# Symmetric Space

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.

### Other articles related to "symmetric space, space, spaces, symmetric, symmetric spaces":

Weakly Symmetric Space
... In mathematics, a weakly symmetric space is a notion introduced by the Norwegian mathematician Atle Selberg in the 1950s as a generalisation of symmetric space, due to Élie Cartan ... Geometrically the spaces are defined as complete Riemannian manifolds such that any two points can be exchanged by an isometry, the symmetric case being when the isometry is required to have ... The classification of weakly symmetric spaces relies on that of periodic automorphisms of complex semisimple Lie algebras ...
Symmetric Space - Applications and Special Cases - Bott Periodicity Theorem
... In the Bott periodicity theorem, the loop spaces of the stable orthogonal group can be interpreted as reductive symmetric spaces ...
Atomic Orbital - Shapes of Orbitals
... to describe the angular forms of regions in space where the electrons occupying the orbital are likely to be found ... of finding the electron anywhere in space ... the orbital is vertical, counter rotating information is unknown, and the orbital is z-axis symmetric ...
List Of Simple Lie Groups - The List of Simple Lie Groups - G2 Complex
... Dimension of symmetric space 14 Compact symmetric space Compact group G2 Non-compact symmetric space ...

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