{
  "id": "1108.5403",
  "version": "v1",
  "published": "2011-08-26T22:20:22.000Z",
  "updated": "2011-08-26T22:20:22.000Z",
  "title": "Characterization of a Banach-Finsler manifold in terms of the algebras of smooth functions",
  "authors": [
    "J. A. Jaramillo",
    "M. Jimenez-Sevilla",
    "L. Sanchez-Gonzalez"
  ],
  "comment": "13 pages",
  "categories": [
    "math.FA",
    "math.DG",
    "math.GN"
  ],
  "abstract": "In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete $C^k$ Finsler manifold $M$ is determined by the normed algebra $C_b^k(M)$ of all real-valued, bounded and $C^k$ smooth functions with bounded derivative defined on $M$. As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete $C^k$ Finsler manifold $M$ is determined by the algebra $C_b^k(M)$; (ii) the weak Finsler structure of a separable and complete $C^k$ Finsler manifold $M$ modeled on a Banach space with a Lipschitz and $C^k$ smooth bump function is determined by the algebra $C^k_b(M)$; (iii) the weak Finsler structure of a $C^k$ uniformly bumpable and complete $C^k$ Finsler manifold $M$ modeled on a Weakly Compactly Generated (WCG) Banach space with an (equivalent) $C^k$ smooth norm is determined by the algebra $C^k_b(M)$; and (iii) the isometric structure of a WCG Banach space $X$ with an $C^1$ smooth bump function is determined by the algebra $C_b^1(X)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-26T22:20:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58B10",
      "58B20",
      "46T05",
      "46T20",
      "46E25",
      "46B20",
      "54C35"
    ],
    "keywords": [
      "smooth functions",
      "weak finsler structure",
      "banach-finsler manifold",
      "smooth bump function",
      "characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.5403J"
    }
  }
}