$\omega $-calibrating almost-complex structures is contractible.

Proof. The first statement of this corollary is a particular case of the proposition when $U=\emptyset $.

To prove the second one, we fix some calibrating structure $J_0$. Denote by $g_{J_0}$ the corresponding Riemannian metric on $X$, , $g_{J_0}(u,v)=\omega (u,J_0v)$. The space of Riemannian metrics is a convex cone $\calc$ in $\hom_{\rr }(TX,T^*X)$. Therefore there exists a contraction $\Psi : \calc \times [0,1]\to \calc $ to $g_{J_0}$, , $\Psi (\cdot , 0)=\id $, $ \Psi (\cdot ,1 )\equiv g_{J_0}$ and $\Psi (g_{J_0},t)\equiv g_{J_0}$.

Consider the following map $\Phi :\calj_{\omega }^c\to \calc $ from the space of $\omega $-calibrating structures to the space of metrics: $$\Phi (J)(u,v)=g_{J}(u,v)=\omega (u,Jv).$$ In the proof of Proposition 1.2.1 (see Remark there) we showed that $$P\scirc \Phi =\id :\calj_{\omega }^c\to \calj_{\omega }^c.$$

Now $P\scirc \Psi (\cdot ,t)\scirc \Phi $ will be a contraction of $\calj_{\omega }^c$ to $J_0$.

Let us prove the following

Proposition 1.2.3. [*For any symplectic manifold $(X,\omega)$, the set $\calj_\omega$ of $\omega$-tame almost complex structures on $X$ is a non-empty contractible manifold*]{}.

The proposition follow immediately from the following result from linear algebra.

Lemma 1.2.4. [*Let $V$ be a (finite-dimensional) real vector space and $\omega$ a linear 