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  "body": "\n## Summary\n\nA pseudo-Riemannian manifold is a [[smooth manifold|F45VQ47GPV]] with a smoothly varying [[nondegenerate|01NHFZF2SK]] [[symmetric bilinear form|0AXZNE5XNH]] on each tangent space, which need not be positive definite.\nIt generalises a [[Riemannian manifold|7CBT4SM83Y]], where the form is positive definite.\nThe most important case is a [[Lorentzian manifold|BY4QYYS788]], the setting of general relativity.\nMuch of Riemannian geometry, such as the [[Levi-Civita connection|CAZYGE1T4E]], geodesics and curvature, works in this setting.\n\n## Explanations\n\n### L1 human\n\nOn a Riemannian manifold every nonzero vector has a positive squared length.\nA pseudo-Riemannian manifold drops this rule: some vectors may have negative or zero squared length.\nWhat is kept is that the form has no hidden directions: no nonzero vector is orthogonal to every vector.\nMinkowski space $\\R^4$ with $-dt^2 + dx^2 + dy^2 + dz^2$ is the basic example.\nThere, a vector such as $(1, 1, 0, 0)$ has squared length $0$ although it is not zero.\n",
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