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arXiv · 2509.09500

On Kanai's conjecture for frame flows over negatively curved manifolds

Louis-Brahim Beaufort
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Abstract

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.

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Louis-Brahim Beaufort. 2025-09-11. On Kanai's conjecture for frame flows over negatively curved manifolds. https://arxiv.org/abs/2509.09500

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