  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.
