encyclopedia
In differential geometry, representation theory
and harmonic analysis
, a symmetric space is a smooth manifold whose group of symmetries contains an inversion symmetry about every point. There are two ways to formulate the inversion symmetry, via Riemannian geometry or via Lie theory. The Lie theoretic definition is more general and more algebraic.
In Riemannian geometry
, the inversions are geodesic
symmetries, and these are required to be isometries
, leading to the notion of a Riemannian symmetric space. More generally, in Lie theory
a symmetric space is a homogeneous space
G/H for a Lie group G such that the stabilizer H of a point is an open subgroup of the fixed point set of an involution of G. This definition includes (globally) Riemannian symmetric spaces and pseudo-Riemannian
symmetric spaces as special cases.
Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. They were first studied extensively and classified by Élie Cartan
. More generally, classifications of irreducible and semisimple symmetric spaces have been given by Marcel Berger
. They are important in representation theory and harmonic analysis as well as differential geometry.
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, and (globally) Riemannian symmetric if in addition its geodesic symmetries are defined on all of M.
its curvature tensor is covariantly constant
, and furthermore that any simply connected, complete
locally Riemannian symmetric space is actually Riemannian symmetric.
Any 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.
, sphere
s, projective space
s, and hyperbolic space
s, each with their standard Riemannian metrics. More examples are provided by compact, semi-simple Lie groups equipped with a bi-invariant Riemannian metric.
Any 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.
. 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 set of an involution σ of G. Thus σ is an automorphism of G with σ2 = idG and H is an open subgroup of the set
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.
acting transitively on M (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 of σ and its identity component (hence an open subgroup).
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 p = hK (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.
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 possibly 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.)
The next step is to show that any irreducible, simply connected Riemannian symmetric space M is of one of the following three types:
1. Euclidean type: M has vanishing curvature, and is therefore isometric to a Euclidean space
.
2. Compact type: M has nonnegative (but not identically zero) sectional curvature
.
3. 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 group
s. 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
, 
, 
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 which 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.
, the most notable examples being Minkowski space
, De Sitter space
and anti de Sitter space
(with zero, positive and negative curvature respectively). De Sitter space of dimension n may be identified with the 1-sheeted hyperboloid in a Minkowski space of dimension n + 1.
Symmetric and locally symmetric spaces in general can be regarded as affine symmetric spaces. If M = G/H is a symmetric space, then Nomizu showed that there is a G-invariant torsion-free
affine connection
on M whose curvature is parallel
. Conversely a manifold with such a connection is locally symmetric (i.e., its universal cover is a symmetric space). Such manifolds can also be described as those affine manifolds whose geodesic symmetries are all globally defined affine diffeomorphisms, generalizing the Riemannian and pseudo-Riemannian case.
is said to be irreducible if
is an irreducible representation of 
. Since 
is not semisimple (or even reductive) in general, it can have indecomposable
representations which are not irreducible.
However, the irreducible symmetric spaces can be classified. As shown by K. Nomizu, there is a dichotomy: an irreducible symmetric space G/H is either flat (i.e., an affine space) or
is semisimple. This is the analogue of the Riemannian dichotomy between Euclidean spaces and those of compact or noncompact type, and it motivated M. Berger to classify semisimple symmetric spaces (i.e., those with 
semisimple) and determine which of these are irreducible. The latter question is more subtle than in the Riemannian case: even if 
is simple, G/H might not be irreducible.
As in the Riemannian case there are semisimple symmetric spaces with G = H × H. Any semisimple symmetric space is a product of symmetric spaces of this form with symmetric spaces such that
is simple. It remains to describe the latter case. For this, one needs to classify involutions σ of a (real) simple Lie algebra 
. If 
is not simple, then 
is a complex simple Lie algebra, and the corresponding symmetric spaces have the form G/H, where H is a real form of G: these are the analogues of the Riemannian symmetric spaces G/K with G a complex simple Lie group, and K a maximal compact subgroup.
Thus we may assume
is simple. The real subalgebra 
may be viewed as the fixed point set of a complex antilinear involution τ of 
, while σ extends to a complex antilinear involution of 
commuting with τ and hence also a complex linear involution σ∘τ.
The classification therefore reduces to the classification of commuting pairs of antilinear involutions of a complex Lie algebra. The composite σ∘τ determines a complex symmetric space, while τ determines a real form. From this it is easy to construct tables of symmetric spaces for any given
, and furthermore, there is an obvious duality given by exchanging σ and τ. This extends the compact/non-compact duality from the Riemannian case, where either σ or τ is a Cartan involution, i.e., its fixed point set is a maximal compact subalgebra.
For exceptional simple Lie groups, the Riemannian case is included explicitly below, by allowing σ to be the identity involution (indicated by a dash). In the above tables this is implicitly covered by the case kl=0.
extended Cartan's definition of symmetric space to that of weakly symmetric Riemannian space, or in current terminology weakly symmetric space. These are defined as Riemannian manifolds M with a transitive connected Lie group of isometries G and an isometry σ normalising G such that given x, y in M there is an isometry s in G such that sx = σy and sy = σx. (Selberg's assumption that s2 should be an element of G was later shown to be unnecessary by Ernest Vinberg.) Selberg proved that weakly symmetric spaces give rise to Gelfand pair
s, so that in particular the unitary representation
of G on L2(M) is multiplicity free.
Selberg's definition can also be phrased equivalently in terms of a generalization of geodesic symmetry. It is required that for every point x in M and tangent vector X at x, there is an isometry s of M, depending on x and X, such that
When s is independent of X, M is a symmetric space.
An account of weakly symmetric spaces and their classification by Akhiezer and Vinberg, based on the classification of periodic automorphisms of complex semisimple Lie algebras, is given in .
. Some examples are complex vector spaces and complex projective spaces, both with their usual Riemannian metric, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric.
An irreducible symmetric space G/K is Hermitian if and only if K contains a central circle. A quarter turn by this circle acts as multiplication by i on the tangent space at the identity coset. Thus the Hermitian symmetric spaces are easily read off of the classification. In both the compact and the non-compact cases it turns out that there are four infinite series, namely AIII, BDI with p=2, DIII and CI, and two exceptional spaces, namely EIII and EVII. The non-compact Hermitian symmetric spaces can be realized as bounded symmetric domains in complex vector spaces.
An irreducible symmetric space G/K is quaternion-Kähler if and only if isotropy representation of K contains an Sp(1) summand acting like the unit quaternions on a quaternionic vector space
. Thus the quaternion-Kähler symmetric spaces are easily read off from the classification. In both the compact and the non-compact cases it turns out that there is exactly one for each complex simple Lie group, namely AI with p=2 or q=2 (these are isomorphic), BDI with p=4 or q=4, CII with p=1 or q=1, EII, EVI, EIX, FI and G.
, the loop spaces of the stable orthogonal group
can be interpreted as reductive symmetric spaces.
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
and harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
, a symmetric space is a smooth manifold whose group of symmetries contains an inversion symmetry about every point. There are two ways to formulate the inversion symmetry, via Riemannian geometry or via Lie theory. The Lie theoretic definition is more general and more algebraic.
In Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
, the inversions are geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
symmetries, and these are required to be isometries
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
, leading to the notion of a Riemannian symmetric space. More generally, in Lie theory
Lie theory
Lie theory is an area of mathematics, developed initially by Sophus Lie.Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896....
a symmetric space is a homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
G/H for a Lie group G such that the stabilizer H of a point is an open subgroup of the fixed point set of an involution of G. This definition includes (globally) Riemannian symmetric spaces and pseudo-Riemannian
Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
symmetric spaces as special cases.
Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. They were first studied extensively and classified by Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
. More generally, classifications of irreducible and semisimple symmetric spaces have been given by Marcel Berger
Marcel Berger
Marcel Berger is a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques , France...
. They are important in representation theory and harmonic analysis as well as differential geometry.
Definition using geodesic symmetries
Let M be a connected Riemannian manifold and p a point of M. A map f defined on a neighborhood of p is said to be a geodesic symmetry, if it fixes the point p and reverses geodesics through that point. It follows that the derivative of the map at p is minus the identity map on the tangent spaceTangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
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, and (globally) Riemannian symmetric if in addition its geodesic symmetries are defined on all of M.
Basic properties
The Cartan–Ambrose–Hicks theorem implies that M is locally Riemannian symmetric if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
its curvature tensor is covariantly constant
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
, and furthermore that any simply connected, complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
locally Riemannian symmetric space is actually Riemannian symmetric.
Any Riemannian symmetric space M is complete and Riemannian homogeneous
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
(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 spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
s, projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
s, and hyperbolic space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
s, each with their standard Riemannian metrics. More examples are provided by compact, semi-simple Lie groups equipped with a bi-invariant Riemannian metric.
Any 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.
General definition
Let G be a connected Lie groupLie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
. 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 set of an involution σ of G. Thus σ is an automorphism of G with σ2 = idG and H is an open subgroup of the set
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 decompositionwith
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 are symmetric spaces
If M is a Riemannian symmetric space, the identity component G of the isometry group of M is a Lie groupLie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
acting transitively on M (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
Jet bundle
In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form...
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
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
such that the isotropy group K is contained between the fixed point group of σ and its identity component (hence an open subgroup).
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 p = hK (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 CartanÉlie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
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 possibly 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:
1. Euclidean type: M has vanishing curvature, and is therefore isometric to a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
.
2. Compact type: M has nonnegative (but not identically zero) sectional curvature
Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...
.
3. 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 group
Simple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
s. 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
, , 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 which 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.| Label | G | K | Dimension | Rank | Geometric interpretation |
|---|---|---|---|---|---|
| AI |  |
 |
 |
n − 1 | Space of real structures on  which leave the complex determinant invariant |
| AII |  |
 |
 |
n − 1 | Space of quaternionic structures on  compatible with the Hermitian metric |
| AIII |  |
 |
 |
min(p,q) | Grassmannian Grassmannian In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological... of complex p-dimensional subspaces of  |
| BDI |  |
 |
 |
min(p,q) | Grassmannian Grassmannian In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological... of oriented real p-dimensional subspaces of  |
| DIII |  |
 |
 |
[n/2] | Space of orthogonal complex structures on  |
| CI |  |
 |
 |
n | Space of complex structures on  compatible with the inner product |
| CII |  |
 |
 |
min(p,q) | Grassmannian Grassmannian In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological... of quaternionic p-dimensional subspaces of  |
| EI |  |
 |
42 | 6 | |
| EII |  |
 |
40 | 4 | Space of symmetric subspaces of  isometric to  |
| EIII |  |
 |
32 | 2 | Complexified Cayley projective plane  |
| EIV |  |
 |
26 | 2 | Space of symmetric subspaces of  isometric to  |
| EV |  |
 |
70 | 7 | |
| EVI |  |
 |
64 | 4 | Rosenfeld's projective plane  over  |
| EVII |  |
 |
54 | 3 | Space of symmetric subspaces of  isomorphic to  |
| EVIII |  |
 |
128 | 8 | Sometimes denoted  |
| EIX |  |
 |
112 | 4 | Space of symmetric subspaces of  isomorphic to  |
| FI |  |
 |
28 | 4 | Space of symmetric subspaces of  isomorphic to  |
| FII |  |
 |
16 | 1 | Cayley projective plane  |
| G |  |
 |
8 | 2 | Space of subalgebras of the octonion algebra Octonion In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...  which are isomorphic to the quaternion algebraQuaternion In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...  |
Symmetric spaces in general
An important class of symmetric spaces generalizing the Riemannian symmetric spaces are pseudo-Riemannian symmetric spaces, in which the Riemannian metric is replaced by a pseudo-Riemannian metric (nondegenerate instead of positive definite on each tangent space). In particular, Lorentzian symmetric spaces, i.e., n dimensional pseudo-Riemannian symmetric spaces of signature (n − 1,1), are important in general relativityGeneral relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, the most notable examples being Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
, De Sitter space
De Sitter space
In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space , denoted dS_n, is the Lorentzian manifold analog of an n-sphere ; it is maximally symmetric, has constant positive curvature,...
and anti de Sitter space
Anti de Sitter space
In mathematics and physics, n-dimensional anti de Sitter space, sometimes written AdS_n, is a maximally symmetric Lorentzian manifold with constant negative scalar curvature...
(with zero, positive and negative curvature respectively). De Sitter space of dimension n may be identified with the 1-sheeted hyperboloid in a Minkowski space of dimension n + 1.
Symmetric and locally symmetric spaces in general can be regarded as affine symmetric spaces. If M = G/H is a symmetric space, then Nomizu showed that there is a G-invariant torsion-free
Torsion-free
In mathematics, the term torsion-free may refer to several unrelated notions:* In abstract algebra, a group is torsion-free if the only element of finite order is the identity....
affine connection
Affine connection
In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
on M whose curvature is parallel
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
. Conversely a manifold with such a connection is locally symmetric (i.e., its universal cover is a symmetric space). Such manifolds can also be described as those affine manifolds whose geodesic symmetries are all globally defined affine diffeomorphisms, generalizing the Riemannian and pseudo-Riemannian case.
Classification results
The classification of Riemannian symmetric spaces does not extend readily to the general case for the simple reason that there is no general splitting of a symmetric space into a product of irreducibles. Here a symmetric space G/H with Lie algebrais said to be irreducible if
is an irreducible representation of . Since is not semisimple (or even reductive) in general, it can have indecomposableIndecomposable module
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.Indecomposable is a weaker notion than simple module:simple means "no proper submodule" N...
representations which are not irreducible.
However, the irreducible symmetric spaces can be classified. As shown by K. Nomizu, there is a dichotomy: an irreducible symmetric space G/H is either flat (i.e., an affine space) or
is semisimple. This is the analogue of the Riemannian dichotomy between Euclidean spaces and those of compact or noncompact type, and it motivated M. Berger to classify semisimple symmetric spaces (i.e., those with semisimple) and determine which of these are irreducible. The latter question is more subtle than in the Riemannian case: even if is simple, G/H might not be irreducible.As in the Riemannian case there are semisimple symmetric spaces with G = H × H. Any semisimple symmetric space is a product of symmetric spaces of this form with symmetric spaces such that
is simple. It remains to describe the latter case. For this, one needs to classify involutions σ of a (real) simple Lie algebra . If is not simple, then is a complex simple Lie algebra, and the corresponding symmetric spaces have the form G/H, where H is a real form of G: these are the analogues of the Riemannian symmetric spaces G/K with G a complex simple Lie group, and K a maximal compact subgroup.Thus we may assume
is simple. The real subalgebra may be viewed as the fixed point set of a complex antilinear involution τ of , while σ extends to a complex antilinear involution of commuting with τ and hence also a complex linear involution σ∘τ.The classification therefore reduces to the classification of commuting pairs of antilinear involutions of a complex Lie algebra. The composite σ∘τ determines a complex symmetric space, while τ determines a real form. From this it is easy to construct tables of symmetric spaces for any given
, and furthermore, there is an obvious duality given by exchanging σ and τ. This extends the compact/non-compact duality from the Riemannian case, where either σ or τ is a Cartan involution, i.e., its fixed point set is a maximal compact subalgebra.Tables
The following table indexes the real symmetric spaces by complex symmetric spaces and real forms, for each classical and exceptional complex simple Lie group.| Gc=SL(n,C) | Gc/SO(n,C) | Gc/S(GL(k,C)×GL(l,C)), k+l=n | Gc/Sp(n,C), n even |
|---|---|---|---|
| G=SL(n,R) | G/SO(k,l) | G/S(GL(k,R)×GL(l,R)) or G/GL(n/2,C), n even |
G/Sp(n,R), n even |
| G=SU(p,q), p+q=n | G/SO(p,q) or SU(p,p)/Sk(p,H) |
G/S(U(kp,kq)×U(lp,lq)) or SU(p,p)/GL(p,C) |
G/Sp(p/2,q/2), p,q even or SU(p,p)/Sp(2p,R) |
| G=SL(n/2,H), n even | G/Sk(n/2,H) | G/S(GL(k/2,H)×GL(l/2,H)), k,l even or G/GL(n/2,C) |
G/Sp(k/2,l/2), k,l even, k+l=n |
| Gc=SO(n,C) | Gc/SO(k,C)×SO(l,C), k+l=n | Gc/GL(n/2,C), n even |
|---|---|---|
| G=SO(p,q) | G/SO(kp,kq)×SO(lp,lq) or SO(n,n)/SO(n,C) |
G/U(p/2,q/2), p,q even or SO(n,n)/GL(n,R) |
| G = Sk(n/2,H), n even | G/Sk(k/2,l/2), k,l even or G/SO(n/2,C) |
G/U(k/2,l/2), k,l even or G/SL(n/4,H) |
| Gc = Sp(2n,C) | Gc/Sp(2k,C)×Sp(2l,C), k + l = n | Gc/GL(n,C) |
|---|---|---|
| G = Sp(p,q), p + q = n | G/Sp(kp,kq)×Sp(lp,lq) or Sp(n,n)/Sp(n,C) |
G/U(p,q) or Sp(p,p)/GL(p,H) |
| G = Sp(2n,R) | G/Sp(2k,R)×Sp(2l,R) or G/Sp(n,C) |
G/U(k,l), k + l = n or G/GL(n,R) |
For exceptional simple Lie groups, the Riemannian case is included explicitly below, by allowing σ to be the identity involution (indicated by a dash). In the above tables this is implicitly covered by the case kl=0.
| G2c | – | G2c/SL(2,C)× SL(2,C) |
|---|---|---|
| G2 | – | G2/SU(2)×SU(2) |
| G2(2) | G2(2)/SU(2)×SU(2) | G2(2)/SL(2,R)× SL(2,R) |
| F4c | – | F4c/Sp(6,C)×Sp(2,C) | F4c/SO(9,C) |
|---|---|---|---|
| F4 | – | F4/Sp(3)×Sp(1) | F4/SO(9) |
| F4(4) | F4(4)/Sp(3)×Sp(1) | F4(4)/Sp(6,R)×Sp(2,R) or F4(4)/Sp(2,1)×Sp(1) |
F4(4)/SO(5,4) |
| F4(-20) | F4(-20)/SO(9) | F4(-20)/Sp(2,1)×Sp(1) | F4(-20)/SO(8,1) |
| E6c | – | E6c/Sp(8,C) | E6c/SL(6,C)×SL(2,C) | E6c/SO(10,C)×SO(2,C) | E6c/F4c |
|---|---|---|---|---|---|
| E6 | – | E6/Sp(4) | E6/SU(6)×SU(2) | E6/SO(10)×SO(2) | E6/F4 |
| E6(6) | E6(6)/Sp(4) | E6(6)/Sp(2,2) or E6(6)/Sp(8,R) |
E6(6)/SL(6,R)×SL(2,R) or E6(6)/SL(3,H)×SU(2) |
E6(6)/SO(5,5)×SO(1,1) | E6(6)/F4(4) |
| E6(2) | E6(2)/SU(6)×SU(2) | E6(2)/Sp(3,1) or E6(2)/Sp(8,R) |
E6(2)/SU(4,2)×SU(2) or E6(2)/SU(3,3)×SL(2,R) |
E6(2)/SO(6,4)×SO(2) or E6(2)/Sk(5,H)×SO(2) |
E6(2)/F4(4) |
| E6(-14) | E6(-14)/SO(10)×SO(2) | E6(-14)/Sp(2,2) | E6(-14)/SU(4,2)×SU(2) or E6(-14)/SU(5,1)×SL(2,R) |
E6(-14)/SO(8,2)×SO(2) or Sk(5,H)×SO(2) |
E6(-14)/F4(-20) |
| E6(-26) | E6(-26)/F4 | E6(-26)/Sp(3,1) | E6(-26)/SL(3,H)×Sp(1) | E6(-26)/SO(9,1)×SO(1,1) | E6(-26)/F4(−20) |
| E7c | – | E7c/SL(8,C) | E7c/SO(12,C)×Sp(2,C) | E7c/E6c×SO(2,C) |
|---|---|---|---|---|
| E7 | – | E7/SU(8) | E7/SO(12)× Sp(1) | E7/E6× SO(2) |
| E7(7) | E7(7)/SU(8) | E7(7)/SU(4,4) or E7(7)/SL(8,R) or E7(7)/SL(4,H) |
E7(7)/SO(6,6)×SL(2,R) or E7(7)/Sk(6,H)×Sp(1) |
E7(7)/E6(6)×SO(1,1) or E7(7)/E6(2)×SO(2) |
| E7(-5) | E7(-5)/SO(12)× Sp(1) | E7(-5)/SU(4,4) or E7(-5)/SU(6,2) |
E7(-5)/SO(8,4)×SU(2) or E7(-5)/Sk(6,H)×SL(2,R) |
E7(-5)/E6(2)×SO(2) or E7(-5)/E6(-14)×SO(2) |
| E7(-25) | E7(-25)/E6× SO(2) | E7(-25)/SL(4,H) or E7(-25)/SU(6,2) |
E7(-25)/SO(10,2)×SL(2,R) or E7(-25)/Sk(6,H)×Sp(1) |
E7(-25)/E6(-14)×SO(2) or E7(-25)/E6(-26)×SO(1,1) |
| E8c | – | E8c/SO(16,C) | E8c/E7c×Sp(2,C) |
|---|---|---|---|
| E8 | – | E8/SO(16) | E8/E7×Sp(1) |
| E8(8) | E8(8)/SO(16) | E8(8)/SO(8,8) or E8(8)/Sk(8,H) | E8(8)/E7(7)×SL(2,R) or E8(8)/E7(-5)×SU(2) |
| E8(-24) | E8(-24)/E7×Sp(1) | E8(-24)/SO(12,4) or E8(-24)/Sk(8,H) | E8(-24)/E7(-5)×SU(2) or E8(-24)/E7(-25)×SL(2,R) |
Weakly symmetric Riemannian spaces
In the 1950s Atle SelbergAtle Selberg
Atle Selberg was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory...
extended Cartan's definition of symmetric space to that of weakly symmetric Riemannian space, or in current terminology weakly symmetric space. These are defined as Riemannian manifolds M with a transitive connected Lie group of isometries G and an isometry σ normalising G such that given x, y in M there is an isometry s in G such that sx = σy and sy = σx. (Selberg's assumption that s2 should be an element of G was later shown to be unnecessary by Ernest Vinberg.) Selberg proved that weakly symmetric spaces give rise to Gelfand pair
Gelfand pair
In mathematics, the expression Gelfand pair is a pair consisting of a group G and a subgroup K that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to...
s, so that in particular the unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
of G on L2(M) is multiplicity free.
Selberg's definition can also be phrased equivalently in terms of a generalization of geodesic symmetry. It is required that for every point x in M and tangent vector X at x, there is an isometry s of M, depending on x and X, such that
- s fixes x;
- the derivative of s at x sends X to –X.
When s is independent of X, M is a symmetric space.
An account of weakly symmetric spaces and their classification by Akhiezer and Vinberg, based on the classification of periodic automorphisms of complex semisimple Lie algebras, is given in .
Symmetric spaces and holonomy
If the identity component of the holonomy group of a Riemannian manifold at a point acts irreducibly on the tangent space, then either the manifold is a locally Riemannian symmetric space, or it is in one of 7 families.Hermitian symmetric spaces
A Riemannian symmetric space which is additionally equipped with a parallel complex structure compatible with the Riemannian metric is called a Hermitian symmetric spaceHermitian symmetric space
In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian...
. Some examples are complex vector spaces and complex projective spaces, both with their usual Riemannian metric, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric.
An irreducible symmetric space G/K is Hermitian if and only if K contains a central circle. A quarter turn by this circle acts as multiplication by i on the tangent space at the identity coset. Thus the Hermitian symmetric spaces are easily read off of the classification. In both the compact and the non-compact cases it turns out that there are four infinite series, namely AIII, BDI with p=2, DIII and CI, and two exceptional spaces, namely EIII and EVII. The non-compact Hermitian symmetric spaces can be realized as bounded symmetric domains in complex vector spaces.
Quaternion-Kähler symmetric spaces
A Riemannian symmetric space which is additionally equipped with a parallel subbundle of End(TM) isomorphic to the imaginary quaternions at each point, and compatible with the Riemannian metric, is called Quaternion-Kähler symmetric space.An irreducible symmetric space G/K is quaternion-Kähler if and only if isotropy representation of K contains an Sp(1) summand acting like the unit quaternions on a quaternionic vector space
Quaternionic vector space
In mathematics, a left quaternionic vector space is a left H-module where H denotes the noncommutative ring of the quaternions....
. Thus the quaternion-Kähler symmetric spaces are easily read off from the classification. In both the compact and the non-compact cases it turns out that there is exactly one for each complex simple Lie group, namely AI with p=2 or q=2 (these are isomorphic), BDI with p=4 or q=4, CII with p=1 or q=1, EII, EVI, EIX, FI and G.
Bott periodicity theorem
In the Bott periodicity theoremBott periodicity theorem
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy...
, the loop spaces of the stable orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
can be interpreted as reductive symmetric spaces.