On the existence of generalized complex space forms (Q1121541)

Let \((M,g,J)\) be an almost Hermitian manifold such that its Riemannian curvature tensor has the expression \(R=f\pi_ 1+h\pi_ 2\) where \(f,h\in C^{\infty}\) and \[ \pi_ 1(X,Y)Z = g(X,Z)Y-g(Y,Z)X, \] \[ \pi_ 2(X,Y)Z = g(JX,Z)JY-g(JY,Z)JX+2g(JX,Y)JZ \] for arbitrary vector fields \(X\), \(Y\), \(Z\). Such expressions appear in a natural way in the works of \textit{F. Tricerri} and the reviewer [Trans. Am. Math. Soc. 267, 365-398 (1981; Zbl 0484.53014)] and the corresponding manifolds are called generalized complex space forms. Moreover, they proved that when h is not identically zero and \(\dim M\geq 6\), then the manifold is a usual complex space form. This result is also true for \(\dim M=4\) if in addition f and h are constant. For this dimension one generally has \(f+h=\)constant and \((M,g,J)\) is Hermitian on the open subset of \(M\) on which \(h\neq 0\). Finally they asked if there exist four- dimensional almost Hermitian manifolds with \(R=f\pi_ 1+h\pi_ 2\) where \(h\) is a nonconstant smooth function. In this paper the author answers this question positively by constructing examples via conformal deformations of Bochner flat manifolds with nonconstant scalar curvature which is nowhere zero. Moreover he proves that any generalized complex space form with nonconstant h which is nowhere zero can be obtained in this way.