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 $h = g$. We show that this extension admits a variational characterization: for fixed $g$, the admissible auxiliary metrics $h$ are precisely the critical points of the conformally invariant mixed Einstein-Hilbert functional \int_M \text{scal}_{\text{g$-$h}} dV_h, where \text{scal}_{\text{g$-$h}} is the $h$-scalar contraction of the curvature tensor of $g$. We also compute the second variation and show that pointwise nondegeneracy of the induced Hessian on trace-free symmetric 2-tensors yields local rigidity and persistence of admissible conformal classes under perturbations of $g$. Finally, we exhibit non-Einstein examples of $(g, h)$ on products of surfaces and on $\mathbb{S}^4$, and, under a shared-orthogonal-frame ansatz, obtain a Berger-type nonnegativity result for the Euler characteristic.