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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] with [[tangent bundle|WY6K3P4YCG]] $TM$.\nAn *almost complex structure* on $M$ is a smooth [[tensor field|WX302T93TW]] $J$ of type $(1, 1)$ such that $J^2 = -\\id$.\nThus $J \\colon TM \\to TM$ is a smooth map that is linear on each tangent space $T_p M$.\nAn *almost complex manifold* is a pair $(M, J)$ of a smooth manifold and an almost complex structure on it.\n\n## Remarks\n\nExample: on $\\R^{2n}$ with coordinates $(x_1, \\ldots, x_n, y_1, \\ldots, y_n)$, put $J \\partial_{x_k} = \\partial_{y_k}$ and $J \\partial_{y_k} = -\\partial_{x_k}$.\nThen $J^2 \\partial_{x_k} = J \\partial_{y_k} = -\\partial_{x_k}$ and likewise for $\\partial_{y_k}$, so $J^2 = -\\id$.\nThis is multiplication by $i$ under $\\R^{2n} \\cong \\C^n$, and every complex manifold carries the structure obtained in this way from its charts.\n\nNon-example: no manifold of odd dimension $m$ has an almost complex structure.\nA real linear map $J$ of $T_p M$ with $J^2 = -\\id$ gives $\\det(J)^2 = \\det(-\\id) = (-1)^m = -1$, which is impossible.\n\nThe *Nijenhuis tensor* of $J$ is\n$$N_J(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] - [X, Y]$$\nfor vector fields $X, Y$, where $[\\cdot, \\cdot]$ is the [[Lie bracket of vector fields|68FGQEBD6J]].\nThe structure $J$ is called *integrable* when $N_J = 0$.\nThe sphere $S^6$ has an almost complex structure coming from the octonions, while $S^4$ has none.\n",
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