λ-biharmonic Riemannian submersions from manifolds with constant sectional curvature
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911687707197440 |
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| author | Maeta, Shun Shito, Miho |
| author_facet | Maeta, Shun Shito, Miho |
| contents | In this paper, we study λ-biharmonic Riemannian submersions, which generalize biharmonic Riemannian submersions. We prove non-existence results for λ-biharmonic Riemannian submersions from (n + 1)-dimensional Riemannian manifolds with constant sectional curvature c to n-dimensional Riemannian manifolds. Our results show that the critical value λ= 2(n - 1)c plays a decisive role. When λ\ne 2(n - 1)c, we prove a nonexistence theorem, although a dimensional assumption is needed in the positive curvature case. On the other hand, when λ= 2(n - 1)c, we prove a non-existence theorem in the nonnegative curvature case, whereas in the negative curvature case, we construct explicit examples. The only remaining local case is the positively curved case with λ\ne 2(n - 1)c and n \ge 5, while in the complete connected positive-curvature setting the theorem of Gromoll and Grove yields harmonicity in all dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15578 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | λ-biharmonic Riemannian submersions from manifolds with constant sectional curvature Maeta, Shun Shito, Miho Differential Geometry 58E20, 53C43 In this paper, we study λ-biharmonic Riemannian submersions, which generalize biharmonic Riemannian submersions. We prove non-existence results for λ-biharmonic Riemannian submersions from (n + 1)-dimensional Riemannian manifolds with constant sectional curvature c to n-dimensional Riemannian manifolds. Our results show that the critical value λ= 2(n - 1)c plays a decisive role. When λ\ne 2(n - 1)c, we prove a nonexistence theorem, although a dimensional assumption is needed in the positive curvature case. On the other hand, when λ= 2(n - 1)c, we prove a non-existence theorem in the nonnegative curvature case, whereas in the negative curvature case, we construct explicit examples. The only remaining local case is the positively curved case with λ\ne 2(n - 1)c and n \ge 5, while in the complete connected positive-curvature setting the theorem of Gromoll and Grove yields harmonicity in all dimensions. |
| title | λ-biharmonic Riemannian submersions from manifolds with constant sectional curvature |
| topic | Differential Geometry 58E20, 53C43 |
| url | https://arxiv.org/abs/2605.15578 |