{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]] of dimension $m$, possibly with boundary, and let $(N, h)$ be a Riemannian manifold of dimension $n$.\nLet $u \\colon M \\to N$ be a smooth map.\nThe *energy density* of $u$ is the function $e(u) = \\frac{1}{2} \\abs{du}^2$ on $M$.\nHere $\\abs{du}^2$ is the squared norm of $du_x \\colon T_x M \\to T_{u(x)} N$ for the metrics $g$ and $h$.\nIn local coordinates $x^i$ on $M$ and $y^\\alpha$ on $N$ it is\n$$\\abs{du}^2 = \\sum_{i,j,\\alpha,\\beta} g^{ij}(x) \\, h_{\\alpha\\beta}(u(x)) \\, \\frac{\\partial u^\\alpha}{\\partial x^i} \\frac{\\partial u^\\beta}{\\partial x^j}.$$\nThe *energy* of $u$ is\n$$E(u) = \\int_M e(u) \\, dV_g = \\frac{1}{2} \\int_M \\abs{du}^2 \\, dV_g,$$\nwhere $dV_g$ is the Riemannian volume measure of $g$.\n\n## Remarks\n\nEquivalently, $\\abs{du}^2 = \\sum_i \\abs{du(e_i)}_h^2$ for an orthonormal basis $(e_i)$ of $T_x M$, which is the trace with respect to $g$ of the pull-back metric $u^* h$.\nThe same formula defines $E(u)$ for maps of Sobolev class $W^{1,2}$, and for non-compact $M$ when the integral converges.\nFor $N = \\R$ it is the Dirichlet energy $\\frac12 \\int_M \\abs{\\nabla u}^2 \\, dV_g$, and for $M$ an interval it is the [[energy of a path|Q12AB1MJ64]].\nWhen $\\dim M = 2$, the energy is invariant under conformal changes of the metric $g$.\nIts critical points are the [[harmonic maps|KZJYG9DS3M]].\n\nExample: for the identity map of a compact $(M, g)$ of dimension $m$, $du(e_i) = e_i$, so $\\abs{du}^2 = m$ and $E(\\id) = \\frac{m}{2} \\operatorname{Vol}(M, g)$.\n\nNon-example: the formula gives no finite energy for the identity map of $\\R$ with the Euclidean metric, since $\\R$ is not compact.\nThere $e = \\frac{1}{2}$ everywhere, and the integral over $\\R$ diverges.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "E6HDE1DFXD",
    "kind": "definition",
    "msc": [
      "58E20",
      "53C43"
    ],
    "name": "Energy of a map between Riemannian manifolds",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Harmonic_map"
      }
    ],
    "tag": "5N1825CM10",
    "uses": [
      "7CBT4SM83Y",
      "F45VQ47GPV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/5N1825CM10/",
  "links": [
    {
      "field": "defines",
      "target": "E6HDE1DFXD"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "F45VQ47GPV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5N1825CM10.md",
  "msc": [
    "58E20",
    "53C43"
  ],
  "name": "Energy of a map between Riemannian manifolds",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5N1/5N1825CM10.md",
  "status": null,
  "tag": "5N1825CM10",
  "uri": "m2://5N1825CM10",
  "url": "https://tamnd.github.io/math2/wiki/energy-of-a-map-between-riemannian-manifolds-definition/",
  "writing": {
    "tag": "5N1825CM10",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}