= Solution
A <compatible almost complex structure> is a smooth bundle map $J:TM\to TM$ such that, for all <tangent vectors> $u,v$,
$$
\boxed{J^2=-I,\qquad\omega(Ju,Jv)=\omega(u,v),\qquad\omega(u,Ju)>0\quad(u\ne0).}
$$
It determines the <Riemannian metric>
$$
\boxed{g(u,v)=\omega(u,Jv).}
$$
Indeed, the invariance of the <symplectic form> under $J$ and its antisymmetry imply $\omega(v,Ju)=\omega(u,Jv)$, so $g$ is symmetric; the positivity condition makes it positive definite. A <compatible triple> consists of this <symplectic form>, this <compatible almost complex structure>, and this <Riemannian metric>. Useful equivalent identities are $\omega(u,v)=g(Ju,v)$ and $g(Ju,Jv)=g(u,v)$.
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