SU(3)-holonomy metrics from nilpotent Lie groups

TL;DR

This paper classifies invariant hypo SU(2)-structures on nilpotent 5D Lie groups and constructs explicit SU(3)-holonomy metrics, revealing limitations on their completeness.

Contribution

It provides a classification of hypo SU(2)-structures on nilpotent Lie groups and analyzes the flow leading to SU(3)-holonomy metrics, including their geometric properties.

Findings

Classified all invariant hypo SU(2)-structures on nilpotent 5D Lie groups.

Constructed families of cohomogeneity one metrics with SU(3) holonomy.

Proved these metrics cannot be extended to complete metrics unless flat.

Abstract

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.