Nearly parallel G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{G}_{2}$$\end{document}-structures with torus

We study nearly parallel G 2-structures with a three-torus symmetry via multi-moment map techniques. An effective three-torus action on a nearly parallel G 2-manifold yields a multi- moment map. The torus acts freely on its regular level sets, so they are torus bundles over smooth three-dimensional manifolds. We show that the geometry of the base spaces is speci- fied by two triples of closed two-forms related by a Riemannian metric. We then describe an inverse construction producing invariant nearly parallel G 2-structures from three-dimensional data. We observe that locally this may produce examples with four-torus symmetry. Mathematics Subject Classification Primary 53C25; Secondary 53C26 · 53D20 · 57S25