A Pontryagin class obstruction for purely electric and purely magnetic Weyl curvature tensors

Abstract

Do all manifolds that admit Lorentzian metrics also admit such metrics that have a purely electric (PE) or purely magnetic (PM) Weyl curvature tensor? To (partially) answer this question, we show that for all algebraic curvature tensors on a $4k$-dimensional scalar product space that are even or odd under the action of a orientation-reversing isometry, the products of Pontryagin forms that land in the top-degree exterior power of the dual vector space vanish. We use this to derive the vanishing of all products of Pontryagin classes that land in the top-degree de Rham cohomology of a $4k$-dimensional pseudo-Riemannian manifold with a PE or PM Riemann or Weyl curvature tensor. For compact manifolds, this gives nontrivial cohomological obstructions to the existence of such pseudo-Riemannian metrics with globally PE or PM Riemann or Weyl curvature tensors. These obstructions can be linked to the existence of Lorentzian metrics of several Petrov subtypes, which play an important role in classifying exact solutions to the Einstein equations. Moreover, they can be applied to foliations by nondegenerate umbilic hypersurfaces, which may appear as timeslices of spacetimes.