Lifting SU(3)-structures to nearly parallel G_{2}-structures

TL;DR

This paper extends Hitchin's method of lifting half-flat SU(3)-structures to construct nearly parallel G2-structures on 7-manifolds using nearly half-flat structures characterized by specific differential form conditions.

Contribution

It introduces a new approach to generate nearly parallel G2-structures by lifting nearly half-flat SU(3)-structures, broadening the scope of Hitchin's original method.

Findings

Constructs nearly parallel G2-structures from nearly half-flat SU(3)-structures.

Defines nearly half-flat structures via stable forms satisfying a specific differential equation.

Demonstrates the applicability of Hitchin's approach to a new class of G2-structures.

Abstract

Hitchin shows that half-flat SU(3)-structures on a 6-dimensional manifold M can be lifted to parallel G_{2}-structure on the product $M\times\mathbb{R}$. We show that Hitchin's approach can also be used to construct nearly parallel G_{2}-structures by lifting so-called nearly half-flat structures. These SU(3)-structures are described by pairs $(\omega,\phi)$ of stable 2- and 3-forms with $d\phi=\lambda\omega^{2}$ for some $\lambda\in\mathbb{R}\setminus\{0\}$.