A new class of almost complex structures (Q1913392)

An almost Hermitian manifold \((M,g,J)\) such that \[ R(JX,JY,Z,W) + R(JY, JZ, X,W) + R(JZ,JX,Y,W)=0, \] where \(R\) denotes the Riemannian curvature of \(g\), is called almost Hermitian \(C\)-manifold. The main results of this paper are: (1) an analogue of the Schur lemma for the holomorphic sectional curvature of an almost Hermitian \(C\)-manifold is proved; and (2) the following theorem is proved: ``Let \((M,g,J)\) be an almost Hermitian \(C\)-manifold with nonzero pointwise constant holomorphic sectional curvature such that \[ \sum_i \biggl\{R \bigl(X,Y, (\nabla_ZJ) e_i,e_i \bigr) + R \bigl(Y,Z, (\nabla_XJ) e_i,e _i \bigr) + R \bigl(Z, X, (\nabla_YJ) e_i,e_i \bigr) \biggr\} = 0\tag{*} \] is satisfied (here \(\nabla\) is the Levi-Civita connection and \(\{e_i\}\) is a local orthonormal frame on \(M)\). Then \((M,g,J)\) is Kählerian.'' No example of an almost Hermitian non-Kähler manifold which satisfies (*) is given.