$f$-Biharmonic submanifolds in space forms and $f$-biharmonic Riemannian submersions from 3-manifolds
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914675131678720 |
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| author | Wang, Ze-Ping Qin, Li-Hua |
| author_facet | Wang, Ze-Ping Qin, Li-Hua |
| contents | $f$-Biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we obtain some descriptions of $f$-biharmonic curves in a space form. We also obtain a complete classification of proper $f$-biharmonic isometric immersions of a developable surface in $\r^3$ by proving that a proper $f$-biharmonic developable surface exists only in the case where the surface is a cylinder. Based on this, we show that a proper biharmonic conformal immersion of a developable surface into $\r^3$ exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as the dual notion of isometric immersions (i.e., submanifolds). We also study $f$-biharmonicity of Riemannian submersions from 3-space forms by using the integrability data. Examples are given of proper $f$-biharmonic Riemannian submersions and $f$-biharmonic surfaces and curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_13910 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $f$-Biharmonic submanifolds in space forms and $f$-biharmonic Riemannian submersions from 3-manifolds Wang, Ze-Ping Qin, Li-Hua Differential Geometry $f$-Biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we obtain some descriptions of $f$-biharmonic curves in a space form. We also obtain a complete classification of proper $f$-biharmonic isometric immersions of a developable surface in $\r^3$ by proving that a proper $f$-biharmonic developable surface exists only in the case where the surface is a cylinder. Based on this, we show that a proper biharmonic conformal immersion of a developable surface into $\r^3$ exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as the dual notion of isometric immersions (i.e., submanifolds). We also study $f$-biharmonicity of Riemannian submersions from 3-space forms by using the integrability data. Examples are given of proper $f$-biharmonic Riemannian submersions and $f$-biharmonic surfaces and curves. |
| title | $f$-Biharmonic submanifolds in space forms and $f$-biharmonic Riemannian submersions from 3-manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2306.13910 |