{
  "id": "2302.03208",
  "version": "v1",
  "published": "2023-02-07T02:32:39.000Z",
  "updated": "2023-02-07T02:32:39.000Z",
  "title": "The sub-Riemannian geometry of screw motions with constant pitch",
  "authors": [
    "Eduardo Hulett",
    "Paola Ruth Moas",
    "Marcos Salvai"
  ],
  "categories": [
    "math.DG",
    "math.OC"
  ],
  "abstract": "We consider a family of Riemannian manifolds M such that for each unit speed geodesic gamma of M there exists a distinguished bijective correspondence L between infinitesimal translations along gamma and infinitesimal rotations around it. The simplest examples are R^3, S^3 and hyperbolic 3-space, with L defined in terms of the cross product. Such an L exists even if the codimension of gamma is not equal to 2, for instance when M is a connected compact semisimple Lie group, like SO(n) and SU(n), or its non-compact dual, or Euclidean space acted on transitively by some group which is contained properly in the full group of rigid motions. Let G be the identity component of the isometry group of M. A curve in G may be thought of as a motion of a body in M with reference state at a point in M. For lambda in R (the pitch), we define a left invariant distribution D^lambda on G in such a way that a curve in G is always tangent to D^lambda if, at each instant, at the infinitesimal level, translating the body along a direction entails rotating it according to lambda L around that direction. We give conditions for the controllability of the control system on G determined by D^lambda and find explicitly all the geodesics of the natural sub-Riemannian structure on (G,D^lambda). We also study a similar system on R^7 \\rtimes SO(7) with L involving the octonionic cross product. In the appendix we give a friendly presentation of the non-compact dual of a compact classical group, as a set of \"small rotations\".",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-07T02:32:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C17",
      "49N10",
      "53C22",
      "70B10",
      "17B25"
    ],
    "keywords": [
      "sub-riemannian geometry",
      "screw motions",
      "constant pitch",
      "connected compact semisimple lie group",
      "non-compact dual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}