The complex Goldberg-Sachs theorem in higher dimensions
This paper presents a comprehensive study of the geometric properties of some holomorphic distributions of totally null \(m\)-planes on complex Riemannian manifolds of certain dimensions. The author finds algebraic conditions for the Weyl tensor and the Cotton-York tensor which guarantee the integrability of the distribution, and in odd dimensions, of its orthogonal complement. It is shown that these results generalize the Petrov classification of the (anti-)self-dual part of the complex Weyl tensor and the complex Goldberg-Sachs theorem. In addition, higher dimensional analogues of the Petrov type \(D\) condition are defined and the results are adapted to real smooth pseudo-Riemannian manifolds. The applications to Hermitian geometry and Robinson (or optical) geometry are discussed.
scientific article; zbMATH DE number 6031296
scientific article