  This note is about the interplay between the almost-hermitian and Riemannian
geometries of a manifold. These geometries can be seen to interact through
curvature. The main result is an obstruction equation to the integrability of
almost-complex structures orthogonal with respect to Riemannian metrics with
constrained sectional curvature. Several geometric consequences ensue, such as
a formula for the norm of the Levi-Civita covariant derivative of a
hypothetical orthogonal complex structure. Our results lead to a partial
recovery of the well-known fact that the round $6$-sphere $S^6$ is not
hermitian. The partial proof is intrinsic in nature, and shows some level of
promise when it comes to generalizing the non-complexity of the round $S^6$
result in new directions.
