Real hypersurfaces with parallel Ricci tensor of a complex space form
Let \(M\) be a real hypersurface of the complex space form \(M_n(c)\) such that the immersion induces on \(M\) a metric contact structure \((J,g,P)\). It is known that there does not exist such a hypersurface \(M\) of \(P_n\mathbb{C}\) with parallel Ricci tensor on which the structure vector field \(P\) is principal, and that there exists no Einstein hypersurface in \(H_n\mathbb{C}\). Now, one says that the Riemannian curvature tensor is harmonic if its Ricci tensor is of Codazzi type. Then it is also known that there does not exist \(M\) with harmonic curvature in \(M_n(c).\) In this paper the author tries to improve these scattered theorems, and presents the more uniformized theorems as follows. There does not exist any \(M\) with parallel Ricci tensor of \(M_n(c)\). Also, there exists no \(M\) with harmonic curvature of \(M_n(c)\) satisfying one of the following conditions: (1) \(P\) is an eigenvector corresponding to the Ricci tensor, (2) the number of Ricci curvatures does not exceed 2.
scientific article; zbMATH DE number 4111505