On the Petrov type of a 4-manifold

On an oriented 4-manifold, we examine the geometry that arises when the curvature operator of a Riemannian or Lorentzian metric [Formula: see text] commutes, not with its own Hodge star operator, but rather with that of another semi-Riemannian metric [Formula: see text] that is a suitable deformation of [Formula: see text]. We classify the case when one of these metrics is Riemannian and the other Lorentzian by generalizing the concept of Petrov Type from general relativity; the case when [Formula: see text] is split-signature is also examined. The “generalized Petrov Types” so obtained relate to the critical points of [Formula: see text]’s sectional curvature, and sometimes yield unique normal forms. They also carry topological information independent of the Hitchin–Thorpe inequality, and yield a direct geometric characterization of “almost-Einstein” metrics via the Ricci or sectional curvature of [Formula: see text].