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Vipul at 12:31, 31 August 2007

2007-08-31T12:31:24Z

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==Statement==

Let <math>M</math> be a [[differential manifold]], <math>J</math> be an [[almost complex structure]] on <math>M</math> (viz a smooth choice of map from each tangent space to itself such that <math>J^2 = -I</math>). Let <math>g</math> be a [[Riemannian metric]] on <math>M</math>, such that:

<math>g(v,Jw) + g(Jv,w) = 0</math>

The tripe <math>(M,g,J)</math> is an [[almost Hermitian structure]] on <math>M</math>. This gives rise to the following [[almost symplectic structure]]: the bilinear form on a tangent space is defined as:

<math>b(v,w) = g(v,Jw)</math>

==Interpretation in terms of structure groups==

An almost Hermitian structure is a reduction ofthe structure group of the <math>2n</math>-dimensional manifold to <math>U(n,\mathbb{C})</math>, while an almost symplectic structure is a reduction of the stucture group to <math>Sp(2n,\mathbb{R})</math>. Since <math>U(n,\mathbb{C})</math> is a subgroup of <math>Sp(2n,\R)</math> an almost Hermitian structure gives an almost symplectic structure.

Almost Hermitian structure gives symplectic form - Revision history

Vipul: 1 revision

Vipul at 12:31, 31 August 2007