almost complex structure

Let $V$ be a vector space over $ℝ$. 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 $V={ℝ}^n\oplus {ℝ}^n$.

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.