Editing Almost complex manifold (section)

== Compatible triples ==
Suppose ''M'' is equipped with a [[symplectic form]] ''ω'', a [[Riemannian metric]] ''g'', and an almost complex structure ''J''. Since ''ω'' and ''g'' are [[Degenerate form|nondegenerate]], each induces a bundle isomorphism ''TM → T*M'', where the first map, denoted ''φ''<sub>''ω''</sub>, is given by the [[interior product]] ''φ''<sub>''ω''</sub>(''u'')&nbsp;=&nbsp;''i''<sub>''u''</sub>''ω''&nbsp;=&nbsp;''ω''(''u'',&nbsp;•) and the other, denoted ''φ''<sub>''g''</sub>, 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'') = (''φ''<sub>''g''</sub>)<sup>−1</sup>(''φ''<sub>''ω''</sub>(''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 if and only if ''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 manifold|almost Kähler structure]] for the Riemannian metric ''ω''(''u'', ''Jv''). Also, if ''J'' is integrable, then (''M'', ''ω'', ''J'') is a [[Kähler manifold]].

These triples are related to the [[Unitary group#2-out-of-3 property|2 out of 3 property of the unitary group]].