A Riemannian metric is said to be Einstein if its Ricci curvature as a function on a unit tangent bundle is constant. A smooth manifold admitting such a metric is called an Einstein manifold. We refer to [7] for a general reference on Einstein metrics. As one of the milestones in the research of Einstein metrics, there are works of Aubin [1] and Yau [76] in the 1970s on the resolution of the Calabi Conjecture concerning the existence of K?hler-Einstein metrics. In Sugaku Expositions, most of the articles concerning Einstein metrics [3,22, 54,61,72] are also closely related with the Calabi Conjecture, and the research on K?hler-Einstein metrics is one of the central subjects in the research on Einstein metrics. However, if there are no special geometric structures such as K?hler structures, many aspects of the existence and nonexistence of Einstein metrics are still not well understood. Hence, a deep study in such a case is needed (cf. [77]). For instance, it is still an open question whether closed oriented smooth manifolds with dimension n≥5 can always admit Einstein metrics. In particular, it is still unknown whether there is an obstruction to the Einstein metrics in the case of dimension n≥5. However, in dimension n=4,it was already known[31,70] that there wass a topological obstruction to the existence of Einstein metrics about 40 years ago. By the obstruction, we are able to know that there exists many 4 manifolds without Einstein metrics.~1 But, of course, we need a ssmooth structure to introdice Riemannian metrics on manifolds. The classical obstruction[31,70] only gives rise to constranits on the topological structuress on Einstein 4 manifolds and hence it gives no information about smooth structures.
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