Contractibility of compatible almost complex structures
= Contractibility of compatible almost complex structures
The <metric construction of a compatible almost complex structure> is continuous in the auxiliary <Riemannian metric>, and recovers $J$ when applied to $g_J=\omega(\cdot,J\cdot)$. Interpolating $g_J$ linearly to a fixed auxiliary metric therefore contracts the space of <compatible almost complex structures>. In particular it is nonempty and path connected.