关于负曲率流形上标架流的Kanai猜想
On Kanai's conjecture for frame flows over negatively curved manifolds
AI总结：
本文证明在严格1/4夹挤的负曲率闭流形上，若标架流遍历且相关丛光滑，则流形位似于实双曲流形，推广了Kanai的三维结果。
AI中文摘要：
设$M$是一个闭的、负曲率黎曼流形，维数$n \neq 4, 8$，且其截面曲率严格$1/4$-夹挤。我们证明，如果标架流是遍历的，并且其不稳定丛与稳定丛之和连同流方向是$\mathcal{C}^2$的，那么$M$与一个实双曲流形是位似的。这将对Kanai在三维情形的先前结果推广到更高维数。该证明推广到测地流在具有紧致结构群的principal丛$P$上的等距扩张，并得到如下二选一结论：要么$P$是平坦的，要么$M$是双曲的。
英文摘要：
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.
