Let "M" be a connected Riemannian manifold and "p" a point of "M". A diffeomorphism "f" of a neighborhood of "p" is said to be a geodesic symmetry if it fixes the point "p" and reverses geodesics through that point, i.e. if "γ" is a geodesic with formula_2 then formula_3 It follows that the derivative of the map "f" at "p" is minus the identity map on the tangent space of "p". On a general Riemannian manifold, "f" need not be isometric, nor can it be extended, in general, from a neighbourhood of "p" to all of "M".