**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*) =

_{u}

*ω*=

*ω*(

*u*, •) and the other, denoted

*φ*, is given by the analogous operation for

_{g}*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.

Read more about this topic: Almost Complex Manifold

### Famous quotes containing the word compatible:

“English general and singular terms, identity, quantification, and the whole bag of ontological tricks may be correlated with elements of the native language in any of various mutually incompatible ways, each *compatible* with all possible linguistic data, and none preferable to another save as favored by a rationalization of the native language that is simple and natural to us.”

—Willard Van Orman Quine (b. 1908)