On the ergodicity of the frame flow on even-dimensional manifolds

Abstract

It is known that the frame flow on a closed n-dimensional Riemannian manifold with negative sectional curvature is ergodic if nis odd and n≠7. In this paper we study its ergodicity in the remaining cases. For neven and n≠8,134, we show that: if n≡2mod 4 or n=4, the frame flow is ergodic if the manifold is ∼0.3-pinched, if n≡0mod 4, it is ergodic if the manifold is ∼0.6-pinched. In the three dimensions n=7,8,134, the respective pinching bounds that we need in order to prove ergodicity are 0.4962..., 0.6212..., and 0.5788.... This is a significant improvement over the previously known results and a step forward towards solving a long-standing conjecture of Brin asserting that 0.25-pinched even-dimensional manifolds have an ergodic frame flow.