{
  "id": "2111.14811",
  "version": "v2",
  "published": "2021-11-29T18:55:20.000Z",
  "updated": "2023-02-05T19:09:15.000Z",
  "title": "On the ergodicity of the frame flow on even-dimensional manifolds",
  "authors": [
    "Mihajlo Cekić",
    "Thibault Lefeuvre",
    "Andrei Moroianu",
    "Uwe Semmelmann"
  ],
  "comment": "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",
  "categories": [
    "math.DS",
    "math.AT",
    "math.DG"
  ],
  "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 $n \\neq 7$. In this paper we study its ergodicity for $n \\geq 4$ even and $n = 7$, and 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, except in the three dimensions $n=7,8,134$, where the respective pinching conditions are $0.4962...$, $0.6212...$, and $0.5788...$. In particular, if $n = 4$ or $n \\equiv 2$ mod $4$, this almost solves a long-standing conjecture of Brin asserting that $1/4$-pinched even-dimensional manifolds have an ergodic frame flow.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-02-05T19:09:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05",
      "37A20",
      "37A25",
      "53C10",
      "53C22"
    ],
    "keywords": [
      "ergodicity",
      "dimensional riemannian manifold",
      "ergodic frame flow",
      "negative sectional curvature",
      "pinched even-dimensional manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}