form of the Hermitian metric.

Definition 1.1.6. [*We say that an exterior $2$-form $\omega $ on $X$ tames an almost-complex structure $J$ if $\omega (v,Jv)>0$ for $v\not= 0$.* ]{}

Remark. [Here and later we will use the convention that for forms $f$ and $h$ on the vector space $V$ the exterior product is defined as $f\wedge g: = f\otimes g - g\otimes f$, , without coefficient ${1\over 2}$. ]{}

1.2. Existence of Calibrating and Tame Structures.

Here we shall prove here the following simple but very important statement:

Proposition 1.2.1. [*Let $X$ be a real manifold and $\omega $ a nowhere degenerate exterior $2$-form on $X$, , $\omega^n\not=0$, where $2n=\dim_{\rr } X$. On the open subset $U\subset X$ a $\omega $-calibrating almost-complex structure $J$ is given. Then for any relatively compact open $U_1\comp U$ there exists a $\omega $-calibrating almost-complex structure $J_1$ on the whole $X$ such that $J_1\mid_{U_1}=J\mid_{U_1}$.* ]{}

Proof. Consider a Riemannian metric $g(u,v):=\omega (u,Jv)$ on $U$. Find a Riemannian metric $g_1$ on $X$ such that $g_1\mid_{U_1}=g\mid_{
U_1}$. Since $\omega $ is not degenerate, there exists a (unique!) isomorphism $A_1:TX\to TX$ such that $$\omega (u,v) = g_1(A_1u,v)\eqno(1.2.1)$$ for all $u,v\in TX$. Further, $\omega $ is antisymmetric, so $$g_1(A_1u,v) 