The complex Goldberg-Sachs theorem in higher dimensions
- 作者：Arman Taghavi-Chabert taghavia@math.muni.cz
- 关键词：Integrability of distributions ; Null foliations ; Curvature prescription ; Complex geometry ; Conformal geometry ; Twistor geometry
- 刊名：Journal of Geometry and Physics
- 出版年：2012
- 期刊代码：29_03930440
- 类别：ph
- 出版时间：May, 2012
- 卷：62
- 期：5
- 页码：981-1012
- 文件大小：443 K
摘要
We study the geometric properties of holomorphic distributions of totally null -planes on a -dimensional complex Riemannian manifold , where and . In particular, given such a distribution , say, we obtain algebraic conditions on the Weyl tensor and the Cotton-York tensor which guarantee the integrability of , 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 condition are defined, and we show that these lead to the integrability of up to holomorphic distributions of totally null -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.