SU(3)-holonomy metrics from nilpotent Lie groups

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.

On the boundary behavior of left-invariant Hitchin and hypo flows

“…In this section, we prove that for suitable classes of split-solvable Lie groups G any proper cohomogeneity-one action of G on a, not necessarily complete, Riemannian manifold M preserving a parallel SU(3)-, G 2 -or Spin (7)-structure on M , respectively, has only regular orbits. In the SU(3)-case, our result generalizes [8,Theorem 25].…”

“…Thanks to Lemma 2.11, we only have to do the proof for H = Spin (7). For that we combine the arguments of [29] and [8]. Recall that for any G 2 -structure ϕ ∈ Ω 3 N on a seven-dimensional manifold N there exists T ∈ End(T M ), called the intrinsic torsion of ϕ, such that ∇ g X ϕ = −T (X) ⋆ ϕ ϕ for any vector field X ∈ X(N ), cf.…”

“…Under mild assumptions, the cohomogeneity-one action extends automatically to any potential completion of M × I, see Proposition 3.4. For initial data which is homogeneous under a compact Lie group these flows and the above extension problem were studied extensively in the literature, see, e.g., [2], [6], [23], [27], [28]. In this paper we focus on the case of left-invariant initial data on a possibly non-compact Lie group, a setting which was previously considered in [5], [8] and [10]. It was shown by Conti [8,Section 8] that this problem has only trivial solutions for the hypo flow on nilpotent Lie groups, in the sense that the resulting manifold is automatically flat.…”

“…In this paper we focus on the case of left-invariant initial data on a possibly non-compact Lie group, a setting which was previously considered in [5], [8] and [10]. It was shown by Conti [8,Section 8] that this problem has only trivial solutions for the hypo flow on nilpotent Lie groups, in the sense that the resulting manifold is automatically flat. His proof uses his classification of hypo SU(2)-structures on 5-dimensional nilpotent Lie groups.…”

Half-flat structures on $$ S^3\times S^3$$

“…The author of [15] uses Lie algebra degenerations to study invariant hypo SU(2)-structures on 5-dimensional nilmanifolds. In a similar way, one could study half-flat structures on the various group contractions of…”