Half-flat structures and special holonomy

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 G(2). 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 G(2)* of G(2)(C). Similar results are obtained for the extension of nearly half-flat structures by nearly parallel G(2)- or G(2)*-structures, as well as for the extension of cocalibrated G(2)- and G(2)*-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 G(2)- or G(2)*-structure. For the group H-3 x H-3, where H-3 is the three-dimensional Heisenberg group, we describe all left-invariant half-flat structures and develop a method to explicitly determine the resulting parallel G(2)- or G(2)*-structure without integrating. In particular, we construct three eight-parameter families of metrics with holonomy equal to G(2) and G(2)*. Moreover, we obtain a strong rigidity result for the metrics induced by a half-flat structure (omega, rho) on H(3)xH(3) satisfying omega(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 Kahler manifolds and one special para-Kahler manifold.