Almost complex manifold

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

• Complex manifold
• Kahler manifold
• Hermitian manifold

Weaker structures

• Almost symplectic manifold: Further information: Almost complex structure gives symplectic form
• Oriented manifold