  The classical theory of $G$-structures, which include almost-complex
structures, explains the relationship between the curvature of compatible
connections and integrability. This note is an effort to understand how the
curvature of Riemannian metrics can obstruct the integrability of
almost-complex structures. It is shown that certain special complex structures
cannot coexist with non-flat constant curvature metrics, and a formal
variational realization of these structures is provided. The approach followed
here is direct, meaning that it bypasses the classical theory. The idea is to
find obstruction equations for the integrability of almost-complex structures
by way of Nijenhuis tensor derivatives. These new equations involve the
curvature of a torsion-free connection, and reveal the interplay between the
almost-complex and Riemannian geometries. Curvature scalars to detect
non-complexity in the compact case then arise in a natural way.
