{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "GA7437914V"
    },
    {
      "field": "specializes",
      "target": "NJ21PN5ZGJ"
    },
    {
      "field": "defines",
      "target": "VZBYXH5995"
    }
  ],
  "body": "\n## Summary\n\nAn Ehresmann connection on a [[fibre bundle|CX4M8K61AB]] is a choice of horizontal directions at every point of the total space.\nThese directions are complementary to the directions along the fibres.\nIt works for any smooth fibre bundle, not only for vector bundles, and needs no linear structure.\nIt lets one lift curves in the base to horizontal curves in the total space, which gives parallel transport between fibres.\nConnections on vector bundles and on principal bundles are Ehresmann connections with extra symmetry.\n\n## Explanations\n\n### L1 human\n\nPicture a bundle as a family of fibres standing over the points of a base.\nAt each point of the total space some directions go along the fibre; these are vertical and are given by the bundle.\nAn Ehresmann connection chooses which of the remaining directions count as horizontal.\nFor the product of a line and a circle, the obvious choice is to call the line directions horizontal.\nThen moving in the base lifts uniquely to moving horizontally upstairs.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "53C05"
    ],
    "name": "Ehresmann connection",
    "same_as": {
      "wikidata": "Q2993471",
      "wikipedia": "Ehresmann connection"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ehresmann_connection"
      }
    ],
    "tag": "TD6VEP3SCE"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/TD6VEP3SCE/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/TD6VEP3SCE.md",
  "msc": [
    "53C05"
  ],
  "name": "Ehresmann connection",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TD6/TD6VEP3SCE.md",
  "status": null,
  "tag": "TD6VEP3SCE",
  "uri": "m2://TD6VEP3SCE",
  "url": "https://math2.tamnd.com/wiki/ehresmann-connection/",
  "writing": {
    "tag": "TD6VEP3SCE",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}