{
  "id": "a-metric-with-no-local-isometric-immersion",
  "title": "A metric with no local isometric immersion",
  "description": "Seven lessons with a complete proof of OpenAI's theorem (2026) that there is a smooth metric on a square such that no neighbourhood of the centre has a smooth isometric immersion into three-dimensional space. The height equation of an immersed surface; the construction of the metric from small squares that accumulate at the boundaries of disks of negative curvature; the patch obstruction, proved in five lessons: regularity of solutions of the height equation across the curve where the curvature changes its sign (an estimate at a free edge for an equation of mixed type), energy estimates where the curvature is negative, and a rapidly oscillating perturbation of the metric whose effect is measured by a moment and excluded by the theorem of Baire.",
  "units": [
    {
      "id": "heights-of-an-immersed-surface",
      "title": "Heights of an immersed surface",
      "reader": "heights-of-an-immersed-surface.html",
      "source": "src/heights-of-an-immersed-surface.md",
      "source_sha256": "6e88f157ca29953f89723bd8a406638e76f9c6bac6e5e05f8f0a18726448244b",
      "status": "author_self_checked"
    },
    {
      "id": "the-metric-and-the-theorem",
      "title": "The metric and the theorem",
      "reader": "the-metric-and-the-theorem.html",
      "source": "src/the-metric-and-the-theorem.md",
      "source_sha256": "13453179e6e57bf97902319d1c46a7e34ec77f9e88584110e77b92642d9f5cd0",
      "status": "author_self_checked"
    },
    {
      "id": "the-differentiated-height-equation",
      "title": "The differentiated height equation",
      "reader": "the-differentiated-height-equation.html",
      "source": "src/the-differentiated-height-equation.md",
      "source_sha256": "461dfa9292f417448b07cb7db221a7c42a304829de214da1492127ea43f077d1",
      "status": "author_self_checked"
    },
    {
      "id": "the-estimate-at-the-free-edge",
      "title": "The estimate at the free edge",
      "reader": "the-estimate-at-the-free-edge.html",
      "source": "src/the-estimate-at-the-free-edge.md",
      "source_sha256": "d17b712cfe8a8ce857669305ba28a70b5c049ca19cb54ed21aede3707ba7d92c",
      "status": "author_self_checked"
    },
    {
      "id": "regularity-on-a-cap",
      "title": "Regularity on a cap",
      "reader": "regularity-on-a-cap.html",
      "source": "src/regularity-on-a-cap.md",
      "source_sha256": "327a22646d0f6f63621453ea8618fe640831c0c54aa32afac221cc6c0581c9da",
      "status": "author_self_checked"
    },
    {
      "id": "energy-estimates-where-the-curvature-is-negative",
      "title": "Energy estimates where the curvature is negative",
      "reader": "energy-estimates-where-the-curvature-is-negative.html",
      "source": "src/energy-estimates-where-the-curvature-is-negative.md",
      "source_sha256": "cec72000bf94bccde45c4a78e1a373c226da970dcbb2542885512968a37f6997",
      "status": "author_self_checked"
    },
    {
      "id": "the-oscillating-perturbation",
      "title": "The oscillating perturbation",
      "reader": "the-oscillating-perturbation.html",
      "source": "src/the-oscillating-perturbation.md",
      "source_sha256": "c7219fa2adbf051231d1246c2a7144d20d15aa25780b6661f50626551bca400b",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "the lesson Submanifolds and hypersurfaces (course Bundles, curvature and holonomy), Proposition A.3: the Gauss formula in coordinates and the Gaussian curvature as det II / det g",
    "core course Real Analysis: differentiation under the integral, uniform convergence of series of derivatives, the inverse function theorem",
    "the lessons prove every other fact they use"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}