Almost Complex Manifold - Compatible Triples

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 nondegenerate, each induces a bundle isomorphism TM → T*M, where the first map, denoted φ_ω, is given by the interior product φ_ω(u) = _uω = ω(u, •) and the other, denoted φ_g, 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_g−1(φ_ω(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.