A Note on Incompressible Vector Fields
In this paper, we use incompressible vector fields for characterizing Killing vector fields. We show that on a compact Riemannian manifold, a nontrivial incompressible vector field has a certain lower bound on the integral of the Ricci curvature in the direction of the incompressible vector field if...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/8/1479 |
| _version_ | 1797583230268866560 |
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| author | Nasser Bin Turki |
| author_facet | Nasser Bin Turki |
| author_sort | Nasser Bin Turki |
| collection | DOAJ |
| description | In this paper, we use incompressible vector fields for characterizing Killing vector fields. We show that on a compact Riemannian manifold, a nontrivial incompressible vector field has a certain lower bound on the integral of the Ricci curvature in the direction of the incompressible vector field if, and only if, the vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is Killing. We also show that a nontrivial incompressible vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> on a compact Riemannian manifold is a Jacobi-type vector field if, and only if, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is Killing. Finally, we show that a nontrivial incompressible vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> on a connected Riemannian manifold has a certain lower bound on the Ricci curvature in the direction of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>, and if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is also a geodesic vector field, it necessarily implies that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is Killing. |
| first_indexed | 2024-03-10T23:32:58Z |
| format | Article |
| id | doaj.art-bae24412fa344fc1bf4d5642dc3e5167 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T23:32:58Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-bae24412fa344fc1bf4d5642dc3e51672023-11-19T03:10:19ZengMDPI AGSymmetry2073-89942023-07-01158147910.3390/sym15081479A Note on Incompressible Vector FieldsNasser Bin Turki0Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaIn this paper, we use incompressible vector fields for characterizing Killing vector fields. We show that on a compact Riemannian manifold, a nontrivial incompressible vector field has a certain lower bound on the integral of the Ricci curvature in the direction of the incompressible vector field if, and only if, the vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is Killing. We also show that a nontrivial incompressible vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> on a compact Riemannian manifold is a Jacobi-type vector field if, and only if, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is Killing. Finally, we show that a nontrivial incompressible vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> on a connected Riemannian manifold has a certain lower bound on the Ricci curvature in the direction of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>, and if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is also a geodesic vector field, it necessarily implies that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> is Killing.https://www.mdpi.com/2073-8994/15/8/1479incompressible vector fieldskilling vector fieldsRicci curvatureEuclidean space |
| spellingShingle | Nasser Bin Turki A Note on Incompressible Vector Fields Symmetry incompressible vector fields killing vector fields Ricci curvature Euclidean space |
| title | A Note on Incompressible Vector Fields |
| title_full | A Note on Incompressible Vector Fields |
| title_fullStr | A Note on Incompressible Vector Fields |
| title_full_unstemmed | A Note on Incompressible Vector Fields |
| title_short | A Note on Incompressible Vector Fields |
| title_sort | note on incompressible vector fields |
| topic | incompressible vector fields killing vector fields Ricci curvature Euclidean space |
| url | https://www.mdpi.com/2073-8994/15/8/1479 |
| work_keys_str_mv | AT nasserbinturki anoteonincompressiblevectorfields AT nasserbinturki noteonincompressiblevectorfields |