to be a (globally) symmetric space if in addition its geodesic symmetries can be extended to isometries on all of M. Basic properties The Cartan–Ambrose–Hicks theorem implies that M is locally Riemannian symmetric if and only if its curvature tensor is covariantly constant, and furthermore that every simply connected, complete locally Riemannian symmetric space is actually Riemannian symmetric. Every Riemannian symmetric space M is complete and Riemannian homogeneous (meaning that the isometry group of M acts transitively on M). In fact, already the identity component of the isometry group acts transitively on M (because M is connected). Locally Riemannian symmetric spaces that are not Riemannian symmetric may be constructed as quotients of Riemannian symmetric spaces by discrete groups of isometries with no fixed points, and as open subsets of (locally) Riemannian symmetric spaces. Examples Basic examples of Riemannian symmetric spaces are Euclidean space, spheres, projective spaces, and hyperbolic spaces, each with their standard Riemannian metrics. More examples are provided by compact, semi-simple Lie groups equipped with a bi-invariant Riemannian metric. Every compact Riemann surface of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric space. Every lens space is locally symmetric but not symmetric, with the exception of which is symmetric. The lens spaces are quotients of the 3-sphere by a discrete isometry that has no fixed points. An example of a non-Riemannian symmetric space is anti-de Sitter space. Algebraic definition Let G be a connected Lie group. Then a symmetric space for G is a homogeneous space G/H where the stabilizer H of a typical point is an open subgroup of the fixed point set of an involution σ in Aut(G). Thus σ is an automorphism of G with σ2 = idG and H is an open subgroup of the invariant set Because H is open, it is a union of components of Gσ (including, of course, the identity component). As an automorphism of G, σ fixes the identity element, and hence, by differentiating at the identity, it induces an automorphism of the Lie algebra of G, also denoted by σ, whose square is the identity. It follows that the eigenvalues of σ are ±1. The +1 eigenspace is the Lie algebra of H (since this is the Lie algebra of Gσ), and the −1 eigenspace will be denoted . Since σ is an automorphism of ,