Symmetric Space - Weakly Symmetric Riemannian Spaces

Weakly Symmetric Riemannian Spaces

In the 1950s Atle Selberg extended Cartan's definition of symmetric space to that of weakly symmetric Riemannian space, or in current terminology weakly symmetric space. These are defined as Riemannian manifolds M with a transitive connected Lie group of isometries G and an isometry σ normalising G such that given x, y in M there is an isometry s in G such that sx = σy and sy = σx. (Selberg's assumption that s2 should be an element of G was later shown to be unnecessary by Ernest Vinberg.) 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.

Selberg's definition can also be phrased equivalently in terms of a generalization of geodesic symmetry. 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

• s fixes x;
• the derivative of s at x sends X to –X.

When s is independent of X, M is a symmetric space.

An account of weakly symmetric spaces and their classification by Akhiezer and Vinberg, based on the classification of periodic automorphisms of complex semisimple Lie algebras, is given in Wolf (2007).