Publication details
The complex Goldberg-Sachs theorem in higher dimensions
| Authors | |
|---|---|
| Year of publication | 2012 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Geometry and Physics |
| MU Faculty or unit | |
| Citation | |
| Web | http://www.sciencedirect.com/science/article/pii/S0393044012000228 |
| Doi | http://dx.doi.org/10.1016/j.geomphys.2012.01.012 |
| Field | General mathematics |
| Keywords | Integrability of distributions; Null foliations; Curvature prescription; Complex geometry; Conformal geometry; Twistor geometry |
| Description | We study the geometric properties of holomorphic distributions of totally null m-planes on a (2m + epsilon)-dimensional complex Riemannian manifold (M, g), where epsilon is an element of {0, 1} and m >= 2. In particular, given such a distribution N, say, we obtain algebraic conditions on the Weyl tensor and the Cotton-York tensor which guarantee the integrability of 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. |
| Related projects: |