A fundamental differential system of Riemannian geometry (Q2280530)

A system of differential forms of degree \(n\) for any oriented Riemannian manifold \(M^{n+1}\) is introduced. The Cartan structural equations are deduced in all dimensions. The author shows that this system of differential forms is related to a special Riemannian geometry of \(G_2\)-twistor space associated with an oriented Riemannian manifold \(M^4\). A new criterion in term of the induced system of differential forms for an oriented Riemannian manifold to be Einstein is established. A new proof of the Hsiung-Minkowski identities in Euclidean space is also given.