Chern-flat and Ricci-flat invariant almost Hermitian structures (Q540461)

Let \((M, g, J)\) be an almost Hermitian manifold and \(\nabla \) its Chern connection, which in the quasi-Kähler case coincides with the so-called second canonical Hermitian connection. Let \(R\) be the curvature tensor and \(\text{Ric}(J)(X, Y):=\frac{1}{2}Tr_{\omega }R(X, Y, \cdot, \cdot )\) be the Ricci form of \(\nabla \), where \(\omega \) is the fundamental form associated to \((g, J)\). Then \((M, g, J)\) is called: (1) Ricci-flat if \(\text{Ric}(J)=0\); (2) Chern-flat if \(R=0\). This very interesting paper is devoted to left-invariant almost Hermitian structures on homogeneous spaces which are either Ricci-flat or Chern-flat. In dimension 4 and 6, a classification is obtained while for the higher dimensional cases several examples are given.