On non-Hopf Ricci-pseudosymmetric hypersurfaces in C P 2 $\mathbb {C}P^{2}$ and C H 2 $\mathbb {C}H^{2}$

Abstract

In this paper, we study an open problem raised by Cecil and Ryan [Geometry of Hypersurfaces, Springer Monographs in Mathematics, p. 531] which asked whether there exist non-Hopf Ricci-pseudosymmetric hypersurfaces in $C{P}^2$ and $C{H}^2$. As our main results, we first prove the nonexistence of non-Hopf Ricci-pseudosymmetric hypersurfaces of the constant type in $C{H}^2$. Then, we prove the existence of non-Hopf Ricci-pseudosymmetric hypersurfaces of the constant type in $C{P}^2$. Finally, applying the preceding results and sharpening Theorem 4.1 of Wang and Zhang [J. Geom. Phys. 181 (2022), 104648], we prove the nonexistence of non-Hopf weakly Einstein hypersurfaces with constant norm of Riemannian curvature tensor in both $C{P}^2$ and $C{H}^2$.