almost complex structure

Let V be a vector spaceMathworldPlanetmath over . Recall that a complex structure on V is a linear operator J on V such that J2=-I, where J2=JJ, and I is the identity operator on V. A prototypical example of a complex structure is given by the map J:VV defined by J(v,w)=(-w,v) where V=nn.

An almost complex structure on a manifoldMathworldPlanetmath M is a differentiable map


on the tangent bundle TM of M, such that

  • J preserves each fiber, that is, the following diagram is commutativePlanetmathPlanetmathPlanetmath:


    or πJ=π, where π is the standard projectionPlanetmathPlanetmath onto M, and i is the identity map on M;

  • J is linear on each fiber, and whose square is minus the identityPlanetmathPlanetmathPlanetmath. This means that, for each fiber Fx:=π-1(x)TM, the restrictionPlanetmathPlanetmath Jx:=JFx is a complex structure on Fx.

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