Symmetric space

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.

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 ${\displaystyle L(2,1)}$ 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.

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.

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 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 ${\displaystyle \mathrm {SO} (n)}$, ${\displaystyle \mathrm {SU} (n)}$, ${\displaystyle \mathrm {Sp} (n)}$ 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 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.

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	${\displaystyle \mathrm {SU} (n)\,}$	${\displaystyle \mathrm {SO} (n)\,}$	${\displaystyle (n-1)(n+2)/2}$	${\displaystyle n-1}$	Space of real structures on ${\displaystyle \mathbb {C} ^{n}}$ which leave the complex determinant invariant
AII	${\displaystyle \mathrm {SU} (2n)\,}$	${\displaystyle \mathrm {Sp} (n)\,}$	${\displaystyle (n-1)(2n+1)}$	${\displaystyle n-1}$	Space of quaternionic structures on ${\displaystyle \mathbb {C} ^{2n}}$ compatible with the Hermitian metric
AIII	${\displaystyle \mathrm {SU} (p+q)\,}$	${\displaystyle \mathrm {S} (\mathrm {U} (p)\times \mathrm {U} (q))\,}$	${\displaystyle 2pq}$	${\displaystyle \min(p,q)}$	Grassmannian of complex p-dimensional subspaces of ${\displaystyle \mathbb {C} ^{p+q}}$
BDI	${\displaystyle \mathrm {SO} (p+q)\,}$	${\displaystyle \mathrm {SO} (p)\times \mathrm {SO} (q)\,}$	${\displaystyle pq}$	${\displaystyle \min(p,q)}$	Grassmannian of oriented real p-dimensional subspaces of ${\displaystyle \mathbb {R} ^{p+q}}$
DIII	${\displaystyle \mathrm {SO} (2n)\,}$	${\displaystyle \mathrm {U} (n)\,}$	${\displaystyle n(n-1)}$	${\displaystyle [n/2]}$	Space of orthogonal complex structures on ${\displaystyle \mathbb {R} ^{2n}}$
CI	${\displaystyle \mathrm {Sp} (n)\,}$	${\displaystyle \mathrm {U} (n)\,}$	${\displaystyle n(n+1)}$	${\displaystyle n}$	Space of complex structures on ${\displaystyle \mathbb {H} ^{n}}$ compatible with the inner product
CII	${\displaystyle \mathrm {Sp} (p+q)\,}$	${\displaystyle \mathrm {Sp} (p)\times \mathrm {Sp} (q)\,}$	${\displaystyle 4pq}$	${\displaystyle \min(p,q)}$	Grassmannian of quaternionic p-dimensional subspaces of ${\displaystyle \mathbb {H} ^{p+q}}$
EI	${\displaystyle E_{6}\,}$	${\displaystyle \mathrm {Sp} (4)/\{\pm I\}\,}$	42	6
EII	${\displaystyle E_{6}\,}$	${\displaystyle \mathrm {SU} (6)\cdot \mathrm {SU} (2)\,}$	40	4	Space of symmetric subspaces of ${\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}$ isometric to ${\displaystyle (\mathbb {C} \otimes \mathbb {H} )P^{2}}$
EIII	${\displaystyle E_{6}\,}$	${\displaystyle \mathrm {SO} (10)\cdot \mathrm {SO} (2)\,}$	32	2	Complexified Cayley projective plane ${\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}$
EIV	${\displaystyle E_{6}\,}$	${\displaystyle F_{4}\,}$	26	2	Space of symmetric subspaces of ${\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}$ isometric to ${\displaystyle \mathbb {OP} ^{2}}$
EV	${\displaystyle E_{7}\,}$	${\displaystyle \mathrm {SU} (8)/\{\pm I\}\,}$	70	7
EVI	${\displaystyle E_{7}\,}$	${\displaystyle \mathrm {SO} (12)\cdot \mathrm {SU} (2)\,}$	64	4	Rosenfeld projective plane ${\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}}$ over ${\displaystyle \mathbb {H} \otimes \mathbb {O} }$
EVII	${\displaystyle E_{7}\,}$	${\displaystyle E_{6}\cdot \mathrm {SO} (2)\,}$	54	3	Space of symmetric subspaces of ${\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}}$ isomorphic to ${\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}}$
EVIII	${\displaystyle E_{8}\,}$	${\displaystyle \mathrm {Spin} (16)/\{\pm \mathrm {vol} \}\,}$	128	8	Rosenfeld projective plane ${\displaystyle (\mathbb {O} \otimes \mathbb {O} )P^{2}}$
EIX	${\displaystyle E_{8}\,}$	${\displaystyle E_{7}\cdot \mathrm {SU} (2)\,}$	112	4	Space of symmetric subspaces of ${\displaystyle (\mathbb {O} \otimes \mathbb {O} )P^{2}}$ isomorphic to ${\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}}$
FI	${\displaystyle F_{4}\,}$	${\displaystyle \mathrm {Sp} (3)\cdot \mathrm {SU} (2)\,}$	28	4	Space of symmetric subspaces of ${\displaystyle \mathbb {O} P^{2}}$ isomorphic to ${\displaystyle \mathbb {H} P^{2}}$
FII	${\displaystyle F_{4}\,}$	${\displaystyle \mathrm {Spin} (9)\,}$	16	1	Cayley projective plane ${\displaystyle \mathbb {O} P^{2}}$
G	${\displaystyle G_{2}\,}$	${\displaystyle \mathrm {SO} (4)\,}$	8	2	Space of subalgebras of the octonion algebra ${\displaystyle \mathbb {O} }$ which are isomorphic to the quaternion algebra ${\displaystyle \mathbb {H} }$

A more modern classification (Huang & Leung 2010) uniformly classifies the Riemannian symmetric spaces, both compact and non-compact, via a Freudenthal magic square construction. The irreducible compact Riemannian symmetric spaces are, up to finite covers, either a compact simple Lie group, a Grassmannian, a Lagrangian Grassmannian, or a double Lagrangian Grassmannian of subspaces of ${\displaystyle (\mathbf {A} \otimes \mathbf {B} )^{n},}$ for normed division algebras A and B. A similar construction produces the irreducible non-compact Riemannian symmetric spaces.

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 algebra

${\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {m}}}$

is said to be irreducible if ${\displaystyle {\mathfrak {m}}}$ is an irreducible representation of ${\displaystyle {\mathfrak {h}}}$. Since ${\displaystyle {\mathfrak {h}}}$ 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 Katsumi Nomizu, there is a dichotomy: an irreducible symmetric space G/H is either flat (i.e., an affine space) or ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}}$ semisimple) and determine which of these are irreducible. The latter question is more subtle than in the Riemannian case: even if ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}}$ is simple. It remains to describe the latter case. For this, one needs to classify involutions σ of a (real) simple Lie algebra ${\displaystyle {\mathfrak {g}}}$. If ${\displaystyle {\mathfrak {g}}^{c}}$ is not simple, then ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}^{c}}$ is simple. The real subalgebra ${\displaystyle {\mathfrak {g}}}$ may be viewed as the fixed point set of a complex antilinear involution τ of ${\displaystyle {\mathfrak {g}}^{c}}$, while σ extends to a complex antilinear involution of ${\displaystyle {\mathfrak {g}}^{c}}$ 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 ${\displaystyle {\mathfrak {g}}^{c}}$, 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.

The following table indexes the real symmetric spaces by complex symmetric spaces and real forms, for each classical and exceptional complex simple Lie group.

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.

The metric tensor on the Riemannian manifold ${\displaystyle M}$ can be lifted to a scalar product on ${\displaystyle G}$ by combining it with the Killing form. This is done by defining

${\displaystyle \langle X,Y\rangle _{\mathfrak {g}}={\begin{cases}\langle X,Y\rangle _{p}\quad &X,Y\in T_{p}M\cong {\mathfrak {m}}\\-B(X,Y)\quad &X,Y\in {\mathfrak {h}}\\0&{\mbox{otherwise}}\end{cases}}}$

Here, ${\displaystyle \langle \cdot ,\cdot \rangle _{p}}$ is the Riemannian metric defined on ${\displaystyle T_{p}M}$, and ${\displaystyle B(X,Y)=\operatorname {trace} (\operatorname {ad} X\circ \operatorname {ad} Y)}$ is the Killing form. The minus sign appears because the Killing form is negative-definite on ${\displaystyle {\mathfrak {h}}~;}$ this makes ${\displaystyle \langle \cdot ,\cdot \rangle _{\mathfrak {g}}}$ positive-definite.

The tangent space ${\displaystyle {\mathfrak {m}}}$ can be further factored into eigenspaces classified by the Killing form.[1] This is accomplished by defining an adjoint map ${\displaystyle {\mathfrak {m}}\to {\mathfrak {m}}}$ taking ${\displaystyle Y\mapsto Y^{\#}}$ as

${\displaystyle \langle X,Y^{\#}\rangle =B(X,Y)}$

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ is the Riemannian metric on ${\displaystyle {\mathfrak {m}}}$ and ${\displaystyle B(\cdot ,\cdot )}$ is the Killing form. This map is sometimes called the generalized transpose, as corresponds to the transpose for the orthogonal groups and the Hermitian conjugate for the unitary groups. It is a linear functional, and it is self-adjoint, and so one concludes that there is an orthonormal basis ${\displaystyle Y_{1},\ldots ,Y_{n}}$ of ${\displaystyle {\mathfrak {m}}}$ with

${\displaystyle Y_{i}^{\#}=\lambda _{i}Y_{i}}$

These are orthogonal with respect to the metric, in that

${\displaystyle \langle Y_{i}^{\#},Y_{j}\rangle =\lambda _{i}\langle Y_{i},Y_{j}\rangle =B(Y_{i},Y_{j})=\langle Y_{j}^{\#},Y_{i}\rangle =\lambda _{j}\langle Y_{j},Y_{i}\rangle }$

since the Killing form is symmetric. This factorizes ${\displaystyle {\mathfrak {m}}}$ into eigenspaces

${\displaystyle {\mathfrak {m}}={\mathfrak {m}}_{1}\oplus \cdots \oplus {\mathfrak {m}}_{d}}$

with

${\displaystyle [{\mathfrak {m}}_{i},{\mathfrak {m}}_{j}]=0}$

for ${\displaystyle i\neq j}$. For the case of ${\displaystyle {\mathfrak {g}}}$ semisimple, so that the Killing form is non-degenerate, the metric likewise factorizes:

${\displaystyle \langle \cdot ,\cdot \rangle ={\frac {1}{\lambda _{1}}}\left.B\right|_{{\mathfrak {m}}_{1}}+\cdots +{\frac {1}{\lambda _{d}}}\left.B\right|_{{\mathfrak {m}}_{d}}}$

In certain practical applications, this factorization can be interpreted as the spectrum of operators, e.g. the spectrum of the hydrogen atom, with the eigenvalues of the Killing form corresponding to different values of the angular momentum of an orbital (i.e. the Killing form being a Casimir operator that can classify the different representations under which different orbitals transform.)

Classification of symmetric spaces proceeds based on whether or not the Killing form is positive/negative definite.

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.

A Riemannian symmetric space which is additionally equipped with a parallel complex structure compatible with the Riemannian metric is called a Hermitian symmetric space. 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.

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. 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.

In the Bott periodicity theorem, the loop spaces of the stable orthogonal group can be interpreted as reductive symmetric spaces.