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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]].\nLet $X$ be a smooth vector field on $M$, that is, a smooth section of the [[tangent bundle|WY6K3P4YCG]].\nIt is a *Killing vector field* if the [[Lie derivative|PJWK0PKN6Q]] of $g$ along $X$ vanishes:\n$$\\mathcal{L}_X g = 0.$$\n\n## Remarks\n\nEquivalently, with the [[Levi-Civita connection|CAZYGE1T4E]] $\\nabla$, the field $X$ is Killing if and only if $g(\\nabla_Y X, Z) + g(Y, \\nabla_Z X) = 0$ for all vector fields $Y, Z$.\nIn coordinates this is Killing's equation $\\nabla_i X_j + \\nabla_j X_i = 0$.\nAnother equivalent form: the local flow of $X$ consists of isometries.\n\nExample: on $\\R^2$ with the Euclidean metric, the rotation field $X = -y \\, \\partial_x + x \\, \\partial_y$ has $\\nabla X$ given by the constant matrix $\\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$, which is antisymmetric, so Killing's equation holds.\n\nNon-example: on $\\R^2$, the field $X = x \\, \\partial_x + y \\, \\partial_y$ has $\\nabla X$ equal to the identity matrix, which is not antisymmetric; indeed $\\mathcal{L}_X g = 2g \\ne 0$.\n\nThe Killing fields of $(M, g)$ form a Lie algebra under the Lie bracket, of dimension at most $n(n + 1)/2$ on a connected manifold of dimension $n$.\n",
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