{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:40:45Z","timestamp":1767339645682,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,21]],"date-time":"2023-01-21T00:00:00Z","timestamp":1674259200000},"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 the 3-dimensional Euclidean space E3, a quadric surface is either ruled or of one of the following two kinds z2=as2+bt2+c,abc\u22600 or z=a2s2+b2t2,a&gt;0,b&gt;0. In the present paper, we investigate these three kinds of surfaces whose Gauss map N satisfies the property \u0394IIN=\u039bN, where \u039b is a square symmetric matrix of order 3, and \u0394II denotes the Laplace operator of the second fundamental form II of the surface. We prove that spheres with the nonzero symmetric matrix \u039b, and helicoids with \u039b as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property.<\/jats:p>","DOI":"10.3390\/sym15020300","type":"journal-article","created":{"date-parts":[[2023,1,25]],"date-time":"2023-01-25T03:21:54Z","timestamp":1674616914000},"page":"300","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Ruled and Quadric Surfaces Satisfying \u0394IIN = \u039bN"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5545-8084","authenticated-orcid":false,"given":"Hassan","family":"Al-Zoubi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, Jordan"}]},{"given":"Tareq","family":"Hamadneh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, Jordan"}]},{"given":"Ma\u2019mon","family":"Abu Hammad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, Jordan"}]},{"given":"Mutaz","family":"Al-Sabbagh","sequence":"additional","affiliation":[{"name":"Department of Basic Engineering Sciences, Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi Arabia"}]},{"given":"Mehmet","family":"Ozdemir","sequence":"additional","affiliation":[{"name":"Department of Basic Engineering Sciences, Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"87","DOI":"10.5556\/j.tkjm.45.2014.1564","article-title":"Some open problems and conjectures on submanifolds of finite type: Recent development","volume":"45","author":"Chen","year":"2014","journal-title":"Tamkang J. 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Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/300\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:12:37Z","timestamp":1760119957000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/300"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,21]]},"references-count":11,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020300"],"URL":"https:\/\/doi.org\/10.3390\/sym15020300","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,1,21]]}}}