In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost complex manifold, but not vice-versa. Almost complex structures have important applications in symplectic geometry.
The concept is due to Ehresmann and Hopf in the 1940s.
Read more about Almost Complex Manifold: Formal Definition, Examples, Differential Topology of Almost Complex Manifolds, Integrable Almost Complex Structures, Compatible Triples, Generalized Almost Complex Structure
Famous quotes containing the words complex and/or manifold:
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—Franz Grillparzer (1791–1872)
“Before abstraction everything is one, but one like chaos; after abstraction everything is united again, but this union is a free binding of autonomous, self-determined beings. Out of a mob a society has developed, chaos has been transformed into a manifold world.”
—Novalis [Friedrich Von Hardenberg] (1772–1801)