New examples of Riemannian manifolds with structure group \(G_ 2\) (Q1822068)

Let W be the class of 7-dimensional manifolds having a 2-fold vector cross product [the first author and \textit{A. Gray}, Ann. Mat. Pura Appl., IV. Ser. 132, 19-45 (1982; Zbl 0524.53023)]. Let W be the space of tensors having the same symmetries as the covariant derivative \(\nabla \phi\) of the fundamental form \(\phi\) of a manifold \(M\in W\). The authors study the represention of \(G_ 2\) on W by using the theory of root systems of semisimple Lie algebras. Moreover, they find new examples of non-compact and compact manifolds belonging to the class W.