A (pseudo-)Riemannian manifold whose geodesics are reversible.
For other uses, see Symmetric space (disambiguation).
Lie groups and Lie algebras
Classical groups
General linear GL(n)
Special linear SL(n)
Orthogonal O(n)
Special orthogonal SO(n)
Unitary U(n)
Special unitary SU(n)
Symplectic Sp(n)
Simple Lie groups
Classical
An
Bn
Cn
Dn
Exceptional
G2
F4
E6
E7
E8
Other Lie groups
Circle
Lorentz
Poincaré
Conformal group
Diffeomorphism
Loop
Euclidean
Lie algebras
Lie group–Lie algebra correspondence
Exponential map
Adjoint representation
Killing form
Index
Simple Lie algebra
Loop algebra
Affine Lie algebra
Semisimple Lie algebra
Dynkin diagrams
Cartan subalgebra
Root system
Weyl group
Real form
Complexification
Split Lie algebra
Compact Lie algebra
Representation theory
Lie group representation
Lie algebra representation
Representation theory of semisimple Lie algebras
Representations of classical Lie groups
Theorem of the highest weight
Borel–Weil–Bott theorem
Lie groups in physics
Particle physics and representation theory
Lorentz group representations
Poincaré group representations
Galilean group representations
Scientists
Sophus Lie
Henri Poincaré
Wilhelm Killing
Élie Cartan
Hermann Weyl
Claude Chevalley
Harish-Chandra
Armand Borel
Glossary
Table of Lie groups
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In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.
In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (M, g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about the endpoints). Both descriptions can also naturally be extended to the setting of pseudo-Riemannian manifolds.
From the point of view of Lie theory, a symmetric space is the quotient G/H of a connected Lie group G by a Lie subgroup H which is (a connected component of) the invariant group of an involution of G. This definition includes more than the Riemannian definition, and reduces to it when H is compact.
Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. Their central role in the theory of holonomy was discovered by Marcel Berger. They are important objects of study in representation theory and harmonic analysis as well as in differential geometry.
curvature −1) is a locally symmetricspace but not a symmetricspace. Every lens space is locally symmetric but not symmetric, with the exception of L (...
notion of Riemannian symmetricspace from real manifolds to complex manifolds. Every Hermitian symmetricspace is a homogeneous space for its isometry group...
product of smaller symmetricspaces). The irreducible simply connected symmetricspaces are the real line, and exactly two symmetricspaces corresponding to...
bilinear form to be symmetric if B(v, w) = B(w, v) for all v, w in V; alternating if B(v, v) = 0 for all v in V; skew-symmetric or antisymmetric if B(v...
space is also called an accessible space or a space with Fréchet topology and an R0 space is also called a symmetricspace. (The term Fréchet space also...
{\displaystyle (0,\ldots ,0)} . Symmetricspace: Hyperbolic n {\displaystyle n} -space can be realised as the symmetricspace of the simple Lie group S O...
mathematical physics, n-dimensional de Sitter space (often abbreviated to dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar...
the metric): in particular, it is a CAT(-1/4) space. Complex hyperbolic spaces are also the symmetricspaces associated with the Lie groups P U ( n , 1 )...
although it is most commonly taken to be either a Lie group or a symmetricspace. The model may or may not be quantized. An example of the non-quantized...
manifold and the corresponding symmetricspaces are called symmetric R-spaces. A flag in a finite dimensional vector space V over a field F is an increasing...
a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{ is symmetric}}\iff...
number is generic for maximally symmetricspaces. Maximally symmetricspaces can be considered as sub-manifolds of flat space, arising as surfaces of constant...
Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. The symmetric difference is equivalent...
Complex projective space carries a (Kähler) metric, called the Fubini–Study metric, in terms of which it is a Hermitian symmetricspace of rank 1. Complex...
was introduced by Élie Cartan (1926) in order to study and classify symmetricspaces. It was not until much later that holonomy groups would be used to...
just symmetric forms when "bilinear" is understood. Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond to symmetric matrices...
programme symmetricspacespace form Maurer–Cartan form Examples hyperbolic space Gauss–Bolyai–Lobachevsky space Grassmannian Complex projective space Real...
characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V. A related concept is that...
algebra is symmetric in A and B, despite the original construction not being symmetric, though Vinberg's symmetric method gives a symmetric construction...
{\displaystyle f.} The most commonly encountered symmetric functions are polynomial functions, which are given by the symmetric polynomials. A related notion is alternating...
{\displaystyle K} and X = G / K {\displaystyle X=G/K} is the associated symmetricspace the conjugacy classes in Γ {\displaystyle \Gamma } can be described...
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