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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold with [[tangent bundle|WY6K3P4YCG]] $TM$.\nA *Riemannian metric* on $M$ is a family $g = (g_p)_{p \\in M}$ such that:\n\n1. for each $p \\in M$, $g_p$ is an [[inner product|0NB0AWGJBW]] on $T_pM$, that is, a symmetric positive definite bilinear form;\n2. for all smooth vector fields $X, Y$ on $M$, the function $p \\mapsto g_p(X_p, Y_p)$ is smooth.\n\n## Remarks\n\nEquivalently, $g$ is a smooth symmetric [[tensor field|WX302T93TW]] of type $(0, 2)$ that is positive definite at every point.\nIn a chart, $g = \\sum_{i,j} g_{ij} \\, dx^i \\otimes dx^j$ where $(g_{ij}(p))$ is a symmetric positive definite matrix depending smoothly on $p$; one often writes $ds^2 = g_{ij} \\, dx^i dx^j$.\nExample: on the upper half-plane $\\set{(x, y) : y \u003e 0}$, $g = (dx^2 + dy^2)/y^2$ is a Riemannian metric; its matrix $y^{-2} I$ is symmetric and positive definite because $y \u003e 0$, and its entries are smooth.\nNon-example: on $\\R^2$, $g = x^2 (dx^2 + dy^2)$ fails clause 1 on the line $x = 0$, where $g_p = 0$ is not positive definite.\nThe length of a piecewise smooth curve $\\gamma : [a, b] \\to M$ is $\\int_a^b \\sqrt{g(\\gamma'(t), \\gamma'(t))} \\, dt$.\nEvery smooth manifold (Hausdorff and second countable) has a Riemannian metric: glue Euclidean metrics of charts with a [[partition of unity|NH739XXVXK]]; positive definiteness survives convex combinations.\n",
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