{
  "id": "2411.09008",
  "version": "v1",
  "published": "2024-11-13T20:22:30.000Z",
  "updated": "2024-11-13T20:22:30.000Z",
  "title": "Integrable sub-Riemannian geodesic flows on the special orthogonal group",
  "authors": [
    "Alejandro Bravo-Doddoli",
    "Philip Arathoon",
    "Anthony M. Bloch"
  ],
  "categories": [
    "math.DG",
    "math.DS"
  ],
  "abstract": "One way to define a sub-Riemannian metric is as the limit of a Riemannian metric. Consider a Riemannian structure depending on a parameter $s$ such that its limit defines a sub-Riemannian metric when $s \\to \\infty$, assuming that the Riemannian geodesic flow is integrable for all $s$. An interesting question is: Can we determine the integrability of the sub-Riemannian geodesic flow as the limit of the integrals of motion of the Riemannian geodesic flow? The paper's main contribution is to provide a positive answer to this question in the special orthogonal group. Theorem 1.1 states that the Riemannian geodesic flow is Liuville integrable: The Manakov integrals' limit suggests the existence of a Lax pair formulation of the Riemannian geodesic equations, and the proof of Theorem 1.1 relies on this Lax pair.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-13T20:22:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J35",
      "17B80",
      "53C17",
      "70G65"
    ],
    "keywords": [
      "special orthogonal group",
      "integrable sub-riemannian geodesic flows",
      "sub-riemannian metric",
      "papers main contribution",
      "lax pair formulation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}