Definition Using Geodesic Symmetries
Let M be a connected Riemannian manifold and p a point of M. A map f defined on 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 and then It follows that the derivative of the map 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.
M is said to be locally Riemannian symmetric if its geodesic symmetries are in fact isometric, and (globally) Riemannian symmetric if in addition its geodesic symmetries are defined on all of M.
Read more about this topic: Symmetric Space
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