Metric construction of a compatible almost complex structure
= Metric construction of a compatible almost complex structure
Choose an auxiliary <Riemannian metric> $h$ and define the skew-adjoint bundle map $A$ by $\omega(u,v)=h(Au,v)$. Then
$$
J=A(-A^2)^{-1/2}
$$
is a <compatible almost complex structure>. The positive square root is defined fiberwise by the <spectral theorem for normal operators>, and its smooth dependence on $A$ makes $J$ smooth.