{
  "id": "2411.12511",
  "version": "v1",
  "published": "2024-11-19T13:52:42.000Z",
  "updated": "2024-11-19T13:52:42.000Z",
  "title": "A Calderón Problem for Beltrami Fields",
  "authors": [
    "Alberto Enciso",
    "Carlos Valero"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "On a $3$-dimensional Riemannian manifold with boundary, we define an analogue of the Dirichlet-to-Neumann map for Beltrami fields, which are the eigenvectors of the curl operator and play a major role in fluid mechanics. This map sends the normal component of a Beltrami field to its tangential component on the boundary. In this paper we establish two results showing how this normal-to-tangential map encodes geometric information on the underlying manifold. First, we show that the normal-to-tangential map is a pseudodifferential operator of order zero on the boundary whose total symbol determines the Taylor series of the metric at the boundary. Second, we go on to show that a real-analytic simply connected $3$-manifold can be reconstructed from its normal-to-tangential map. Interestingly, since Green's functions do not exist for the Beltrami field equation, a key idea of the proof is to find an appropriate substitute, which turn out to have a natural physical interpretation as the magnetic fields generated by small current loops.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-19T13:52:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "calderón problem",
      "normal-to-tangential map encodes geometric information",
      "dimensional riemannian manifold",
      "total symbol determines",
      "beltrami field equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}