{"title":"On non-Hopf Ricci-pseudosymmetric hypersurfaces in \u0000 \u0000 \u0000 C\u0000 \u0000 P\u0000 2\u0000 \u0000 \u0000 $mathbb {C}P^{2}$\u0000 and \u0000 \u0000 \u0000 C\u0000 \u0000 H\u0000 2\u0000 \u0000 \u0000 $mathbb {C}H^{2}$","authors":"Qianshun Cui, Zejun Hu","doi":"10.1002/mana.202300463","DOIUrl":"https://doi.org/10.1002/mana.202300463","url":null,"abstract":"<p>In this paper, we study an open problem raised by Cecil and Ryan [<i>Geometry of Hypersurfaces</i>, Springer Monographs in Mathematics, p. 531] which asked whether there exist non-Hopf Ricci-pseudosymmetric hypersurfaces in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}P^{2}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}H^{2}$</annotation>\u0000 </semantics></math>. As our main results, we first prove the nonexistence of non-Hopf Ricci-pseudosymmetric hypersurfaces of the constant type in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}H^{2}$</annotation>\u0000 </semantics></math>. Then, we prove the existence of non-Hopf Ricci-pseudosymmetric hypersurfaces of the constant type in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}P^{2}$</annotation>\u0000 </semantics></math>. Finally, applying the preceding results and sharpening Theorem 4.1 of Wang and Zhang [<i>J. Geom. Phys</i>. <b>181</b> (2022), 104648], we prove the nonexistence of non-Hopf weakly Einstein hypersurfaces with constant norm of Riemannian curvature tensor in both <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}P^{2}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {C}H^{2}$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"527-547"},"PeriodicalIF":0.8,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}