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almost complex structure
Let $V$ be a vector space over $\mathbb{R}$. Recall that a complex structure on $V$ is a linear operator $J$ on $V$ such that $J^{2}=I$, where $J^{2}=J\circ J$, and $I$ is the identity operator on $V$. A prototypical example of a complex structure is given by the map $J:V\to V$ defined by $J(v,w)=(w,v)$ where $V=\mathbb{R}^{n}\oplus\mathbb{R}^{n}$.
An almost complex structure on a manifold $M$ is a differentiable map
$J:TM\to TM$ 
on the tangent bundle $TM$ of $M$, such that

$J$ preserves each fiber, that is, the following diagram is commutative:
$\xymatrix{{TM}\ar[r]^{{J}}\ar[d]_{{\pi}}&{TM}\ar[d]^{{\pi}}\\ {M}\ar[r]_{{i}}&{M}}$ or $\pi\circ J=\pi$, where $\pi$ is the standard projection onto $M$, and $i$ is the identity map on $M$;

$J$ is linear on each fiber, and whose square is minus the identity. This means that, for each fiber $F_{x}:=\pi^{{1}}(x)\subseteq TM$, the restriction $J_{x}:=J\mid_{{F_{x}}}$ is a complex structure on $F_{x}$.
Remark. If $M$ is a complex manifold, then multiplication by $i$ on each tangent space gives an almost complex structure.
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