Pinching and Tensorial Rigidity for Ergodicity of Frame Flows

Let $(M^n,g)$ be a closed, connected, oriented Riemannian manifold with negative sectional curvature and $n\ge 2$. It is known, due to Brin and Gromov, that the oriented frame flow is ergodic in every odd dimension $n\ge 3$, $n\mathrm{\ne }7$. We prove that if a closed, connected, oriented negatively curved Riemannian manifold is strictly quarter-pinched, then its oriented frame flow is ergodic.