K, where h ∈ G) and define where σ is the involution of G fixing K. Then one can check that sp is an isometry with (clearly) sp(p) = p and (by differentiating) dsp equal to minus the identity on TpM. Thus sp is a geodesic symmetry and, since p was arbitrary, M is a Riemannian symmetric space. If one starts with a Riemannian symmetric space M, and then performs these two constructions in sequence, then the Riemannian symmetric space yielded is isometric to the original one. This shows that the "algebraic data" (G,K,σ,g) completely describe the structure of M. Classification of Riemannian symmetric spaces The algebraic description of Riemannian symmetric spaces enabled Élie Cartan to obtain a complete classification of them in 1926. For a given Riemannian symmetric space M let (G,K,σ,g) be the algebraic data associated to it. To classify the possible isometry classes of M, first note that the universal cover of a Riemannian symmetric space is again Riemannian symmetric, and the covering map is described by dividing the connected isometry group G of the covering by a subgroup of its center. Therefore, we may suppose without loss of generality that M is simply connected. (This implies K is connected by the long exact sequence of a fibration, because G is connected by assumption.) Classification scheme A simply connected Riemannian symmetric space is said to be irreducible if it is not the product of two or more Riemannian symmetric spaces. It can then be shown that any simply connected Riemannian symmetric space is a Riemannian product of irreducible ones. Therefore, we may further restrict ourselves to classifying the irreducible, simply connected Riemannian symmetric spaces. The next step is to show that any irreducible, simply connected Riemannian symmetric space M is of one of the following three types: 1. Euclidean type: M has vanishing curvature, and is therefore isometric to a Euclidean space. 2. Compact type: M has nonnegative (but not identically zero) sectional curvature. 3. Non-compact type: M has nonpositive (but not identically zero) sectional curvature. A more refined invariant is the rank, which is the maximum dimension of a subspace of the tangent space (to any point) on which the curvature is identically zero. The rank is always at least one, with equality if the sectional curvature is positive or negative. If the curvature is positive, the space is of compact type, and if negative, it