Eigenvectors of the De-Rham Operator
We aim to examine the influence of the existence of a nonzero eigenvector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> of the de-Rham ope...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/24/4942 |
| _version_ | 1797380147024756736 |
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| author | Nasser Bin Turki Sharief Deshmukh Gabriel-Eduard Vîlcu |
| author_facet | Nasser Bin Turki Sharief Deshmukh Gabriel-Eduard Vîlcu |
| author_sort | Nasser Bin Turki |
| collection | DOAJ |
| description | We aim to examine the influence of the existence of a nonzero eigenvector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> of the de-Rham operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> on a <i>k</i>-dimensional Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>. If the vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> annihilates the de-Rham operator, such a vector field is called a de-Rham harmonic vector field. It is shown that for each nonzero vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, there are two operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>ζ</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Ψ</mi><mi>ζ</mi></msub></semantics></math></inline-formula> associated with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>, called the basic operator and the associated operator of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>, respectively. We show that the existence of an eigenvector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> on a compact manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, such that the integral of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ric</mi><mo>(</mo><mi>ζ</mi><mo>,</mo><mi>ζ</mi><mo>)</mo></mrow></semantics></math></inline-formula> admits a certain lower bound, forces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> to be isometric to a <i>k</i>-dimensional sphere. Moreover, we prove that the existence of a de-Rham harmonic vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> on a connected and complete Riemannian space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, having <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>div</mi><mfenced open="(" close=")"><mi>ζ</mi></mfenced><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula> and annihilating the associated operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Ψ</mi><mi>ζ</mi></msub></semantics></math></inline-formula>, forces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> to be isometric to the <i>k</i>-dimensional Euclidean space, provided that the squared length of the covariant derivative of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> possesses a certain lower bound. |
| first_indexed | 2024-03-08T20:34:08Z |
| format | Article |
| id | doaj.art-b07b18b1ced54210995ec1e6d01b0d0f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T20:34:08Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-b07b18b1ced54210995ec1e6d01b0d0f2023-12-22T14:23:22ZengMDPI AGMathematics2227-73902023-12-011124494210.3390/math11244942Eigenvectors of the De-Rham OperatorNasser Bin Turki0Sharief Deshmukh1Gabriel-Eduard Vîlcu2Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Mathematics and Informatics, National University of Science and Technology Politehnica Bucharest, 313 Splaiul Independenţei, 060042 Bucharest, RomaniaWe aim to examine the influence of the existence of a nonzero eigenvector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> of the de-Rham operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> on a <i>k</i>-dimensional Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>. If the vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> annihilates the de-Rham operator, such a vector field is called a de-Rham harmonic vector field. It is shown that for each nonzero vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, there are two operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>ζ</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Ψ</mi><mi>ζ</mi></msub></semantics></math></inline-formula> associated with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>, called the basic operator and the associated operator of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>, respectively. We show that the existence of an eigenvector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula> on a compact manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, such that the integral of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ric</mi><mo>(</mo><mi>ζ</mi><mo>,</mo><mi>ζ</mi><mo>)</mo></mrow></semantics></math></inline-formula> admits a certain lower bound, forces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> to be isometric to a <i>k</i>-dimensional sphere. Moreover, we prove that the existence of a de-Rham harmonic vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> on a connected and complete Riemannian space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, having <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>div</mi><mfenced open="(" close=")"><mi>ζ</mi></mfenced><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula> and annihilating the associated operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="sans-serif">Ψ</mi><mi>ζ</mi></msub></semantics></math></inline-formula>, forces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>N</mi><mi>k</mi></msup><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> to be isometric to the <i>k</i>-dimensional Euclidean space, provided that the squared length of the covariant derivative of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> possesses a certain lower bound.https://www.mdpi.com/2227-7390/11/24/4942de-Rham operatoreigenvector<i>k</i>-sphere <named-content content-type="inline-formula"><inline-formula><mml:math display="block" id="mm900"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>S</mml:mi><mml:mi>c</mml:mi><mml:mi>k</mml:mi></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula></named-content>Ricci curvaturemanifoldharmonic vector field |
| spellingShingle | Nasser Bin Turki Sharief Deshmukh Gabriel-Eduard Vîlcu Eigenvectors of the De-Rham Operator Mathematics de-Rham operator eigenvector <i>k</i>-sphere <named-content content-type="inline-formula"><inline-formula><mml:math display="block" id="mm900"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>S</mml:mi><mml:mi>c</mml:mi><mml:mi>k</mml:mi></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula></named-content> Ricci curvature manifold harmonic vector field |
| title | Eigenvectors of the De-Rham Operator |
| title_full | Eigenvectors of the De-Rham Operator |
| title_fullStr | Eigenvectors of the De-Rham Operator |
| title_full_unstemmed | Eigenvectors of the De-Rham Operator |
| title_short | Eigenvectors of the De-Rham Operator |
| title_sort | eigenvectors of the de rham operator |
| topic | de-Rham operator eigenvector <i>k</i>-sphere <named-content content-type="inline-formula"><inline-formula><mml:math display="block" id="mm900"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>S</mml:mi><mml:mi>c</mml:mi><mml:mi>k</mml:mi></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula></named-content> Ricci curvature manifold harmonic vector field |
| url | https://www.mdpi.com/2227-7390/11/24/4942 |
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