Half‐flat structures and special holonomy

By Vicente Cort'es, Thomas Leistner, Lars Schafer, Fabian Schulte-Hengesbach. 2009

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It was proved by Hitchin that any solution of his evolution equations for a half‐flat SU (3)‐structure on a compact six‐manifold M defines an extension of M to a seven‐manifold with holonomy in G2. We give a new proof, which does not require the compactness of M. More generally, we prove that the evolution of any half‐flat G‐structure on a six‐manifold M defines an extension of M to a Ricci‐flat seven‐manifold N, for any real form G of SL (3, ℂ). If G is non‐compact, then the holonomy group of N is a subgroup of the non‐compact form G2* of G2ℂ. Similar results are obtained for the extension of nearly half‐flat structures by nearly parallel G2‐ or G2*‐structures, as well as for the extension of cocalibrated G2‐ and G2*‐structures by parallel Spin (7)‐ and Spin 0(3, 4)‐structures, respectively. As an application, we obtain that any six‐dimensional homogeneous manifold with an invariant half‐flat structure admits a canonical extension to a seven‐manifold with a parallel G2‐ or G2*‐structure. For the group H3 × H3, where H3 is the three‐dimensional Heisenberg group, we describe all left‐invariant half‐flat structures and develop a method to explicitly determine the resulting parallel G2‐ or G2*‐structure without integrating. In particular, we construct three eight‐parameter families of metrics with holonomy equal to G2 and G2*. Moreover, we obtain a strong rigidity result for the metrics induced by a half‐flat structure (ω, ρ) on H3×H3 satisfying ω(z,z)=0 , where z denotes the centre. Finally, we describe the special geometry of the space of stable three‐forms satisfying a reality condition. Considering all possible reality conditions, we find four different special Kähler manifolds and one special para‐Kähler manifold.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/806056a46f9705c1a6d99e79ad0d67862f7a21c4