{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T15:48:25Z","timestamp":1765295305306,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,14]],"date-time":"2022-05-14T00:00:00Z","timestamp":1652486400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007446","name":"Deanship of Scientific Research at King Khalid University, Saudi Arabia","doi-asserted-by":"publisher","award":["R. G. P. 2\/130\/43","PNURSP2022R27"],"award-info":[{"award-number":["R. G. P. 2\/130\/43","PNURSP2022R27"]}],"id":[{"id":"10.13039\/501100007446","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","doi-asserted-by":"publisher","award":["R. G. P. 2\/130\/43","PNURSP2022R27"],"award-info":[{"award-number":["R. G. P. 2\/130\/43","PNURSP2022R27"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we study the properties of PR-pseudo-slant submanifold of para-Kenmotsu manifold and obtain the integrability conditions for the slant distribution and anti-invariant distribution of such submanifold. We derived the necessary and sufficient conditions for a PR-pseudo-slant submanifold of para-Kenmotsu manifold to be a PR-pseudo-slant warped product which are in terms of warping functions and shape operator. Some examples of PR-pseudo-slant warped products of para-Kenmotsu manifold are also illustrated in the article.<\/jats:p>","DOI":"10.3390\/sym14051001","type":"journal-article","created":{"date-parts":[[2022,5,15]],"date-time":"2022-05-15T09:48:22Z","timestamp":1652608102000},"page":"1001","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Characterizations of PR-Pseudo-Slant Warped Product Submanifold of Para-Kenmotsu Manifold with Slant Base"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5992-1902","authenticated-orcid":false,"given":"Sachin Kumar","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176215, Himachal Pradesh, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2116-7382","authenticated-orcid":false,"given":"Fatemah","family":"Mofarreh","sequence":"additional","affiliation":[{"name":"Mathematical Science Department, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11546, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anuj","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Central University of Himachal Pradesh, Dharamshala 176215, Himachal Pradesh, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6053-3031","authenticated-orcid":false,"given":"Akram","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1017\/S0004972700017925","article-title":"Slant immersions","volume":"41","author":"Chen","year":"1990","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_2","unstructured":"Carriazo, A. (2000, January 20\u201322). Bi-slant immersions. Proceedings of the ICARAMS 2000, Kharagpur, India."},{"key":"ref_3","unstructured":"Carriazo, A. (2002). New developments in slant submanifolds theory. Applicable Mathematics in the Golden Age, Narosa Publishing House."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s00025-009-0386-9","article-title":"Classification of Quasi-Minimal Surfaces with Parallel Mean Curvature Vector in Pseudo-Euclidean 4-Space E24","volume":"55","author":"Chen","year":"2009","journal-title":"Results Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s10474-008-8033-6","article-title":"Classification of quasi-minimal slant surfaces in Lorentzian complex space forms","volume":"122","author":"Chen","year":"2009","journal-title":"Acta Math. Hung."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"561","DOI":"10.11650\/twjm\/1500405807","article-title":"On slant submanifolds of neutral Kaehler manifolds","volume":"14","author":"Arslan","year":"2010","journal-title":"Taiwan. J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.difgeo.2017.02.002","article-title":"Slant submanifolds in neutral almost contact pseudo-metric manifolds","volume":"54","author":"Carriazo","year":"2017","journal-title":"Differ. Geom. Its Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0002-9947-1969-0251664-4","article-title":"Manifolds of negative curvature","volume":"145","author":"Bishop","year":"1969","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_9","first-page":"149","article-title":"On motions in semi-reducible Riemann space","volume":"12","author":"Kruchkovich","year":"1957","journal-title":"Uspekhi Mat. Nauk."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Chen, B.Y. (2017). Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific.","DOI":"10.1142\/10419"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"20","DOI":"10.2307\/1969989","article-title":"The imbedding problem for Riemannian manifolds","volume":"63","author":"Nash","year":"1956","journal-title":"Ann. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/s006050170019","article-title":"Geometry of warped product CR-submanifolds in Kaehler manifolds","volume":"133","author":"Chen","year":"2001","journal-title":"Monatshefte F\u00fcR Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s006050170002","article-title":"Geometry of warped product CR-submanifolds in Kaehler manifolds, II","volume":"134","author":"Chen","year":"2001","journal-title":"Monatshefte F\u00fcR Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Alkhaldi, A.H., and Ali, A. (2019). Classification of Warped Product Submanifolds in Kenmotsu Space Forms Admitting Gradient Ricci Solitons. Mathematics, 7.","DOI":"10.3390\/math7020112"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ali, A., and Alkhaldi, A.H. (2019). Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications. Symmetry, 11.","DOI":"10.3390\/sym11020200"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ali, A., and Mofarreh, F. (2020). Geometric inequalities of bi-warped product submanifolds of nearly Kenmotsu manifolds and their applications. Mathematics, 8.","DOI":"10.3390\/math8101805"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1007\/s00009-016-0832-3","article-title":"Pointwise pseudo-slant warped product submanifolds in a Ka\u00a8hler manifold","volume":"14","author":"Srivastava","year":"2017","journal-title":"Mediterr. J. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1950031","DOI":"10.1142\/S0219887819500312","article-title":"Geometric classification of warped product submanifolds of nearly Kaehler manifolds with a slant fiber","volume":"16","author":"Ali","year":"2019","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3521","DOI":"10.2298\/FIL1911521B","article-title":"Chen\u2019s type inequality forwarped product pseudo-slant submanifolds of Kenmotsu f-manifolds","volume":"33","author":"Balkan","year":"2019","journal-title":"Filomat"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1186\/s13660-015-0825-y","article-title":"An inequality for warped product pseudo-slant submanifolds of nearly cosymplectic manifolds","volume":"2015","year":"2015","journal-title":"J. Inequalities Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1186\/s13660-015-0802-5","article-title":"Some inequalities for warped product pseudo-slant submanifolds of nearly Kenmotsu manifolds","volume":"2015","author":"Ali","year":"2015","journal-title":"J. Inequalities Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5833","DOI":"10.2298\/FIL1718833K","article-title":"Pointwise pseudo-slant submanifolds of a Kenmotsu manifold","volume":"31","author":"Khan","year":"2017","journal-title":"Filomat"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2115","DOI":"10.2298\/FIL1806115N","article-title":"Warped product submanifolds of Kenmotsu manifolds with slant fiber","volume":"32","author":"Naghi","year":"2018","journal-title":"Filomat"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"373","DOI":"10.2989\/16073606.2018.1452800","article-title":"Geometry of warped product pseudo-slant submanifolds of Kenmotsu manifolds","volume":"42","author":"Naghi","year":"2019","journal-title":"Quaest. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.11650\/twjm\/1500406736","article-title":"Geometry of PR-warped products in para-Ka\u00a8hler manifolds","volume":"16","author":"Chen","year":"2012","journal-title":"Taiwan. J. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s00022-016-0325-3","article-title":"Geometry of PR-semi-invariant warped product submanifolds in paracosymplectic manifold","volume":"108","author":"Srivastava","year":"2017","journal-title":"J. Geom."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Srivastava, S.K., Sharma, A., and Tiwari, S.K. (2017). On PR-Pseudo-Slant Warped Product Submanifolds in a Nearly Paracosymplectic Manifold, Alexandru Ioan Cuza University of Ia\u015fi.","DOI":"10.1007\/s00009-016-0832-3"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/s40065-018-0234-4","article-title":"Nonexistence of PR-semi-slant warped product submanifolds in paracosymplectic manifolds","volume":"9","author":"Sharma","year":"2020","journal-title":"Arab. J. Math."},{"key":"ref_29","first-page":"231","article-title":"Pointwise PR-pseudo-slant submanifold of para-Kaehler manifold","volume":"14","author":"Sharma","year":"2021","journal-title":"Bull. Transilv. Univ. Bras. Math. Inform. Phys. Ser. III"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5408","DOI":"10.3934\/math.2022300","article-title":"Conformal \u03b7-Ricci solitons within the framework of indefinite Kenmotsu manifolds","volume":"7","author":"Li","year":"2022","journal-title":"AIMS Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/math-2022-0048","article-title":"Geometry of conformal \u03b7-Ricci solitons and conformal \u03b7-Ricci almost solitons on Paracontact geometry","volume":"20","author":"Li","year":"2022","journal-title":"Open Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"8554738","DOI":"10.1155\/2021\/8554738","article-title":"Homology groups in warped product submanifolds in hyperbolic spaces","volume":"2021","author":"Li","year":"2021","journal-title":"J. Math."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Li, Y.L., Alkhaldi, A.H., Ali, A., and Laurian-Ioan, P. (2021). On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature. Mathematics, 9.","DOI":"10.3390\/math9243156"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"9309","DOI":"10.3934\/math.2021541","article-title":"Biharmonic submanifolds of K\u00e4hler product manifolds","volume":"6","author":"Li","year":"2021","journal-title":"AIMS Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5777554","DOI":"10.1155\/2021\/5777554","article-title":"A general inequality for CR-warped products in generalized Sasakian space form and its applications","volume":"2021","author":"Li","year":"2021","journal-title":"Adv. Math. Phys."},{"key":"ref_36","first-page":"6195939","article-title":"Some eigenvalues estimate for the \u03d5-Laplace operator on slant submanifolds of Sasakian space forms","volume":"2021","author":"Li","year":"2021","journal-title":"J. Funct. Space"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"12077","DOI":"10.3934\/math.2022671","article-title":"Yamabe constant evolution and monotonicity along the conformal Ricci flow","volume":"7","author":"Li","year":"2022","journal-title":"AIMS Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00022-018-0423-5","article-title":"The decomposition of almost paracontact metric manifolds in eleven classes revisited","volume":"109","author":"Zamkovoy","year":"2018","journal-title":"J. Geom."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/s00009-014-0496-9","article-title":"On a class of \u03b1-Para Kenmotsu Manifolds","volume":"13","author":"Srivastava","year":"2016","journal-title":"Mediterr. J. Math."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00009-017-1018-3","article-title":"Slant submanifolds of para-Hermitian manifolds","volume":"14","author":"Alegre","year":"2017","journal-title":"Mediterr. J. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/BF01421206","article-title":"Eine innere Kennzeichnung der verzerrten Produkte","volume":"241","author":"Hiepko","year":"1979","journal-title":"Math. Ann."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/1001\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:10:40Z","timestamp":1760137840000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/1001"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,14]]},"references-count":41,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14051001"],"URL":"https:\/\/doi.org\/10.3390\/sym14051001","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,5,14]]}}}