On Kanai's conjecture for frame flows over negatively curved manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912583297007616 |
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| author | Beaufort, Louis-Brahim |
| author_facet | Beaufort, Louis-Brahim |
| contents | Let $M$ be a closed, negatively curved Riemannian manifold of dimension $n \neq 4, 8$ with strictly $1/4$-pinched sectional curvature. We prove, that if the frame flow is ergodic and the sum of its unstable and stable bundles together with its flow direction is $\mathcal{C}^2$, then $M$ is homothetic to a real hyperbolic manifold. This extends to higher dimensions a previous result of Kanai in dimension 3. The proof generalises to isometric extensions of geodesic flows to a principal bundle $P$ with compact structure group and yields the following alternative : either $P$ is flat, or $M$ is hyperbolic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Kanai's conjecture for frame flows over negatively curved manifolds Beaufort, Louis-Brahim Dynamical Systems Differential Geometry 37D30, 37D40, 57R22 Let $M$ be a closed, negatively curved Riemannian manifold of dimension $n \neq 4, 8$ with strictly $1/4$-pinched sectional curvature. We prove, that if the frame flow is ergodic and the sum of its unstable and stable bundles together with its flow direction is $\mathcal{C}^2$, then $M$ is homothetic to a real hyperbolic manifold. This extends to higher dimensions a previous result of Kanai in dimension 3. The proof generalises to isometric extensions of geodesic flows to a principal bundle $P$ with compact structure group and yields the following alternative : either $P$ is flat, or $M$ is hyperbolic. |
| title | On Kanai's conjecture for frame flows over negatively curved manifolds |
| topic | Dynamical Systems Differential Geometry 37D30, 37D40, 57R22 |
| url | https://arxiv.org/abs/2509.09500 |