PlanetMath: almost complex structure

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almost complex structure

(Definition)

Let be a vector space over $\mathbb{R}$ . Recall that a complex structure on is a linear operator on such that , where $J^2=J\circ J$ , and is the identity operator on . A prototypical example of a complex structure is given by the map $J:V\to V$ defined by where $V=\mathbb{R}^n\oplus \mathbb{R}^n$ .

An almost complex structure on a manifold is a differentiable map

$\displaystyle J:TM\to TM$

on the tangent bundle

of

, such that

preserves each fiber, that is, the following diagram is commutative:
$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {TM}\ar[r]^{J}\ar[d]_{\pi}&{TM}\ar[d]^{\pi}\ {M}\ar[r]_{i}&{M} } } \end{xy}$
or $\pi\circ J=\pi$ , where $\pi$ is the standard projection onto , and is the identity map on ;
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 .

Remark. If is a complex manifold, then multiplication by on each tangent space gives an almost complex structure.

"almost complex structure" is owned by rspuzio. [ full author list (3) | owner history (8) ]

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See Also: Kähler manifold, hyperkähler manifold, $\mathbb{C}$ as a Kähler manifold

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Cross-references: tangent space, multiplication, complex manifold, restriction, identity, square, onto, projection, commutative, fiber, preserves, tangent bundle, differentiable map, manifold, map, identity operator, linear operator, complex structure, vector space
There are 6 references to this entry.

This is version 4 of almost complex structure, born on 2002-12-12, modified 2007-09-04.
Object id is 3739, canonical name is AlmostComplexStructure.
Accessed 2983 times total.

Classification:

53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general)

Pending Errata and Addenda

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