Generalized Einstein conditions on holomorphic Riemannian manifolds

Summary At first, a necessary and sufficient condition for a K?hler-Norden manifold to be holomorphic Einstein is found. Next, it is shown that the so-called (real) generalized Einstein conditions for K?hler-Norden manifolds are not essential since the scalarcurvature of such manifolds is constant. In this context, we study generalized holomorphic Einstein conditions. Using the one-to-one correspondence between K?hler-Norden structures and holomorphic Riemannian metrics, we establish necessary and sufficient conditions for K?hler-Norden manifolds to satisfy the generalized holomorphic Einstein conditions. And a class of new examples of such manifolds is presented. Finally, in virtue of the obtained results, we mention that Theorems 1 and 2 of H. Kim and J. Kim [10] are not true in general.