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arXiv · 1212.2689

Spinors and the Weyl Tensor Classification in Six Dimensions

Carlos Batista·Bruno Carneiro da Cunha
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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.

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Carlos Batista, Bruno Carneiro da Cunha. 2012-12-12. Spinors and the Weyl Tensor Classification in Six Dimensions. https://doi.org/10.1063/1.4804991

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