Exceptional Holonomy on Vector Bundles with Two-Dimensional Fibers

Abstract

An SU(3)- or SU(1,2)-structure on a 6-dimensional manifold N 6 can be defined as a pair of a 2-form ω and a 3-form ρ. We prove that any analytic SU(3)- or SU(1,2)-structure on N 6 with dω∧ω=0 can be extended to a parallel $\operatorname{Spin}(7)$ - or $\operatorname{Spin}_{0}(3,4)$ -structure Φ that is defined on the trivial disc bundle N 6×B ϵ (0) for a sufficiently small ϵ>0. Furthermore, we show by an example that Φ is not uniquely determined by (ω,ρ) and discuss if our result can be generalized to non-trivial bundles.