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 $n\neq7$. We prove Brin's quarter-pinching conjecture when $n=4$, and when $n\equiv2\pmod4$ with $n\neq134$: strict $1/4$-pinching implies ergodicity of the oriented frame flow. We also prove that if $n\equiv0\pmod4$ and $n\geq12$, then $5/13$-pinching implies ergodicity. In the exceptional dimensions $7$, $8$, and $134$, we prove ergodicity under strict $0.4661...$-, $0.5358...$-, and $2/5$-pinching, respectively. These results substantially improve the corresponding bounds obtained by Cekić--Lefeuvre--Moroianu--Semmelmann.