Curvature decomposition of [formula omitted]-manifolds

Abstract

Explicit formulas for the G 2 -components of the Riemannian curvature tensor on a manifold with a G 2 -structure are given in terms of Ricci contractions. We define a conformally invariant Ricci-type tensor that determines the 27-dimensional part of the Weyl tensor and show that its vanishing on compact G 2 -manifold with closed fundamental form forces the three-form to be parallel. A topological obstruction for the existence of a G 2 -structure with closed fundamental form is obtained in terms of the integral norms of the curvature components. We produce integral inequalities for closed G 2 -manifold and investigate limiting cases. We make a study of warped products and cohomogeneity-one G 2 -manifolds. As a consequence every Fernandez–Gray type of G 2 -structure whose scalar curvature vanishes may be realized such that the metric has holonomy contained in G 2 .