{
  "id": "1806.05440",
  "version": "v1",
  "published": "2018-06-14T10:00:54.000Z",
  "updated": "2018-06-14T10:00:54.000Z",
  "title": "A new geometric structure on tangent bundles",
  "authors": [
    "Nikos Georgiou",
    "Brendan Guilfoyle"
  ],
  "comment": "23 pages, AMS-Tex",
  "categories": [
    "math.DG"
  ],
  "abstract": "For a Riemannian manifold $(N,g)$, we construct a scalar flat metric $G$ in the tangent bundle $TN$. It is locally conformally flat if and only if either, $N$ is a 2-dimensional manifold or, $(N,g)$ is a real space form. It is also shown that $G$ is locally symmetric if and only if $g$ is locally symmetric. We then study submanifolds in $TN$ and, in particular, find the conditions for a curve to be geodesic. The conditions for a Lagrangian graph to be minimal or Hamiltonian minimal in the tangent bundle $T{\\mathbb R}^n$ of the Euclidean real space ${\\mathbb R}^n$ are studied. Finally, using the cross product in ${\\mathbb R}^3$ we show that the space of oriented lines in ${\\mathbb R}^3$ can be minimally isometrically embedded in $T{\\mathbb R}^3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-14T10:00:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tangent bundle",
      "geometric structure",
      "euclidean real space",
      "locally symmetric",
      "real space form"
    ],
    "note": {
      "typesetting": "AMS-TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}