On \(G_ 2\)-manifolds
Let \((M,J,g)\) be a \((2m)\)-dimensional almost Hermitian manifold. Let \(\nabla\) denote the natural metric connection on \(M\) and \(R(X,Y)Z\) the curvature operator associated with \(\nabla\). The fundamental 2-form of \(M\) is \(F(X,Y)= g(\overline X, Y)\) for all vector fields \(X\) and \(Y\), where \(\overline X= JX\). An almost Hermitian manifold \((M,J,g)\) is said to be a \(G_ 2\)-manifold if and only if, for all vector fields \(X\), \(Y\) and \(Z\) on \(M\), holds \[ \underset{X,Y,Z} \sigma(\nabla_ X F)(Y,Z)= \underset{X,Y,Z} \sigma(\nabla_{\overline X} F)(\overline Y,Z), \] where by \(\sigma\) denote the cyclic sum [see also \textit{L. Hervella} and \textit{E. Vidal}, Nouvelles géométries pseudo-kählériennes \(G_ 1\) et \(G_ 2\), C. R. Acad. Sci., Paris, Ser. A 283, 115-118 (1976; Zbl 0331.53026)]. The authors study several properties of a \(G_ 2\)-manifold concerning with the curvature identities, the Chern classes and some results of type ``Schur's theorem''.
scientific article; zbMATH DE number 605488