A metric with no local isometric immersion

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.

Seven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Heights of an immersed surface
2. The metric and the theorem
3. The differentiated height equation
4. The estimate at the free edge
5. Regularity on a cap
6. Energy estimates where the curvature is negative
7. The oscillating perturbation

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Submanifolds and hypersurfaces (Bundles, curvature and holonomy)
• The Gauss formula and the Gaussian curvature of a surface in space (lesson Submanifolds and hypersurfaces, course Bundles, curvature and holonomy); the lesson proves the other facts it uses.