In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold ("M", "g") is said to be symmetric if and only if, for each point "p" of "M", there exists an isometry of "M" fixing "p" and acting on the tangent space formula_1 as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about the endpoints). Both descriptions can also naturally be extended to the setting of pseudo-Riemannian manifolds.