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almost complex structure (Definition)

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

$\displaystyle J:TM\to TM$
on the tangent bundle $ TM$ of $ M$, such that

Remark. If $ M$ is a complex manifold, then multiplication by $ i$ 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:
AMS MSC53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general)

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