On the ergodicity of the frame flow on even-dimensional manifolds
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| Format: | Preprint |
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2021
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| _version_ | 1866913623797923840 |
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| author | Cekić, Mihajlo Lefeuvre, Thibault Moroianu, Andrei Semmelmann, Uwe |
| author_facet | Cekić, Mihajlo Lefeuvre, Thibault Moroianu, Andrei Semmelmann, Uwe |
| contents | 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 $n \neq 7$. In this paper we study its ergodicity in the remaining cases. For $n$ even and $n \neq 8, 134$, 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 $0.4962...$, $0.6212...$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_14811 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the ergodicity of the frame flow on even-dimensional manifolds Cekić, Mihajlo Lefeuvre, Thibault Moroianu, Andrei Semmelmann, Uwe Dynamical Systems Algebraic Topology Differential Geometry 37A05, 37A20, 37A25, 53C10, 53C22 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 $n \neq 7$. In this paper we study its ergodicity in the remaining cases. For $n$ even and $n \neq 8, 134$, 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 $0.4962...$, $0.6212...$, 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. |
| title | On the ergodicity of the frame flow on even-dimensional manifolds |
| topic | Dynamical Systems Algebraic Topology Differential Geometry 37A05, 37A20, 37A25, 53C10, 53C22 |
| url | https://arxiv.org/abs/2111.14811 |