Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 16 / 6 / a / Solution

An almost complex structure is a smooth bundle endomorphism $J:TM\to TM$ with $J^2=-I$. It is an $\omega$-compatible almost complex structure whenThese conditions make $g_J(v,w)=\omega (v,Jw)$ a Riemannian metric. In particular it is symmetric: invariance and $J^2=-I$ give $\omega (Jv,w)=-\omega (v,Jw)$, and skew-symmetry then gives $g_J(w,v)=g_J(v,w)$. Positivity is the second condition. Moreover $J$ is an isometry for $g_J$. Compatibility does not require that the almost complex structure be integrable.