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

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.