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  "body": "\n## Summary\n\nA Riemannian manifold is a smooth manifold together with a [[Riemannian metric|Q61VFHCH2J]].\nIt is the setting of Riemannian geometry, where one studies lengths, angles, geodesics, volume and curvature on curved spaces.\nEuclidean space, spheres and hyperbolic spaces are the basic examples, and every submanifold of Euclidean space inherits a Riemannian structure.\nReplacing positive definiteness by a Lorentzian signature gives the [[pseudo-Riemannian manifolds|GH6CXWFNDJ]] that model spacetime in general relativity.\n\n## Explanations\n\n### L1 human\n\nA Riemannian manifold is a curved space where you can measure.\nNear each point it looks like a small piece of ordinary space, and you know how to measure lengths and angles there.\nFrom this you can find the shortest paths, compute areas, and detect how curved the space is.\nThe surface of the Earth, with distances measured along the surface, is a two-dimensional example.\n",
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