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

It is known that the frame flow on a closed $n$ $n$ -dimensional Riemannian manifold with negative sectional curvature is ergodic if $n$ $n$ is odd and $n\ne 7$ $n \neq 7$ . In this paper we study its ergodicity in the remaining cases. For $n$ $n$ even and $n\ne 8,134$ $n \neq 8, 134$ , we show that: (1)

if $n\equiv 2$ $n \equiv 2$ mod 4 or $n=4$ $n=4$ , the frame flow is ergodic if the manifold is $\sim 0.3$ $\sim 0.3$ -pinched,

(2)

if $n\equiv 0$ $n \equiv 0$ mod 4, it is ergodic if the manifold is $\sim 0.6$ $\sim 0.6$ -pinched.

In the three dimensions $n=7,8,134$ $n=7,8,134$ , 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.