Let M be a manifold with a nondegenetate 2-form \(\omega\). Let \(\mathcal{J}\) denote the set of \(\omega\)-tamed almost complex structures. In other words, an almost complex structure J belongs to \(\mathcal{J}\) if \(\omega(X,JX)>0\) for all nonzero \(X\in TM\). I don’t see why this space is contractile. I can show that the space of \(\omega\)-tamed almost complex structures on a symplectic vector space is contractible but this argument depends on a choice of symplectic basis. This argument wouldn’t seem to work when extended to the manifold case since at best one has local symplectic frames and there is no reason to expect that the contraction over one open set should agree with the contraction on another open set on the overlap of the two open sets. Any thoughts? Supposedly this is true according to a remark in a book by McDuff and Salamon.