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  "body": "\n## Summary\n\nA symmetric space is a Riemannian manifold that has, at each point, an isometry reversing all geodesics through that point.\nEuclidean spaces, round spheres, hyperbolic spaces and projective spaces are symmetric spaces.\nEach symmetric space is a quotient $G/K$ of a Lie group by a closed subgroup, so its geometry reduces to Lie theory.\n[[Cartan's classification|9P8VK3H4YZ]] lists all of them, which makes them a large and well understood family of model spaces.\n\n## Explanations\n\n### L1 human\n\nStand at a point of the space and look in some direction.\nIn a symmetric space there is a symmetry of the whole space that keeps you fixed and swaps every direction with its opposite.\nOn a round sphere, this is the rotation by a half turn about the axis through you and the antipodal point.\nSo the space looks the same from every point and in every pair of opposite directions.\n",
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