September 10, 2017
TITLE: SO(4)-structures on 7-manifolds

ABSTRACT: I will talk about the geometry of SO(4)-structures on 7-manifolds, under the restriction that the SO(4)-structure induces a metric with holonomy contained in . This amounts to a condition on the torsion of the SO(4)-structure and I will describe what happens in various cases where the torsion is restricted further. Part of this description gives a characterisation of the Bryant-Salamon examples as the unique examples with torsion lying in particular subspaces. If a -manifold is foliated by associative or coassociative submanifolds, then it carries a naturally defined SO(4)-structure. I will give an interpretation of a result of Baraglia about ‘semi-flat’ coassociative fibrations in this language, and talk about the case of ‘semi-flat’ associative fibrations.