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

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  • Qianshun Cui
  • Zejun Hu

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 CP2$\mathbb {C}P^{2}$ and CH2$\mathbb {C}H^{2}$. As our main results, we first prove the nonexistence of non‐Hopf Ricci‐pseudosymmetric hypersurfaces of the constant type in CH2$\mathbb {C}H^{2}$. Then, we prove the existence of non‐Hopf Ricci‐pseudosymmetric hypersurfaces of the constant type in CP2$\mathbb {C}P^{2}$. 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 CP2$\mathbb {C}P^{2}$ and CH2$\mathbb {C}H^{2}$.

Suggested Citation

  • Qianshun Cui & Zejun Hu, 2025. "On non‐Hopf Ricci‐pseudosymmetric hypersurfaces in CP2$\mathbb {C}P^{2}$ and CH2$\mathbb {C}H^{2}$," Mathematische Nachrichten, Wiley Blackwell, vol. 298(2), pages 527-547, February.
    • Handle: RePEc:bla:mathna:v:298:y:2025:i:2:p:527-547
    DOI: 10.1002/mana.202300463
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