Characteristic properties of almost Hermitian structures on reductive homogeneous spaces. (Q1889462)

Let \(G/H\) be a homogeneous reductive almost Hermitian space with invariant almost Hermitian structure \(J\). The reductive decomposition of the Lie algebra \(\mathfrak g\) of the group \(G\) has the form \(\mathfrak g=\mathfrak h\oplus\mathfrak m\), where \(\mathfrak h\) is the Lie algebra of the group \(H\). Let \(X_{\mathfrak h}\) and \(X_{\mathfrak m}\) denote the projections of the vector \(X\in \mathfrak g\) on the subspaces \(\mathfrak h\) and \(\mathfrak m\), respectively, with respect to the reductive decomposition \(\mathfrak g=\mathfrak h\oplus\mathfrak m\), and let \(J_0\) be the operator of the almost Hermitian structure \(J\) on \(\mathfrak m\). The author obtains criteria which allow him to give a simple description of the Kähler, nearly Kähler, and \(G_1\) structures on \(G/H\), provided that \([X,J_0X]_{\mathfrak m}=0\) for all \(X\in \mathfrak m\). Some conditions for \([X,J_0X]_{\mathfrak m}=0\) to be satisfied are studied as well as related examples.