On non‐Hopf Ricci‐pseudosymmetric hypersurfaces in CP2$\mathbb {C}P^{2}$ and CH2$\mathbb {C}H^{2}$

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 and . As our main results, we first prove the nonexistence of non‐Hopf Ricci‐pseudosymmetric hypersurfaces of the constant type in . Then, we prove the existence of non‐Hopf Ricci‐pseudosymmetric hypersurfaces of the constant type in . 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 and .