On \(4\)-dimensional almost Hermitian manifolds with pointwise constant holomorphic sectional curvature (Q2782766)

Let \((M,g,J)\) be an almost Hermitian manifold. Denote by \({\mathcal {AH}}_2\) the class of such manifolds whose curvature tensor \(R\) satisfies the identity \(R(X,Y,Z,W)= R(JX, JY, Z,W)+ R(JX, Y, JZ, W)+ R(JX, Y,Z, JW)\) and by \({\mathcal {AH}}_3\) that where the identity \(R(X,Y,Z,W)= R(JX,JY,JZ,JW)\) holds. Note that \({\mathcal {AH}}_2\subset {\mathcal {AH}}_3\) and that on a Hermitian manifold, we have \({\mathcal H}_2= {\mathcal H}_3\). NEWLINENEWLINENEWLINEIn this paper, the authors prove, by giving an explicit expression for \(R\), that a 4-dimensional manifold of the class \({\mathcal {AH}}_2\) which has pointwise constant holomorphic sectional curvature is a generalized complex space form [see \textit{F. Tricerri} and \textit{L. Vanhecke}, Trans. Am . Math. Soc. 267, 365-398 (1981; Zbl 0484.53014)]. They also prove that if a 4-dimensional manifold \((M,g,J)\) has pointwise constant holomorphic sectional curvature, then \(M\in {\mathcal {AH}}_3\) if and only if the manifold is Einsteinian and weakly \(*\)-Einsteinian, i.e., the Ricci tensor \(\rho\) and the Ricci \(*\)-tensor \(\rho^*\) satisfy \(\rho= \lambda g\), \(\rho^* =\mu g\), respectively. Since \({\mathcal H}_2= {\mathcal H}_3\), it then follows that a 4-dimensional Hermitian manifold of pointwise constant holomorphic sectional curvature which is also an Einstein and weakly \(*\)-Einstein space, is a generalized complex space form.