Hodge Splittings and Einstein 4-manifolds

On an oriented $4$-manifold, we study pairs of Riemannian metrics $(g,h)$ for which the curvature tensor of $g$ preserves the Hodge splitting determined by $h$.

This extends the Einstein condition in dimension four, which is recovered when $h$ is conformal to $g$.

We prove that such pairs satisfy a generalized Hitchin-Thorpe inequality, which reduces to the classical one when $h$ is conformal to $g$.

We then exhibit a pair $(g,h)$ on $\#_5\mathbb{CP}^2$, which violates Hitchin-Thorpe and hence admits no Einstein metric, thus showing that our condition is indeed broader than the Einstein condition.