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 $n$ is odd and $n\ne 7$. In this paper we study its ergodicity in the remaining cases. For $n$ even and $n\ne 8,134$, we show that: if $n\equiv 2$ mod $4$ or $n=4$, the frame flow is ergodic if the manifold is $\sim 0.3$-pinched, if $n\equiv 0$ mod $4$, it is ergodic if the manifold is $\sim 0.6$-pinched. In the three dimensions $n=7,8,134$, the respective pinching bounds that we need in order to prove ergodicity are $\mathrm{0.4962...}$, $\mathrm{0.6212...}$, and $\mathrm{0.5788...}$. 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.

36 pages, 1 figure; new version containing an improvement of the pinching bound in dimension 134, using some general result about the Fourier degree of sections of vector bundles over the sphere; final version incorporating comments of the referees, accepted in Inventiones Mathematicae

References in corpus (2)

• Isometric extensions of Anosov flows via microlocal analysis
• On the ergodicity of unitary frame flows on Kähler manifolds

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