Petrov types for the Weyl tensor via the Riemannian-to-Lorentzian bridge

Abstract

We analyze oriented Riemannian 4-manifolds whose Weyl tensors W satisfy the conformally invariant condition \(W(T,\cdot ,\cdot ,T) = 0\) for some nonzero vector T. While this can be algebraically classified via W’s normal form, we find a further geometric classification by deforming the metric into a Lorentzian one via T. We show that such a W will have the analogue of Petrov Types from general relativity, that only Types I and D can occur, and that each is completely determined by the number of critical points of W’s associated Lorentzian quadratic form. A similar result holds for the Lorentzian version of this question, with T timelike.