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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, with [[tangent spaces|WY6K3P4YCG]] $T_pM$.\nA *pseudo-Riemannian metric* on $M$ is a smooth symmetric $(0, 2)$-[[tensor field|WX302T93TW]] $g$ on $M$ such that:\n\n1. for each $p \\in M$, the map $g_p \\colon T_pM \\times T_pM \\to \\R$ is a [[symmetric bilinear form|0AXZNE5XNH]];\n2. for each $p \\in M$, the form $g_p$ is [[nondegenerate|01NHFZF2SK]]: if $g_p(v, w) = 0$ for all $w \\in T_pM$, then $v = 0$;\n3. the [[signature|07EZA80NEH]] of $g_p$ is the same for all $p \\in M$.\n\nA *pseudo-Riemannian manifold* is a pair $(M, g)$ of a smooth manifold $M$ and a pseudo-Riemannian metric $g$ on $M$.\n\n## Remarks\n\nExample: on $\\R^2$ with coordinates $(t, x)$, let $g = -dt^2 + dx^2$.\nAt each point its matrix is $\\operatorname{diag}(-1, 1)$, which is symmetric with nonzero determinant, so $g$ is nondegenerate with constant signature: $(\\R^2, g)$ is pseudo-Riemannian.\n\nNon-example: on $\\R^2$, let $g = dt^2 + t \\, dx^2$.\nIts matrix $\\operatorname{diag}(1, t)$ is degenerate where $t = 0$, so clause 2 fails on the line $t = 0$.\n\nOn a connected manifold, clause 3 follows from clause 2 and continuity, since the signature can only change where the determinant of $g$ vanishes.\nA Riemannian metric is the case where $g_p$ is positive definite at every point.\nIn local coordinates $(x^1, \\ldots, x^n)$ one writes $g = \\sum_{i,j} g_{ij} \\, dx^i \\, dx^j$ with a symmetric invertible matrix $(g_{ij})$.\n",
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