On the Existence of \(\eta \) -Einstein Contact Metric Structures

The goal of this paper is two-fold: firstly, we give a necessary and sufficient condition for the existence of a transverse Einstein metric on a given co-oriented contact manifold \((M , \eta )\) . Secondly, we apply this result to obtain important consequences. In particular, we give another proof of a theorem by Boyer and Galicki (Sasakian Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2008). xii+613pp.) on solutions to Goldberg’s conjecture for \(\eta \) -Einstein K-contact manifolds. An example illustrating our construction result is provided.