Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms
Let M be a Lorentz surface and F:M→N a time-like and conformal immersion of M into a 4-dimensional neutral space form N with zero mean curvature vector. We show that the curvature K of the induced metric on M by F is identically equal to the constant sectional curvature L0 of N if and only if the co...
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| Format: | Article |
| Language: | English |
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Odesa National University of Technology
2024-01-01
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| Series: | Pracì Mìžnarodnogo Geometričnogo Centru |
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| Online Access: | https://journals.ontu.edu.ua/index.php/geometry/article/view/2585 |
| _version_ | 1797231767312138240 |
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| author | Naoya Ando |
| author_facet | Naoya Ando |
| author_sort | Naoya Ando |
| collection | DOAJ |
| description | Let M be a Lorentz surface and F:M→N a time-like and conformal immersion of M into a 4-dimensional neutral space form N with zero mean curvature vector. We show that the curvature K of the induced metric on M by F is identically equal to the constant sectional curvature L0 of N if and only if the covariant derivatives of both of the time-like twistor lifts are zero or light-like. If K≡L0, then the normal connection ∇⟂ of F is flat, while the converse is not necessarily true. We also prove that a holomorphic paracomplex quartic differential Q on M defined by F is zero or null if and only if the covariant derivative of at least one of the time-like twistor lifts is zero or light-like. In addition, we get that K is identically equal to L0 if and only if not only ∇⟂ is flat but also Q is zero or null |
| first_indexed | 2024-04-24T15:49:37Z |
| format | Article |
| id | doaj.art-bfd4dc8364f34a54978bb30936149f2d |
| institution | Directory Open Access Journal |
| issn | 2072-9812 2409-8906 |
| language | English |
| last_indexed | 2024-04-24T15:49:37Z |
| publishDate | 2024-01-01 |
| publisher | Odesa National University of Technology |
| record_format | Article |
| series | Pracì Mìžnarodnogo Geometričnogo Centru |
| spelling | doaj.art-bfd4dc8364f34a54978bb30936149f2d2024-04-01T13:02:58ZengOdesa National University of TechnologyPracì Mìžnarodnogo Geometričnogo Centru2072-98122409-89062024-01-01171365510.15673/pigc.v17i1.25852585Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space formsNaoya Ando0Kumamoto University, JapanLet M be a Lorentz surface and F:M→N a time-like and conformal immersion of M into a 4-dimensional neutral space form N with zero mean curvature vector. We show that the curvature K of the induced metric on M by F is identically equal to the constant sectional curvature L0 of N if and only if the covariant derivatives of both of the time-like twistor lifts are zero or light-like. If K≡L0, then the normal connection ∇⟂ of F is flat, while the converse is not necessarily true. We also prove that a holomorphic paracomplex quartic differential Q on M defined by F is zero or null if and only if the covariant derivative of at least one of the time-like twistor lifts is zero or light-like. In addition, we get that K is identically equal to L0 if and only if not only ∇⟂ is flat but also Q is zero or nullhttps://journals.ontu.edu.ua/index.php/geometry/article/view/2585time-like surfacezero mean curvature vectortime-like twistor liftparacomplex quartic differentialnormal connection |
| spellingShingle | Naoya Ando Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms Pracì Mìžnarodnogo Geometričnogo Centru time-like surface zero mean curvature vector time-like twistor lift paracomplex quartic differential normal connection |
| title | Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms |
| title_full | Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms |
| title_fullStr | Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms |
| title_full_unstemmed | Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms |
| title_short | Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms |
| title_sort | time like surfaces with zero mean curvature vector in 4 dimensional neutral space forms |
| topic | time-like surface zero mean curvature vector time-like twistor lift paracomplex quartic differential normal connection |
| url | https://journals.ontu.edu.ua/index.php/geometry/article/view/2585 |
| work_keys_str_mv | AT naoyaando timelikesurfaceswithzeromeancurvaturevectorin4dimensionalneutralspaceforms |