{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T18:17:09Z","timestamp":1780078629371,"version":"3.54.0"},"reference-count":34,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,8,28]],"date-time":"2021-08-28T00:00:00Z","timestamp":1630108800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we prove that, for compact warped product submanifolds Mn in an Euclidean space En+k, there are no stable p-currents, homology groups are vanishing, and M3 is homotopic to the Euclidean sphere S3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature, and the Hessian tensor. The results in this paper can be considered an extension of Xin\u2019s work in the framework of a compact warped product submanifold, when the base manifold is minimal in ambient manifolds.<\/jats:p>","DOI":"10.3390\/sym13091587","type":"journal-article","created":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T22:58:15Z","timestamp":1630450695000},"page":"1587","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Null Homology Groups and Stable Currents in Warped Product Submanifolds of Euclidean Spaces"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1614-3228","authenticated-orcid":false,"given":"Yanlin","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2269-718X","authenticated-orcid":false,"given":"Pi\u015fcoran","family":"Laurian-Ioan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science Victoriei 76, North University Center of Baia Mare Technical University of Cluj Napoca, 430122 Baia Mare, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Akram","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ali H.","family":"Alkhaldi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"427","DOI":"10.2307\/1970913","article-title":"On stable currents and their application to global problems in real and complex geometry","volume":"98","author":"Lawson","year":"1973","journal-title":"Ann. Math."},{"key":"ref_2","first-page":"233","article-title":"An application of integral currents to the vanishing theorems","volume":"27","author":"Xin","year":"1984","journal-title":"Sci. China Ser. A."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1093\/qmath\/hag045","article-title":"The Ricci curvature of submanifolds and its applications","volume":"55","author":"Vlachos","year":"2003","journal-title":"Quart. J. Math."},{"key":"ref_4","unstructured":"Wallah, N.R. (1972). Minimal Immersion of Symmetric Spaces into Spheres, Symmetric Spaces, Dekker."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2607","DOI":"10.1090\/S0002-9939-07-08901-0","article-title":"Homology vanishing theorems for submanifolds","volume":"135","author":"Vlachos","year":"2007","journal-title":"Proc. Amer. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1684","DOI":"10.1007\/s00039-013-0231-x","article-title":"Geometric, topological and differentiable rigidity of submanifolds in space forms","volume":"23","author":"Xu","year":"2013","journal-title":"Geom. Funct. Anal."},{"key":"ref_7","first-page":"357","article-title":"The topology of four-dimensional manifolds","volume":"17","author":"Freedman","year":"1982","journal-title":"J. Diff. Geom."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1142\/S0129167X0800490X","article-title":"Vanishing and topological sphere theorems for submanifolds of Hyperbolic space","volume":"19","author":"Fu","year":"2008","journal-title":"Intern. J. Math."},{"key":"ref_9","first-page":"779","article-title":"Minimal submanifolds of sphere with bounded second fundamental form","volume":"79","author":"Gauchman","year":"1993","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s00022-010-0024-4","article-title":"Non-existence of stable currents in submanifolds of the Euclidean spaces","volume":"96","author":"Liu","year":"2009","journal-title":"J. Geom."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1850032","DOI":"10.1142\/S0219887818500329","article-title":"On the topology of CR-warped product submanifolds","volume":"15","author":"Sahin","year":"2018","journal-title":"Int. J. Geom. Meth. Mod. Phy."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1023\/A:1000189116072","article-title":"The topological sphere theorem for complete submanifolds","volume":"107","author":"Shiohama","year":"1997","journal-title":"Comp. Maths."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1007\/s00229-011-0508-z","article-title":"Differentiable sphere theorems for submanifolds of positive k-th ricci curvature","volume":"138","author":"Xu","year":"2012","journal-title":"Manu. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1017\/S0004972700029762","article-title":"Non-existence of stable currents in submanifolds of a product of two spheres","volume":"44","author":"Zhang","year":"1991","journal-title":"Bull. Austral. Math. Soc."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0926-2245(96)00004-6","article-title":"Isometric immersions of warped products","volume":"6","author":"Nolker","year":"1996","journal-title":"Diff. Geom. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1186\/s13660-016-1177-y","article-title":"Estimation of inequalities for warped product semi-slant submanifolds of Kenmotsu space forms","volume":"1","author":"Liaqat","year":"2016","journal-title":"J. Inequal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Alluhaibi, N., Mofarreh, F., Ali, A., and Mior Othman, W.A. (2020). Geometric inequalities of warped product submanifolds and their applications. Mathematics, 8.","DOI":"10.3390\/math8050759"},{"key":"ref_19","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. Amer. Math. Soc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1017\/S001309150100075X","article-title":"On isometric minimal immersions from warped products into real space forms","volume":"45","author":"Chen","year":"2002","journal-title":"Proc. Edinb. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2050121","DOI":"10.1142\/S0219887820501212","article-title":"Homology of warped product submanifolds in the unit sphere and its applications","volume":"17","author":"Ali","year":"2020","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"283","DOI":"10.5488\/CMP.9.2.283","article-title":"Homotopy in statistical physics","volume":"9","author":"Kenna","year":"2006","journal-title":"Condens. Matter Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"9309","DOI":"10.3934\/math.2021541","article-title":"Biharmonic submanifolds of Kaehler product manifolds","volume":"6","author":"Li","year":"2021","journal-title":"AIMS Math."},{"key":"ref_24","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_25","doi-asserted-by":"crossref","first-page":"175008","DOI":"10.1088\/0264-9381\/26\/17\/175008","article-title":"Stable homology as an indicator of manifoldlikeness in causal set topology","volume":"26","author":"Major","year":"2009","journal-title":"Class. Quantum Grav."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.tcs.2008.06.033","article-title":"Causal set topology","volume":"405","author":"Surya","year":"2008","journal-title":"Theor. Comput. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"391","DOI":"10.2307\/1970239","article-title":"Generalized Poincare\u2019s conjecture in dimensions greater than four","volume":"74","author":"Smale","year":"1961","journal-title":"Ann. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1112\/jlms\/s2-6.2.333","article-title":"Homology spheres which are covered by spheres","volume":"6","author":"Sjerve","year":"1973","journal-title":"J. Lond. Math. Soc."},{"key":"ref_29","first-page":"255","article-title":"Three-manifolds with positive Ricci curvature","volume":"17","author":"Hamilton","year":"1982","journal-title":"J. Diff. Geom."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Berger, M., Gauduchon, P., and Mazet, E. (1971). Le spectre d\u2019une vari\u00e9t\u00e9 Riemannienne, Springer.","DOI":"10.1007\/BFb0064643"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Li, Y.L., Liu, S.Y., and Wang, Z.G. (2020). Tangent developables and Darboux developables of framed curves. Topology Appl., 107526.","DOI":"10.1016\/j.topol.2020.107526"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"104188","DOI":"10.1016\/j.geomphys.2021.104188","article-title":"Lightlike tangent developables in de Sitter 3-space","volume":"164","author":"Li","year":"2021","journal-title":"J. Geom. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s00006-020-01097-1","article-title":"Geometric Algebra of Singular Ruled Surfaces","volume":"31","author":"Li","year":"2021","journal-title":"Adv. Appl. Clifford Algebras."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2150008-68","DOI":"10.1142\/S0219887821500080","article-title":"Singularities and dualities of pedal curves in pseudo-hyperbolic and de Sitter space","volume":"18","author":"Li","year":"2021","journal-title":"Int. J. Geom. Meth. Mod. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/9\/1587\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:54:23Z","timestamp":1760165663000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/9\/1587"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,28]]},"references-count":34,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["sym13091587"],"URL":"https:\/\/doi.org\/10.3390\/sym13091587","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,28]]}}}