SU(3)-holonomy metrics from nilpotent Lie groups

One way of producing explicit Riemannian 6-manifolds with holonomy SU(3) is by integrating a flow of SU(2)-structures on a 5-manifold, called the hypo evolution flow. In this paper we classify invariant hypo SU(2)-structures on nilpotent 5-dimensional Lie groups. We characterize the hypo evolution flow in terms of gauge transformations, and study the flow induced on the variety of frames on a Lie algebra taken up to automorphisms. We classify the orbits of this flow for all hypo nilpotent structures, obtaining several families of cohomogeneity one metrics with holonomy contained in SU(3). We prove that these metrics cannot be extended to a complete metric, unless they are flat.

Keywords

SU(3) holonomy, nilmanifold, cohomogeneity one

2010 Mathematics Subject Classification

Primary 53C25. Secondary 17B30, 53C29, 53C42.

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Published 13 May 2014