Symmetric space

In mathematics symmetric spaces are a class of Riemannian manifolds with a particularly high degree of symmetries.

They are an important class of examples in geometry and topology and are used for example in representation theory, harmonic analysis, number theory, modular forms and physics.

2.1 Example

3.1 Definitions
3.2 Examples
3.3 Product decomposition
3.4 sectional curvature
3.5 duality

4.1 Symmetric spaces of rank 1

Definition

A connected Riemannian manifold is a symmetric space if, for every a reflection, ie an isometry with and.

Examples

The Euclidean space is a symmetric to each one defines the mirroring

The unit sphere is a symmetric space. At is the uniquely determined point on the great circle through and, for the well (if and no antipodal points ) applies.

Geodesic symmetry

Let be a geodesic with. It follows for all.

Conversely, one can define a geodesic mirroring in any Riemannian manifold locally ( in a sufficiently small neighborhood of a point ). A Riemannian manifold is called locally symmetric if is on its domain of definition is an isometry. It is a symmetric space, if an isometry is and can be defined completely.

Homogeneous space

Every symmetric space is a homogeneous space, ie of the mold for a continuous Lie group and a compact subassembly, so that the Riemannian metric of the effect of left- invariant. Cartan symmetric spaces characterized as follows: Let a connected Lie group, a compact subgroup and a Liegruppenhomomorphismus with as well. ( Here, the fixed points of and the connected component of the neutral element is called. Cartan involution. ) Then transmits a -invariant Riemannian metric and is a symmetric space.

Cartan decomposition

Be a symmetric space and the Cartan involution. Be the Lie algebras of.

Be. Because the only eigenvalues , is the eigenspace corresponding to the eigenvalue. We denote the eigenspace corresponding to the eigenvalue, it corresponds to the tangent space in. Then and

The defined using the Killing form form

Is positive semidefinite.

Conversely, there is a decomposition with these properties always an involution, which is on and on. Be the simply connected Lie group with Lie algebra, then there is an involution with too and thus a symmetric space.

Example

With is a Cartan decomposition.

Types of symmetric spaces

Definitions

A symmetric space is of compact type if the Killing form is semi-definite negative.

A symmetric space is of Euclidean type if is abelian.

A symmetric space is of type nichtkompaktem if the Killing form non- degenerate, but not negative semi-definite and a Cartan decomposition is. (In this case is semisimple and maximal compact subgroup. )

Examples

The sphere is a symmetric space of compact type.
The Euclidean space is a symmetric space of Euclidean type, as is the n-dimensional torus.
The hyperbolic space is a symmetric space of nichtkompaktem type. and are symmetric spaces of nichtkompaktem type.

Product decomposition

A symmetric space is called irreducible if it can not be decomposed as a product of two non-trivial symmetric spaces. Every symmetric space can be decomposed as a product of irreducible symmetric spaces of compact, Euclidean and nichtkompaktem type.

Sectional curvature

Symmetric spaces of compact type have sectional curvature, symmetric spaces of Euclidean type have sectional curvature, symmetric spaces of type nichtkompaktem have sectional curvature.

Symmetric spaces of type nichtkompaktem are CAT (0 )-spaces and contractible.