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

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

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$J$ 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 

Canonical name  AlmostComplexStructure 
Date of creation  20130322 13:15:34 
Last modified on  20130322 13:15:34 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  7 
Author  rspuzio (6075) 
Entry type  Definition 
Classification  msc 53D05 
Related topic  KahlerManifold 
Related topic  HyperkahlerManifold 
Related topic  MathbbCIsAKahlerManifold 