On the proportionality of Chern and Riemannian scalar curvatures

On a Kahler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kahler condition. While such a link is not so obvious in the non-Kahler setting, one can seek to understand extensions of these characterizations to general Hermitian manifolds. This idea has been the subject of much study from the cohomological side, however, the focus here is to address such a question from the perspective of curvature relationships. In particular, on compact manifolds the K"ahler condition is characterized by the relationship that the Chern scalar curvature is equal to half the Riemannian scalar curvature. What we study here is the existence, or lack thereof, of non-Kahler Hermitian metrics for which a more general proportionality relationship between these scalar curvatures holds.