Almost Complex Structure

Definition

An almost complex structure $J$ on a smooth real manifold $M$ is a linear complex structure on each tangent space of the manifold. Equivalently, is an automorphism

$$J:TM\to TM$$

of the tangent bundle such that $J^2=-1$.

Tame

Let $M$ be a smooth manifold and let $\omega \in {\Omega }^2(M)$ be a nondegenerate $2$-form. An almost complex structure $J$ on $M$ is called $\omega$-tame if

$$\omega (v,Jv)>0\text{for every nonzero tangent vector }v\in TM.$$

Property

Proposition