Half-flat structures and special holonomy

Abstract

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, ). If G is non-compact, then the holonomy group of N is a subgroup of the non-compact form G2* of G2c2. 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, where 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. © 2010 London Mathematical Society.