Extrinsic characterizations of biconservative surfaces in the $4$-dimensional hyperbolic space
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913152684261376 |
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| author | Nistor, Simona Rusu, Mihaela |
| author_facet | Nistor, Simona Rusu, Mihaela |
| contents | Biconservative submanifolds arise as a natural relaxation of the biharmonic condition and play an important role in the submanifold theory. In this paper, we study non-CMC biconservative surfaces with parallel normalized mean curvature vector field (PNMC surfaces) in the four-dimensional hyperbolic space $\mathbb{H}^4$, for which we consider the hyperboloid model. We provide a local extrinsic description of such surfaces, showing that they are generated by a directrix curve lying in a totally geodesic hypersurface $\mathbb{H}^3$ of $\mathbb{H}^4$, through a certain normal flow. This extrinsic classification of non-CMC, PNMC biconservative surfaces in $\mathbb{H}^4$ splits naturally into three cases according to the type of a certain vector field, which can be non-zero null, spacelike or timelike. Together with the previous results, the classification of non-CMC, PNMC surfaces in four-dimensional space forms is now completed, from intrinsic and extrinsic point of view. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22396 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extrinsic characterizations of biconservative surfaces in the $4$-dimensional hyperbolic space Nistor, Simona Rusu, Mihaela Differential Geometry 53C43, 53B25 Biconservative submanifolds arise as a natural relaxation of the biharmonic condition and play an important role in the submanifold theory. In this paper, we study non-CMC biconservative surfaces with parallel normalized mean curvature vector field (PNMC surfaces) in the four-dimensional hyperbolic space $\mathbb{H}^4$, for which we consider the hyperboloid model. We provide a local extrinsic description of such surfaces, showing that they are generated by a directrix curve lying in a totally geodesic hypersurface $\mathbb{H}^3$ of $\mathbb{H}^4$, through a certain normal flow. This extrinsic classification of non-CMC, PNMC biconservative surfaces in $\mathbb{H}^4$ splits naturally into three cases according to the type of a certain vector field, which can be non-zero null, spacelike or timelike. Together with the previous results, the classification of non-CMC, PNMC surfaces in four-dimensional space forms is now completed, from intrinsic and extrinsic point of view. |
| title | Extrinsic characterizations of biconservative surfaces in the $4$-dimensional hyperbolic space |
| topic | Differential Geometry 53C43, 53B25 |
| url | https://arxiv.org/abs/2605.22396 |