{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T02:33:49Z","timestamp":1767926029044,"version":"3.49.0"},"reference-count":38,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T00:00:00Z","timestamp":1686873600000},"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 study, the authors focus on quasi-hemi-slant submanifolds (qhs-submanifolds) of (\u03b1,\u03b2)-type almost contact manifolds, also known as trans-Sasakian manifolds. Essentially, we give sufficient and necessary conditions for the integrability of distributions using the concept of quasi-hemi-slant submanifolds of trans-Sasakian manifolds. We also consider the geometry of foliations dictated by the distribution and the requirements for submanifolds of trans-Sasakian manifolds with quasi-hemi-slant factors to be totally geodesic. Lastly, we give an illustration of a submanifold with a quasi-hemi-slant factor and discuss its application to number theory.<\/jats:p>","DOI":"10.3390\/sym15061270","type":"journal-article","created":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T08:56:01Z","timestamp":1686905761000},"page":"1270","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Aspects of Submanifolds on (\u03b1, \u03b2)-Type Almost Contact Manifolds with Quasi-Hemi-Slant Factor"],"prefix":"10.3390","volume":"15","author":[{"given":"Ali H.","family":"Hakami","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 277, Jazan 4512, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1713-6831","authenticated-orcid":false,"given":"Mohd Danish","family":"Siddiqi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 277, Jazan 4512, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"O\u01e7uzhan","family":"Bahadir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Letters, Kahramanmaras Sutcu Imam University, Kahrmanmaras 46040, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1176-4666","authenticated-orcid":false,"given":"Toukeer","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Liberal Education, Era University, Lucknow 226003, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/BF01218650","article-title":"Immersions with parallel second fundamental form","volume":"140","author":"Ferus","year":"1974","journal-title":"Math. 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