On some compact almost Kähler locally symmetric space
An almost Hermitian manifold \((M^{2n},g,J)\) is said to be almost Kählerian if its Kähler form \(\Omega\) is closed. S. Goldberg conjectured that a compact almost Kähler Einstein space is necessarily Kählerian, i.e., the almost complex structure is integrable. K. Sekigawa showed that this conjecture holds for non-negative scalar curvature. In this paper the author considers a \(*\)-version of this result and proves: a compact almost Kähler weakly \(*\)-Einstein space which is locally symmetric and has non-negative \(*\)-scalar curvature is Kählerian. \((M,g,J)\) is said to be a weakly \(*\)-Einstein space if the Ricci \(*\)-tensor \(\rho^*\) satisfies \(2n\rho^* = \tau^*g\) where \(\tau^*\) is the \(*\)-scalar curvature and \(\rho^*\) is defined by \(\rho^*(x,y)=\sum^{2n}_{i=1} R(x,e_i,Jy,Je_i)\) for an arbitrary orthonormal basis \(\{e_i, i=1,\dots,2n\}\) of the tangent space at each point.
scientific article; zbMATH DE number 1127741