The complex Goldberg-Sachs theorem in higher dimensions

TL;DR

This paper extends the Goldberg-Sachs theorem to higher-dimensional complex Riemannian manifolds, establishing algebraic conditions for integrability of null distributions and exploring implications for Petrov classification and real pseudo-Riemannian geometries.

Contribution

It generalizes the Goldberg-Sachs theorem and Petrov classification to higher dimensions, providing algebraic criteria for integrability of null distributions.

Findings

Algebraic conditions on Weyl and Cotton-York tensors for integrability.

Higher-dimensional Petrov type D analogues lead to multiple integrable null distributions.

Applications to Hermitian and Robinson geometries in pseudo-Riemannian contexts.

Abstract

We study the geometric properties of holomorphic distributions of totally null $m$-planes on a $(2m+\epsilon)$-dimensional complex Riemannian manifold $(\mathcal{M}, \bm{g})$, where $\epsilon \in {0,1}$ and $m \geq 2$. In particular, given such a distribution $\mathcal{N}$, say, we obtain algebraic conditions on the Weyl tensor and the Cotton-York tensor which guarrantee the integrability of $\mathcal{N}$, and in odd dimensions, of its orthogonal complement. These results generalise the Petrov classification of the (anti-)self-dual part of the complex Weyl tensor, and the complex Goldberg-Sachs theorem from four to higher dimensions. Higher-dimensional analogues of the Petrov type D condition are defined, and we show that these lead to the integrability of up to $2^m$ holomorphic distributions of totally null $m$-planes. Finally, we adapt these findings to the category of real smooth pseudo-Riemannian manifolds, commenting notably on the applications to Hermitian geometry and Robinson (or optical) geometry.