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

It is known that the frame flow on a closed n -dimensional Riemannian manifold with negative sectional curvature is ergodic if n is odd and n ≠ 7 . In this paper we study its ergodicity in the remaining cases. For n even and n ≠ 8 , 134 , we show that: if n ≡ 2 mod 4 or n = 4 , the frame flow is ergodic if the manifold is ∼ 0.3 -pinched, if n ≡ 0 mod 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.