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 g commutes, not with its own Hodge star operator, but rather with that of another semi-Riemannian metric h that is a suitable deformation of g. 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 h is split-signature is also examined. The “generalized Petrov Types” so obtained relate to the critical points of g’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 g.