{
  "id": "1007.0979",
  "version": "v1",
  "published": "2010-07-06T18:49:02.000Z",
  "updated": "2010-07-06T18:49:02.000Z",
  "title": "Inverse problems for differential forms on Riemannian manifolds with boundary",
  "authors": [
    "Katsiaryna Krupchyk",
    "Matti Lassas",
    "Gunther Uhlmann"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Consider a real-analytic orientable connected complete Riemannian manifold $M$ with boundary of dimension $n\\ge 2$ and let $k$ be an integer $1\\le k\\le n$. In the case when $M$ is compact of dimension $n\\ge 3$, we show that the manifold and the metric on it can be reconstructed, up to an isometry, from the set of the Cauchy data for harmonic $k$-forms, given on an open subset of the boundary. This extends a result of [13] when $k=0$. In the two-dimensional case, the same conclusion is obtained when considering the set of the Cauchy data for harmonic $1$-forms. Under additional assumptions on the curvature of the manifold, we carry out the same program when $M$ is complete non-compact. In the case $n\\ge 3$, this generalizes the results of [12] when $k=0$. In the two-dimensional case, we are able to reconstruct the manifold from the set of the Cauchy data for harmonic $1$-forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-06T18:49:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R30",
      "58A10",
      "58J10",
      "58J32"
    ],
    "keywords": [
      "inverse problems",
      "differential forms",
      "cauchy data",
      "two-dimensional case",
      "orientable connected complete riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.0979K"
    }
  }
}