{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1YKCH218MN"
    },
    {
      "field": "uses",
      "target": "45P2SEGGR1"
    },
    {
      "field": "uses",
      "target": "F4WDYPQ9VP"
    },
    {
      "field": "uses",
      "target": "SSCJWAQ7AX"
    },
    {
      "field": "about",
      "target": "45P2SEGGR1"
    },
    {
      "field": "defines",
      "target": "E8TQZG66XM"
    }
  ],
  "body": "\n## Summary\n\nThe heat kernel of a Riemannian manifold is the fundamental solution of the heat equation $\\partial_t u + \\Delta u = 0$, where $\\Delta$ is the [[Laplace–Beltrami operator|F4NWZNTRMD]].\nOn a closed manifold it is the integral kernel of $e^{-t\\Delta}$ and is the sum of $e^{-t\\lambda_j} \\phi_j(x) \\phi_j(y)$ over an orthonormal basis of eigenfunctions.\nIts behaviour for small time encodes local geometry, such as volume and curvature, and its trace encodes the spectrum.\nIt is the main tool of the heat equation approach to [[index theory|675MAXBQN6]] and [[spectral geometry|QKAFRDPGE4]].\n\n## Explanations\n\n### L1 human\n\nPut a unit of heat at one point $y$ of a curved surface and let it spread.\nThe heat kernel $K(t, x, y)$ is the temperature at $x$ after time $t$.\nFor short times heat has not yet felt the shape of the surface, and $K$ looks like the Gaussian of flat space; the curvature appears in correction terms.\nFor long times on a closed surface, the heat spreads out evenly and $K$ tends to $1/\\mathrm{area}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "58J35",
      "35K08"
    ],
    "name": "Heat kernel on a Riemannian manifold",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Heat_kernel"
      }
    ],
    "specializes": [
      "KD404MF042"
    ],
    "tag": "327NP6S7H5"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/327NP6S7H5/",
  "links": [
    {
      "field": "specializes",
      "target": "KD404MF042"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/327NP6S7H5.md",
  "msc": [
    "58J35",
    "35K08"
  ],
  "name": "Heat kernel on a Riemannian manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/327/327NP6S7H5.md",
  "status": null,
  "tag": "327NP6S7H5",
  "uri": "m2://327NP6S7H5",
  "url": "https://tamnd.github.io/math2/wiki/heat-kernel-on-a-riemannian-manifold/",
  "writing": {
    "tag": "327NP6S7H5",
    "kind": "concept",
    "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
      }
    ]
  }
}