Almost complex manifold: Difference between revisions
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* [[Complex manifold]] | * [[Complex manifold]] | ||
* [[Kahler manifold]] | |||
* [[Hermitian manifold]] | |||
===Weaker structures=== | ===Weaker structures=== | ||
* [[Almost symplectic manifold]] | * [[Almost symplectic manifold]]: {{further|[[Almost complex structure gives symplectic form]]}} | ||
* [[Oriented manifold]] | * [[Oriented manifold]] | ||
Latest revision as of 19:33, 18 May 2008
This article defines a differential manifold with the following additional structure -- the structure group is reduced to: general linear group over complex numbers
Definition
An almost complex manifold is the following data:
- A differential manifold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M}
- At each point, a complex structure to the vector space at that point. In other words, for every point, we give a linear map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J^2 = -I} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} varying smoothly with the point.
Relation with other structures
Stronger structures
Weaker structures
- Almost symplectic manifold: Further information: Almost complex structure gives symplectic form
- Oriented manifold