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" = σ"y" and "sy" = σ"x". (Selberg's assumption that σ2 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 "L"2("M") is multiplicity free.