Ricci pseudo \(\eta\)-parallel real hypersurfaces of a complex space form (Q372581)

Recall that a complex space form \(M^n(c)\) with constant holomorphic sectional curvature \(4c\) is the complex Euclidean space with Euclidean metric if \(c=0\), the complex the projective space with Fubini-Study metric if \(c>0\) and the complex hyperbolic space with Poincaré metric if \(c<0\). Here, \(n\) stands for the complex dimension. It is known that complex space forms with \(c\not= 0\) cannot contain real hypersurfaces with parallel Ricci tensor. In the literature several notions of semi-parallelity or pseudo-parallelity can be found. The author of the present paper introduces a notion which he calls pseudo \(\eta\)-parallelity for the Ricci tensor and proves a criterion for the pseudo \(\eta\)-parallelity of the Ricci tensor of a real hypersurface in a complex space form \(M^n(c)\) with \(c\not=0\) and \(n\geq 3\). The criterion is the following: The Ricci tensor \(S\) of a real hypersurface \(M\) is pseudo \(\eta\)-parallel if and only if \(M\) is pseudo-Einstein.