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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]].\nLet $J \\colon TM \\to TM$ be a smooth [[tensor field|WX302T93TW]] of type $(1,1)$, that is, a smooth field of linear endomorphisms of the tangent spaces.\nLet $[\\cdot, \\cdot]$ be the [[Lie bracket of vector fields|68FGQEBD6J]].\nThe *Nijenhuis tensor* of $J$ is the map $N_J$ that sends vector fields $X, Y$ to the vector field\n$$N_J(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] + J^2 [X, Y].$$\n\n## Remarks\n\nThe map $N_J$ is $C^\\infty(M)$-bilinear and antisymmetric, so it is a tensor field of type $(1,2)$, a $2$-form with values in $TM$.\nIt equals $\\frac{1}{2}[J, J]$, where $[J, J]$ is the [[Frölicher–Nijenhuis bracket|D46QECMC10]] of $J$ with itself.\nFor an [[almost complex structure|9SYWRGVTRS]], where $J^2 = -\\id$, the formula reads $N_J(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] - [X, Y]$.\nSome authors use a different constant factor.\n\nExample: on $\\R^2$ let $J \\partial_x = \\partial_y$ and $J \\partial_y = -\\partial_x$.\nFor $X, Y$ among $\\partial_x, \\partial_y$, all of $X, Y, JX, JY$ are constant combinations of $\\partial_x$ and $\\partial_y$, so all brackets vanish.\nBy bilinearity over functions, $N_J = 0$.\nMore generally, for an almost complex structure one has $N_J(JX, Y) = -J N_J(X, Y)$, so $N_J(JX, X) = 0$; on a $2$-manifold $X$ and $JX$ span each tangent space where $X \\ne 0$, hence $N_J = 0$ on every surface.\n\nNon-example: the expression $[JX, JY] - [X, Y]$ alone is not a tensor.\nReplacing $X$ by $fX$ for a function $f$ gives $f([JX, JY] - [X, Y]) - (JY)(f)\\, JX + Y(f)\\, X$, since $[fA, B] = f[A, B] - B(f) A$.\nThe last two terms are not zero in general, so the expression is not $C^\\infty(M)$-linear in $X$; the two middle terms of $N_J$ cancel them.\n",
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