If "M" is a Riemannian symmetric space, the identity component "G" of the isometry group of "M" is a Lie group acting transitively on "M" (that is, "M" is Riemannian homogeneous). Therefore, if we fix some point "p" of "M", "M" is diffeomorphic to the quotient "G/K", where "K" denotes the isotropy group of the action of "G" on "M" at "p". By differentiating the action at "p" we obtain an isometric action of "K" on T"p""M". This action is faithful (e.g., by a theorem of Kostant, any isometry in the identity component is determined by its 1-jet at any point) and so "K" is a subgroup of the orthogonal group of T"p""M", hence compact. Moreover, if we denote by "s""p": M → M the geodesic symmetry of "M" at "p", the map