Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.
In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport.
More generally, a Riemannian manifold (M, g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space
as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about the endpoints).
Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics.
They are important objects of study in representation theory and harmonic analysis as well as in differential geometry.
It follows that the derivative of the map f at p is minus the identity map on the tangent space of p. On a general Riemannian manifold, f need not be isometric, nor can it be extended, in general, from a neighbourhood of p to all of M. M is said to be locally Riemannian symmetric if its geodesic symmetries are in fact isometric.
More examples are provided by compact, semi-simple Lie groups equipped with a bi-invariant Riemannian metric.
, this gives a direct sum decomposition with The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer
with a direct sum decomposition satisfying these three conditions, the linear map σ, equal to the identity on
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.
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 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.
The next step is to show that any irreducible, simply connected Riemannian symmetric space M is of one of the following three types: 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.
Therefore, it remains to classify the irreducible, simply connected Riemannian symmetric spaces of compact and non-compact type.
For non-compact type, G is a simply connected complex simple Lie group and K is its maximal compact subgroup.
The examples of class A are completely described by the classification of noncompact simply connected real simple Lie groups.
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.
A more modern classification (Huang & Leung 2010) uniformly classifies the Riemannian symmetric spaces, both compact and non-compact, via a Freudenthal magic square construction.
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.
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).
Selberg proved that weakly symmetric spaces give rise to Gelfand pairs, so that in particular the unitary representation of G on L2(M) is multiplicity free.
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.
The metric tensor on the Riemannian manifold M can be lifted to a scalar product on G by combining it with the Killing form.
semisimple, so that the Killing form is non-degenerate, the metric likewise factorizes: 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.)
A quarter turn by this circle acts as multiplication by i on the tangent space at the identity coset.
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.
A Riemannian symmetric space that 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.
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.