Real hypersurfaces in complex space forms with \(\eta\)-recurrent Ricci tensor (Q2758279)

Let \(M_n(c)\), \(c \neq 0\), be a complex space form. The induced almost contact metric structure on a real hypersurface \(M\) of \(M_n(c)\) is usually denoted by \((\phi, \xi, \eta, g)\). Following the definition of \textit{recurrent} tensor field given in [\textit{S. Kobayashi} and \textit{K. Nomizu}, Foundations of Differential Geometry. I., Wiley, New York (1963; Zbl 0119.37502)], the authors say that a real hypersurface \(M\) of \(M_n(c)\) has \(\eta\)-\textit{recurrent} Ricci tensor if and only if there exists a 1-form \(\alpha\) such that \(g((\nabla_X S)Y,Z)=\alpha(X)g(SY,Z)\), \(S\) being the Ricci tensor of \(M\). In this paper, the authors classify such kinds of real hypersurfaces in both complex projective and complex hyperbolic spaces with the additional hypothesis that the structure vector field \(\xi\) is principal. In the first case, \(M\) must also be of constant mean curvature, though the authors remark that this assumption could be deleted.