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:
  
  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.