{
  "id": "1009.5841",
  "version": "v1",
  "published": "2010-09-29T11:08:46.000Z",
  "updated": "2010-09-29T11:08:46.000Z",
  "title": "On a construction of Burago and Zalgaller",
  "authors": [
    "Emil Saucan"
  ],
  "comment": "24 pages",
  "categories": [
    "math.DG",
    "math.CV"
  ],
  "abstract": "The purpose of this note is to scrutinize the proof of Burago and Zalgaller regarding the existence of $PL$ isometric embeddings of $PL$ compact surfaces into $\\mathbb{R}^3$. We conclude that their proof does not admit a direct extension to higher dimensions. Moreover, we show that, in general, $PL$ manifolds of dimension $n \\geq 3$ admit no nontrivial $PL$ embeddings in $\\mathbb{R}^{n+1}$ that are close to conformality. We also extend the result of Burago and Zalgaller to a large class of noncompact $PL$ 2-manifolds. The relation between intrinsic and extrinsic curvatures is also examined, and we propose a $PL$ version of the Gauss compatibility equation for smooth surfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-29T11:08:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B70",
      "57R40",
      "53C42",
      "30C65"
    ],
    "keywords": [
      "construction",
      "gauss compatibility equation",
      "smooth surfaces",
      "extrinsic curvatures",
      "isometric embeddings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.5841S"
    }
  }
}