Orientable hypersurfaces of Riemannian manifolds with Spin (7) structure
A Spin(7) structure is given by an \(8\)-dimensional Riemannian manifold \(\overline M\) and a nowhere vanishing \(4\)-form \(\overline \varphi\) on \(\overline M\). An orientable manifold \(M\) with a \(G_2\)-structure (the exceptional Lie group of authomorphisms of the Cayley numbers \(O\)) is \(7\)-dimensional, it has an induced Riemannian metric and there exists on it a nowhere vanishing \(3\)-form \(\varphi\). In the present paper \(\overline M\) is an ambient space of \(M\). Then the influence of \(\overline M\) on the type of the \(G_2\)-structure induced on \(M\) is studied and some types of these induced \(G_2\)-structures are characterized by means of the geometrical conditions satisfied by \(M\). One of these cases has been considered by A. Gray. Finally, some interesting examples are given.
scientific article; zbMATH DE number 1225539