Introduction

Abstract

Following Hitchin [76], a k-form φ on a differentiable manifold M is called stable if the orbit of φ( p) under GL(T _p M) is open in $\Lambda ^{k}T_{p}^{{\ast}}M$ for all p ∈ M. In this text we are mainly concerned with six-dimensional manifolds M endowed with a stable two-form ω and a stable three-form ρ. A stable three-form defines an endomorphism field J _ρ on M such that J _ρ ² = ɛid, see (2.6). We will assume the following algebraic compatibility equations between ω and ρ:

$$\displaystyle{\omega \wedge \rho = 0,\quad J_{\rho }^{{\ast}}\rho \wedge \rho = \frac{2} {3}\omega ^{3}.}$$

The pair (ω, ρ) defines an SU( p, q)-structure if ɛ = −1 and an $\mathrm{SL}(3, \mathbb{R})$-structure if ɛ = +1. In the former case, the pseudo-Riemannian metric ω( J _ρ ⋅ , ⋅ ) has signature (2p, 2q). In the latter case it has signature (3, 3). The structure is called half-flat if the pair (ω, ρ) satisfies the following exterior differential system:

$$\displaystyle{d\omega ^{2} = 0,\quad d\rho = 0.}$$

In [76], Hitchin introduced the following evolution equations for a time-dependent pair of stable forms (ω(t), ρ(t)) evolving from a half-flat SU(3)-structure (ω(0), ρ(0)):

$$\displaystyle{ \frac{\partial } {\partial t}\rho = d\omega,\quad \frac{\partial } {\partial t}\hat{\omega } = d\hat{\rho },}$$

where $\hat{\omega }= \frac{\omega ^{2}} {2}$ and $\hat{\rho }= J_{\rho }^{{\ast}}\rho$. For compact manifolds M, he showed that these equations are the flow equations of a certain Hamiltonian system and that any solution defined on some interval $0 \in I \subset \mathbb{R}$ defines a Riemannian metric on M × I with holonomy group in G₂. We give a new proof of this theorem, which does not use the Hamiltonian system and does not assume that M is compact. Moreover, our proof yields a similar result for all three types of half-flat G-structures: G = SU(3), SU(1, 2) and $\mathrm{SL}(3, \mathbb{R})$. For the noncompact groups G we obtain a pseudo-Riemannian metric of signature (3, 4) and holonomy group in G₂ ^∗ on M × I (see Theorem 4.1.3 of Chap. 4).