{
  "id": "1411.5425",
  "version": "v1",
  "published": "2014-11-20T02:39:55.000Z",
  "updated": "2014-11-20T02:39:55.000Z",
  "title": "Tangent spaces and tangent bundles for diffeological spaces",
  "authors": [
    "J. Daniel Christensen",
    "Enxin Wu"
  ],
  "comment": "24 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "Tangent spaces and tangent bundles of smooth manifolds are the building blocks for differential geometry. We study how these structures can be extended to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. There are several equivalent ways of defining the tangent space of a smooth manifold at a point which give rise to inequivalent definitions in the case of diffeological spaces. In this paper we focus on two ways. The internal tangent space of a diffeological space is defined using smooth curves into the space, and the external tangent space is defined using smooth derivations on germs of smooth functions. After proving basic results about these tangent spaces, we compare them by calculating many examples and observe that while they agree for smooth manifolds and most of the examples, they do not agree in general. After this, we recall Hector's definition of the tangent bundle of a diffeological space as the union of the internal tangent spaces with a certain diffeology. We show that both scalar multiplication and addition can fail to be smooth, revealing errors in several references. We then give an improved definition of the tangent bundle, using what we call the dvs diffeology, which ensures that scalar multiplication and addition are smooth. We establish basic facts about these tangent bundles, compute them in many examples, and study the question of whether the fibres of the tangent bundles are fine diffeological vector spaces. Our examples include singular spaces, spaces whose natural topology is non-Hausdorff (e.g., irrational tori), infinite-dimensional vector spaces and diffeological groups, and spaces of smooth maps between smooth manifolds (including diffeomorphism groups).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-20T02:39:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tangent bundle",
      "diffeological space",
      "smooth manifold",
      "internal tangent space",
      "scalar multiplication"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.5425C"
    }
  }
}