Ruled and Quadric Surfaces Satisfying Δ<sup><i>II</i></sup><i>N</i> = <i>ΛN</i>
In the 3-dimensional Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>E</mi><mn>3</mn></msup></semantics></math></inline-formula>, a quadr...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/2/300 |
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| author | Hassan Al-Zoubi Tareq Hamadneh Ma’mon Abu Hammad Mutaz Al-Sabbagh Mehmet Ozdemir |
| author_facet | Hassan Al-Zoubi Tareq Hamadneh Ma’mon Abu Hammad Mutaz Al-Sabbagh Mehmet Ozdemir |
| author_sort | Hassan Al-Zoubi |
| collection | DOAJ |
| description | In the 3-dimensional Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>E</mi><mn>3</mn></msup></semantics></math></inline-formula>, a quadric surface is either ruled or of one of the following two kinds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><mi>a</mi><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mo>,</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>=</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mfrac><mi>b</mi><mn>2</mn></mfrac><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="4pt"></mspace><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. In the present paper, we investigate these three kinds of surfaces whose Gauss map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">N</mi></semantics></math></inline-formula> satisfies the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>Δ</mo><mrow><mi>I</mi><mi>I</mi></mrow></msup><mi mathvariant="bold-italic">N</mi><mo>=</mo><mi>Λ</mi><mi mathvariant="bold-italic">N</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Λ</mi></semantics></math></inline-formula> is a square symmetric matrix of order 3, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mo>Δ</mo><mrow><mi>I</mi><mi>I</mi></mrow></msup></semantics></math></inline-formula> denotes the Laplace operator of the second fundamental form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>I</mi></mrow></semantics></math></inline-formula> of the surface. We prove that spheres with the nonzero symmetric matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Λ</mi></semantics></math></inline-formula>, and helicoids with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Λ</mi></semantics></math></inline-formula> as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property. |
| first_indexed | 2024-03-11T08:04:59Z |
| format | Article |
| id | doaj.art-7efa37c9e95948dfbd411e4eced511fb |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T08:04:59Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-7efa37c9e95948dfbd411e4eced511fb2023-11-16T23:31:29ZengMDPI AGSymmetry2073-89942023-01-0115230010.3390/sym15020300Ruled and Quadric Surfaces Satisfying Δ<sup><i>II</i></sup><i>N</i> = <i>ΛN</i>Hassan Al-Zoubi0Tareq Hamadneh1Ma’mon Abu Hammad2Mutaz Al-Sabbagh3Mehmet Ozdemir4Department of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JordanDepartment of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JordanDepartment of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, JordanDepartment of Basic Engineering Sciences, Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi ArabiaDepartment of Basic Engineering Sciences, Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi ArabiaIn the 3-dimensional Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>E</mi><mn>3</mn></msup></semantics></math></inline-formula>, a quadric surface is either ruled or of one of the following two kinds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><mi>a</mi><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mo>,</mo><mi>a</mi><mi>b</mi><mi>c</mi><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>=</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mfrac><mi>b</mi><mn>2</mn></mfrac><msup><mi>t</mi><mn>2</mn></msup><mo>,</mo><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="4pt"></mspace><mi>b</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. In the present paper, we investigate these three kinds of surfaces whose Gauss map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">N</mi></semantics></math></inline-formula> satisfies the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mo>Δ</mo><mrow><mi>I</mi><mi>I</mi></mrow></msup><mi mathvariant="bold-italic">N</mi><mo>=</mo><mi>Λ</mi><mi mathvariant="bold-italic">N</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Λ</mi></semantics></math></inline-formula> is a square symmetric matrix of order 3, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mo>Δ</mo><mrow><mi>I</mi><mi>I</mi></mrow></msup></semantics></math></inline-formula> denotes the Laplace operator of the second fundamental form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>I</mi></mrow></semantics></math></inline-formula> of the surface. We prove that spheres with the nonzero symmetric matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Λ</mi></semantics></math></inline-formula>, and helicoids with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>Λ</mi></semantics></math></inline-formula> as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property.https://www.mdpi.com/2073-8994/15/2/300ruled surfacesquadric surfacessurfaces of coordinate finite type in Euclidean 3-spaceLaplace operator |
| spellingShingle | Hassan Al-Zoubi Tareq Hamadneh Ma’mon Abu Hammad Mutaz Al-Sabbagh Mehmet Ozdemir Ruled and Quadric Surfaces Satisfying Δ<sup><i>II</i></sup><i>N</i> = <i>ΛN</i> Symmetry ruled surfaces quadric surfaces surfaces of coordinate finite type in Euclidean 3-space Laplace operator |
| title | Ruled and Quadric Surfaces Satisfying Δ<sup><i>II</i></sup><i>N</i> = <i>ΛN</i> |
| title_full | Ruled and Quadric Surfaces Satisfying Δ<sup><i>II</i></sup><i>N</i> = <i>ΛN</i> |
| title_fullStr | Ruled and Quadric Surfaces Satisfying Δ<sup><i>II</i></sup><i>N</i> = <i>ΛN</i> |
| title_full_unstemmed | Ruled and Quadric Surfaces Satisfying Δ<sup><i>II</i></sup><i>N</i> = <i>ΛN</i> |
| title_short | Ruled and Quadric Surfaces Satisfying Δ<sup><i>II</i></sup><i>N</i> = <i>ΛN</i> |
| title_sort | ruled and quadric surfaces satisfying δ sup i ii i sup i n i i λn i |
| topic | ruled surfaces quadric surfaces surfaces of coordinate finite type in Euclidean 3-space Laplace operator |
| url | https://www.mdpi.com/2073-8994/15/2/300 |
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