Curvature decomposition of G2G2-manifolds

Explicit formulas for the G2G2-components of the Riemannian curvature tensor on a manifold with a G2G2-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 G2G2-manifold with closed fundamental form forces the three-form to be parallel. A topological obstruction for the existence of a G2G2-structure with closed fundamental form is obtained in terms of the integral norms of the curvature components. We produce integral inequalities for closed G2G2-manifold and investigate limiting cases. We make a study of warped products and cohomogeneity-one G2G2-manifolds. As a consequence every Fernandez–Gray type of G2G2-structure whose scalar curvature vanishes may be realized such that the metric has holonomy contained in G2G2.