Einstein manifolds and contact geometry (Q2719018)

The Goldberg conjecture states that a compact almost Kähler-Einstein manifold is Kähler-Einstein. This conjecture was proved by \textit{K. Sekigawa} [J. Math. Soc. Japan 39, 677-684 (1987; Zbl 0637.53053)] in the case of nonnegative scalar curvature. Now, let \((M,\eta,\xi,\varphi,g)\) be a contact metric manifold and \((M,\eta,J)\) the corresponding strongly pseudoconvex CR manifold. \(M\) is a \(K\)-contact manifold if the Reeb vector field \(\xi\) is Killing with respect to the metric \(g\). \(M\) is a Sasakian manifold if it is \(K\)-contact and the CR structure \((\eta,J)\) is integrable. Hence, the \(K\)-contact notion is intermediate between the notions of contact metric structure and Sasakian structure. If the Reeb vector field \(\xi\) of a \(K\)-contact manifold is quasi-regular, under a compactness assumption there is an orbifold fibration over a compact almost Kähler orbifold. Since \(K\)-contact Einstein manifolds have positive scalar curvature, it is natural to believe that an odd dimensional Goldberg conjecture is true. In fact, in this paper the authors obtain the following important theorem: every compact \(K\)-contact Einstein manifold is Sasakian-Einstein. NEWLINENEWLINENEWLINEThe proof of this result is divided in two part. In the first part, the authors consider the assumption that the contact form \(\eta\) is quasi-regular; in the second part, they consider the case where \(\eta\) is not quasi-regular and construct a sequence of quasi-regular \(K\)-contact structures \((\eta_j, \xi_j,\varphi_j, g_j)\) on \(M\) whose limit with respect to the compact-open topology is the original \(K\)-contact Einstein structure \((\eta,\xi, \varphi, g)\). NEWLINENEWLINENEWLINEThe authors conclude the paper discussing some interesting consequences of the main theorem to almost Kähler structures on cones, and to some related work on \(\eta\)-Einstein \(K\)-contact manifolds.