A generalization of the Goldberg conjecture for coKähler manifolds
A celebrated conjecture in Kähler geometry is the Goldberg conjecture: any compact Einstein almost Kähler manifold is Kähler. The best result for proving Goldberg's conjecture was obtained by Sekigawa who proved that any compact Einstein almost Kähler manifold with non-negative scalar curvature is Kähler. Since Sasakian manifolds are the odd-dimensional analogues of Kähler manifolds, \textit{C. P. Boyer} and \textit{K. Galicki} [Proc. Am. Math. Soc. 129, No. 8, 2419--2430 (2001; Zbl 0981.53027)] first considered Goldberg's conjecture in the framework of contact metric geometry and proved that any compact Einstein K-contact manifold is Sasakian. However, co-Kähler manifolds can also be considered as the odd-dimensional version of Kähler manifolds. The aim of the paper under review is just to study the existence of a Goldberg-like conjecture in the framework of co-Kähler geometry.
scientific article; zbMATH DE number 6661796
scientific article