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 . Properties Some properties and forms of symmetric spaces can be noted. Lifting the metric tensor The metric tensor on the Riemannian manifold can be lifted to a scalar product on by combining it with the Killing form. This is done by defining Here, is the Riemannian metric defined on , and is the Killing form. The minus sign appears because the Killing form is negative-definite on this makes positive-definite. Factorization The tangent space can be further factored into eigenspaces classified by the Killing form. This is accomplished by defining an adjoint map taking as where is the Riemannian metric on and is the Killing form. This map is sometimes called the generalized transpose, as corresponds to the transpose for the orthogonal groups and the Hermitian conjugate for the unitary groups. It is a linear functional, and it is self-adjoint, and so one concludes that there is an orthonormal basis of with These are orthogonal with respect to the metric, in that since the Killing form is symmetric. This factorizes into eigenspaces with for . For the case of semisimple, so that the Killing form is non-degenerate, the metric likewise factorizes: In certain practical applications, this factorization can be interpreted as the spectrum of operators, e.g. the spectrum of the hydrogen atom, with the eigenvalues of the Killing form corresponding to different values of the angular momentum of an orbital (i.e. the Killing form being a Casimir operator that can classify the different representations under which different orbitals transform.) Classification of symmetric spaces proceeds based on whether or not the Killing form is positive/negative definite. Applications and special cases Symmetric spaces and holonomy If the identity component of the holonomy group of a Riemannian manifold at a point acts irreducibly