In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a weakly symmetric space is a notion introduced by the Norwegian mathematician Atle Selberg

Atle Selberg

Atle Selberg was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory...

 in the 1950s as a generalisation of symmetric space

Symmetric space

A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...

, due to Élie Cartan

Élie Cartan

Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...

. Geometrically the spaces are defined as complete Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

s such that any two points can be exchanged by an isometry

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

, the symmetric case being when the isometry is required to have period two. The classification of weakly symmetric spaces relies on that of periodic automorphisms of complex semisimple Lie algebras. They provide examples of Gelfand pair

Gelfand pair

In mathematics, the expression Gelfand pair is a pair consisting of a group G and a subgroup K that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to...

s, although the corresponding theory of spherical functions

Zonal spherical function

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G...

 in harmonic analysis

Harmonic analysis

Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

, known for symmetric spaces, has not yet been developed.