Suppose M is equipped with a symplectic form ω, a Riemannian metric g, and an almost-complex structure J. Since ω and g are nondegenerate, each induces a bundle isomorphism TM → T*M, where the first map, denoted φω, is given by the interior product φω(u) = uω = ω(u, •) and the other, denoted φg, is given by the analogous operation for g. With this understood, the three structures (g,ω,J) form a compatible triple when each structure can be specified by the two others as follows:
- g(u, v) = ω(u, Jv)
- ω(u, v) = g(Ju, v)
- J(u) = φg−1(φω(u)).
In each of these equations, the two structures on the right hand side are called compatible when the corresponding construction yields a structure of the type specified. For example, ω and J are compatible iff ω( •,J • ) is a Riemannian metric. The bundle on M whose sections are the almost complex structures compatible to ω has contractible fibres: the complex structures on the tangent fibres compatible with the restriction to the symplectic forms.
Using elementary properties of the symplectic form ω, one can show that a compatible almost-complex structure J is an almost Kähler structure for the Riemannian metric ω(u,Jv). It is also a fact that if J is integrable, then (M,ω,J) is a Kähler manifold.
These triples are related to the 2 out of 3 property of the unitary group.
Read more about this topic: Almost Complex Manifold
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