Metric surface realization research library

Family 333: Smooth isometric immersions of surfaces into ℝ⁴. First application triage, 9 October 2026 Australia/Brisbane.

Problem and potential new use

Geometry researchers can organize four-dimensional smooth isometric immersions for arbitrary closed surface metrics.

Applicability and commercial boundary

An immersion may self-intersect and is different from an embedding. The existence statement supplies no practical 3D fabrication or 4D numerical construction.

Initial business decision

Research or conditional engineering only. A practical implementation, recurring buyer problem and willingness to pay have not been established. Mathematical usefulness alone is insufficient evidence for a standalone lucrative product.

Next verification action

Inspect the constructive proof and separately certify injectivity if an application needs an embedding.

Evidence scope

The catalog statement was individually reviewed. Main-paper abstracts and available selected formal scope notes were also inspected; family 354’s explicit main profile formula was read. No independent Lean verification was run. Source revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. See source metadata.

Smooth isometric immersions of closed surfaces into Euclidean four-space.