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 $T M$ 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
Canonical name	AlmostComplexStructure
Date of creation	2013-03-22 13:15:34
Last modified on	2013-03-22 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