On the Petrov Type of a 4-manifold

TL;DR

This paper generalizes Petrov classification to 4-manifolds by examining curvature operators commuting with different metrics, revealing new geometric and topological insights, including relations to curvature critical points and almost-Einstein metrics.

Contribution

It introduces a generalized Petrov type classification for 4-manifolds based on commuting curvature operators with deformed metrics, extending concepts from general relativity.

Findings

Relation of generalized Petrov types to sectional curvature critical points

Identification of unique normal forms in certain cases

Topological information independent of Hitchin-Thorpe inequality

Abstract

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 are shown to 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 formulation of "almost-Einsten" metric via the Ricci or sectional curvature of $g$.