Almost complex manifold

Remarks

Example: on $\R^{2n}$ with coordinates $(x_1, \ldots, x_n, y_1, \ldots, y_n)$, put $J \partial_{x_k} = \partial_{y_k}$ and $J \partial_{y_k} = -\partial_{x_k}$. Then $J^2 \partial_{x_k} = J \partial_{y_k} = -\partial_{x_k}$ and likewise for $\partial_{y_k}$, so $J^2 = -\id$. This is multiplication by $i$ under $\R^{2n} \cong \C^n$, and every complex manifold carries the structure obtained in this way from its charts.

Non-example: no manifold of odd dimension $m$ has an almost complex structure. A real linear map $J$ of $T_p M$ with $J^2 = -\id$ gives $\det(J)^2 = \det(-\id) = (-1)^m = -1$, which is impossible.

The Nijenhuis tensor of $J$ is $$N_J(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] - [X, Y]$$ for vector fields $X, Y$, where $[\cdot, \cdot]$ is the Lie bracket of vector fields. The structure $J$ is called integrable when $N_J = 0$. The sphere $S^6$ has an almost complex structure coming from the octonions, while $S^4$ has none.