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.