Almost Hermitian manifold: Difference between revisions
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* A [[differential manifold]] <math>M</math> | * A [[differential manifold]] <math>M</math> | ||
* A [[Riemannian metric]] <math>g</math> on <math>M</math> | * A [[Riemannian metric]] <math>g</math> on <math>M</math> | ||
* | * An [[almost complex structure]] viz a map <math>J: TM \to TM</math> such that <math>J^2 = -I</math> with the further property that <math>g(Jv,w) + g(v,Jw) = 0</math> | ||
Note that this gives a reduction of the structure group to the unitary group over complex numbers. | Note that this gives a reduction of the structure group to the unitary group over complex numbers. | ||
Revision as of 12:21, 31 August 2007
This article defines a differential manifold with the following additional structure -- the structure group is reduced to: unitary group
Definition
An almost Hermitian manifold is the following data:
- A differential manifold
- A Riemannian metric on
- An almost complex structure viz a map such that with the further property that
Note that this gives a reduction of the structure group to the unitary group over complex numbers.