Spinors and the Weyl Tensor Classification in Six Dimensions
arXiv:1212.2689 · doi:10.1063/1.4804991
Abstract
A spinorial approach to 6-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the 4-dimensional case, making full use of the SO(6) symmetry to uncover results not easily seen in the tensorial approach. Using spinors, we propose a classification of the Weyl tensor by reinterpreting it as a map from 3-vectors to 3-vectors. This classification is shown to be intimately related to the integrability of maximally isotropic subspaces, establishing a natural framework to generalize the Goldberg-Sachs theorem. We work in complexified spaces, showing that the results for any signature can be obtained by taking the desired real slice.
23 pages; This version matches the published one
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Cited by in corpus (7)
- Integrability Conditions for Killing-Yano Tensors and Conformal Killing-Yano Tensors
- Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes
- On the Weyl Tensor Classification in All Dimensions and its Relation with Integrability Properties
- Killing Horizons and Spinors
- Conformally Invariant Spinorial Equations in Six Dimensions
- Killing Spinors and Related Symmetries in Six Dimensions
- On the Pursuit of Generalizations for the Petrov Classification and the Goldberg-Sachs Theorem