Thijs de Kok | Searcharxiv

arXiv2026

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

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.

math.DG↗

arXiv2025

A partitioned manifold index theorem for noncompact hypersurfaces

Roe's partitioned manifold index theorem applies when a complete Riemannian manifold $M$ is cut into two pieces along a compact hypersurface $N$. It states that a version of the index of a Dirac operator on $M$ localized to $N$ equals the index of the corresponding Dirac operator on $N$. This yields obstructions to positive scalar curvature, and implies cobordism invariance of the index of Dirac operators on compact manifolds. We generalize this result to cases where $N$ may be noncompact, under assumptions on the way it is embedded into $M$. This results in an equality between two classes in the $K$-theory of the Roe algebra of $N$. Bunke and Ludewig, and Engel and Wulff, have recently obtained related results based on different approaches.

math.DG↗