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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a symmetric space is a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
(or more generally, a
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of
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 that varies smoothly from point to point ...
, leading to consequences in the theory of
holonomy In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
; or algebraically through
Lie theory In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is ...
, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
,
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 studies modules over these abstract algebraic structures. In essen ...
and
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
. 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_pM as minus the identity (every symmetric space is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
, 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 manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
s. From the point of view of Lie theory, a symmetric space is the quotient ''G''/''H'' of a connected
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
''G'' by a Lie subgroup ''H'' which is (a connected component of) the invariant group of an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
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 \gamma(0)=p then 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 In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
its curvature tensor is covariantly constant, and furthermore that every
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
,
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
locally Riemannian symmetric space is actually Riemannian symmetric. Every Riemannian symmetric space ''M'' is complete and Riemannian
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
(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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
,
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
s,
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s, and
hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
s, each with their standard Riemannian metrics. More examples are provided by compact, semi-simple
Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
equipped with a bi-invariant Riemannian metric. Every compact
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
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 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 In mathematics and physics, ''n''-dimensional anti-de Sitter space (AdS''n'') is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872� ...
.


Algebraic definition

Let ''G'' be a connected
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
. 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 Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
''σ'' in Aut(''G''). Thus ''σ'' is an automorphism of ''G'' with ''σ''2 = id''G'' and ''H'' is an open subgroup of the invariant set : G^\sigma=\. Because ''H'' is open, it is a union of components of ''G''''σ'' (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 \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 \mathfrak h of ''H'' (since this is the Lie algebra of ''G''''σ''), and the −1 eigenspace will be denoted \mathfrak m. Since ''σ'' is an automorphism of \mathfrak g, this gives a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
decomposition : \mathfrak g = \mathfrak h\oplus\mathfrak m with : mathfrak h,\mathfrak hsubset \mathfrak h,\; mathfrak h,\mathfrak msubset \mathfrak m,\; mathfrak m,\mathfrak msubset \mathfrak h. The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer \mathfrak h is a Lie subalgebra of \mathfrak g. The second condition means that \mathfrak m is an \mathfrak h-invariant complement to \mathfrak h in \mathfrak g. Thus any symmetric space is a
reductive homogeneous space In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space ''X'' together with a transitive action on ''X'' by a Lie group ''G'', which acts ...
, but there are many reductive homogeneous spaces which are not symmetric spaces. The key feature of symmetric spaces is the third condition that \mathfrak m brackets into \mathfrak h. Conversely, given any Lie algebra \mathfrak g with a direct sum decomposition satisfying these three conditions, the linear map ''σ'', equal to the identity on \mathfrak h and minus the identity on \mathfrak m, 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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
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 T''p''''M''. 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 T''p''''M'', hence compact. Moreover, if we denote by ''s''''p'': M → M the geodesic symmetry of ''M'' at ''p'', the map :\sigma: G \to G, h \mapsto s_p \circ h \circ s_p is an involutive Lie group
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 automorphis ...
such that the isotropy group ''K'' is contained between the fixed point group G^\sigma and its identity component (hence an open subgroup) (G^\sigma)_o\,, 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 ''p'' = ''hK'' (a coset of ''K'', where ''h'' ∈ ''G'') and define :s_p: M \to M,\quad h'K \mapsto h \sigma(h^h')K where ''σ'' is the involution of ''G'' fixing ''K''. Then one can check that ''s''''p'' is an isometry with (clearly) ''s''''p''(''p'') = ''p'' and (by differentiating) d''s''''p'' equal to minus the identity on T''p''''M''. Thus ''s''''p'' 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 Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
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 A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
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 In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, hom ...
, 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 is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. 2. Compact type: ''M'' has nonnegative (but not identically zero)
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a 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 In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
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 \mathrm(n), \mathrm(n), \mathrm(n) and the five exceptional Lie groups ''E''6, ''E''7, ''E''8, ''F''4, ''G''2. 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.


As Grassmannians

A more modern classification uniformly classifies the Riemannian symmetric spaces, both compact and non-compact, via a
Freudenthal magic square In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea i ...
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 A double is a look-alike or doppelgänger; one person or being that resembles another. Double, The Double or Dubble may also refer to: Film and television * Double (filmmaking), someone who substitutes for the credited actor of a character * ' ...
of subspaces of (\mathbf A \otimes \mathbf B)^n, for normed division algebras A and B. A similar construction produces the irreducible non-compact Riemannian symmetric spaces.


General symmetric spaces

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 In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
(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 relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the most notable examples being
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
,
De Sitter space In mathematical physics, ''n''-dimensional de Sitter space (often abbreviated to dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canoni ...
and
anti-de Sitter space In mathematics and physics, ''n''-dimensional anti-de Sitter space (AdS''n'') is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872� ...
(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 In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
(i.e. an affine connection whose
torsion tensor In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a cur ...
vanishes) on ''M'' whose
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
is
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
. Conversely a manifold with such a connection is locally symmetric (i.e., its
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
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 algebra :\mathfrak g = \mathfrak h\oplus \mathfrak m is said to be irreducible if \mathfrak m is an
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _ ...
of \mathfrak h. Since \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 \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 \mathfrak g semisimple) and determine which of these are irreducible. The latter question is more subtle than in the Riemannian case: even if \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 \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 \mathfrak g. If \mathfrak g^c is not simple, then \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 \mathfrak g^c is simple. The real subalgebra \mathfrak g may be viewed as the fixed point set of a complex
antilinear In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y ...
involution ''τ'' of \mathfrak g^c, while ''σ'' extends to a complex antilinear involution of \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 \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.


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. 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.


Weakly symmetric Riemannian spaces

In the 1950s
Atle Selberg Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded ...
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'' = σ''y'' and ''sy'' = σ''x''. (Selberg's assumption that σ2 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 In mathematics, a Gelfand pair is a pair ''(G,K)'' consisting of a group ''G'' and a subgroup ''K'' (called an Euler subgroup of ''G'') that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related ...
s, so that in particular the
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(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
of ''G'' on ''L''2(''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 algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
s, is given in .


Properties

Some properties and forms of symmetric spaces can be noted.


Lifting the metric tensor

The
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
on the Riemannian manifold M can be lifted to a scalar product on G by combining it with the
Killing form In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) sho ...
. This is done by defining :\langle X,Y\rangle_\mathfrak=\begin \langle X,Y\rangle_p \quad & X,Y\in T_pM\cong \mathfrak \\ -B(X,Y) \quad & X,Y\in \mathfrak \\ 0 & \mbox \end Here, \langle\cdot,\cdot\rangle_p is the Riemannian metric defined on T_pM, and B(X,Y)=\operatorname ( \operatorname X \circ \operatorname Y) is the
Killing form In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) sho ...
. The minus sign appears because the Killing form is negative-definite on \mathfrak~; this makes \langle \cdot,\cdot\rangle_\mathfrak positive-definite.


Factorization

The tangent space \mathfrak can be further factored into eigenspaces classified by the Killing form.Jurgen Jost, (2002) "Riemannian Geometry and Geometric Analysis", Third edition, Springer ''(See section 5.3, page 256)'' This is accomplished by defining an adjoint map \mathfrak\to\mathfrak taking Y\mapsto Y^\# as :\langle X,Y^\# \rangle = B(X,Y) where \langle \cdot,\cdot \rangle is the Riemannian metric on \mathfrak and 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 Y_1,\ldots,Y_n of \mathfrak with :Y^\#_i=\lambda_iY_i These are orthogonal with respect to the metric, in that :\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 \mathfrak into eigenspaces :\mathfrak=\mathfrak_1\oplus\cdots\oplus\mathfrak_d with : mathfrak_i,\mathfrak_j0 for i\ne j. For the case of \mathfrak semisimple, so that the Killing form is non-degenerate, the metric likewise factorizes: :\langle\cdot,\cdot\rangle=\frac\left.B\_+\cdots +\frac\left.B\_ 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 In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
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.


Applications and special cases


Symmetric spaces and holonomy

If the identity component of the
holonomy group In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
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 space In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian ...
. 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(T''M'') isomorphic to the imaginary quaternions at each point, and compatible with the Riemannian metric, is called
quaternion-Kähler symmetric space In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature is ...
. 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 quaternion In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
s on a
quaternionic vector space In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H is the (non-commutative) division ring of quaternions. The space H''n'' of ''n''-tuples of quaternions is both a left and right H-module using the co ...
. 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 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 comp ...
, the
loop spaces In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topol ...
of the stable
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
can be interpreted as reductive symmetric spaces.


See also

*
Orthogonal symmetric Lie algebra In mathematics, an orthogonal symmetric Lie algebra is a pair (\mathfrak, s) consisting of a real Lie algebra \mathfrak and an automorphism s of \mathfrak of order 2 such that the eigenspace \mathfrak of ''s'' corresponding to 1 (i.e., the set \mat ...
*
Relative root system In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a s ...
*
Satake diagram In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a D ...
* Cartan involution


References

* * * * Contains a compact introduction and many tables. * * * * * The standard book on Riemannian symmetric spaces. * * * Chapter XI contains a good introduction to Riemannian symmetric spaces. * * * * * *{{citation, title=Harmonic Analysis on Commutative Spaces, first=Joseph A., last= Wolf, publisher=American Mathematical Society, year= 2007 , isbn=978-0-8218-4289-8 Differential geometry Riemannian geometry Lie groups Homogeneous spaces