The complex Goldberg-Sachs theorem in higher dimensions

Abstract

We study the geometric properties of holomorphic distributions of totally null $m$-planes on a $(2m+\epsilon )$-dimensional complex Riemannian manifold $(\mathcal{M},\mathbf{g})$, where $\epsilon \in 0,1$ and $m\ge 2$. 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.

Section 2 partly rewritten: issue regarding self-duality clarified. Section 5.2 clarified. Some remarks added. Lemma 3.7 (previously 3.7) corrected. A few mathematical and notational inaccuracies corrected, and typos and sign mistakes fixed throughout. Some references added

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• On the Goldberg-Sachs theorem in higher dimensions in the non-twisting case
• A Goldberg-Sachs theorem in dimension three
• A Generalization of the Goldberg-Sachs Theorem and its Consequences
• On the uniqueness of the Myers-Perry spacetime as a type II(D) solution in six dimensions
• Twisting non-shearing congruences of null geodesics, almost CR structures, and Einstein metrics in even dimensions
• On the Weyl Tensor Classification in All Dimensions and its Relation with Integrability Properties
• Spinors and the Weyl Tensor Classification in Six Dimensions
• Pure spinors, intrinsic torsion and curvature in odd dimensions
• Twistor Geometry of Null Foliations in Complex Euclidean Space
• The curvature of almost Robinson manifolds
• Pure Subspaces, Generalizing the Concept of Pure Spinors
• On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces
• Almost Robinson geometries
• Conformally Invariant Spinorial Equations in Six Dimensions
• On the Pursuit of Generalizations for the Petrov Classification and the Goldberg-Sachs Theorem