{
  "id": "2308.07089",
  "version": "v1",
  "published": "2023-08-14T11:38:51.000Z",
  "updated": "2023-08-14T11:38:51.000Z",
  "title": "Covariant Derivatives on Homogeneous Spaces -- Horizontal Lifts and Parallel Transport",
  "authors": [
    "Markus Schlarb"
  ],
  "comment": "36 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We consider invariant covariant derivatives on reductive homogeneous spaces corresponding to the well-known invariant affine connections. These invariant covariant derivatives are expressed in terms of horizontally lifted vector fields on the Lie group. This point of view allows for a characterization of parallel vector fields along curves. Moreover, metric invariant covariant derivatives on a reductive homogeneous space equipped with an invariant pseudo-Riemannian metric are characterized. As a by-product, a new proof for the existence of invariant covariant derivatives on reductive homogeneous spaces and their the one-to-one correspondence to certain bilinear maps is obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-14T11:38:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53B05",
      "53C30",
      "53C22"
    ],
    "keywords": [
      "horizontal lifts",
      "parallel transport",
      "reductive homogeneous space",
      "metric invariant covariant derivatives",
      "well-known invariant affine connections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}