Pinching and Tensorial Rigidity for Ergodicity of Frame Flows

Abstract

Let $(M^n,g)$ denote a closed oriented negatively curved Riemannian manifold. For such manifolds, the oriented frame flow is known to be ergodic for all odd dimensions $n\ne 7$. We prove Brin's quarter-pinching conjecture when $n=4$, and when $n\equiv 2(\mathrm{mod}4)$ with $n\ne 134$: strict $1/4$-pinching implies ergodicity of the oriented frame flow. We also prove that if $n\equiv 0(\mathrm{mod}4)$ and $n\ge 12$, then $5/13$-pinching implies ergodicity. In the exceptional dimensions $7$, $8$, and $134$, we prove ergodicity under strict $\mathrm{0.4661...}$-, $\mathrm{0.5358...}$-, and $2/5$-pinching, respectively. These results substantially improve the corresponding bounds obtained by Cekić--Lefeuvre--Moroianu--Semmelmann.

46 pages