# 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.

Title almost complex structure AlmostComplexStructure 2013-03-22 13:15:34 2013-03-22 13:15:34 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Definition msc 53D05 KahlerManifold HyperkahlerManifold MathbbCIsAKahlerManifold