Symmetric space

Summary

A symmetric space is a Riemannian manifold that has, at each point, an isometry reversing all geodesics through that point. Euclidean spaces, round spheres, hyperbolic spaces and projective spaces are symmetric spaces. Each symmetric space is a quotient $G/K$ of a Lie group by a closed subgroup, so its geometry reduces to Lie theory. Cartan's classification lists all of them, which makes them a large and well understood family of model spaces.