oai-math-revised beta

Browse › Partial differential equations › 365 Joint metric and connection recovery from one boundary patch

Determination of a metric and a unitary connection from one boundary patch

OpenAI · October 5, 2026

Lean

Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

We prove that zero-frequency boundary measurements on any nonempty open boundary patch determine both a smooth Riemannian metric and a smooth unitary connection on the trivial Hermitian rank-two bundle over a compact connected smooth manifold of dimension at least three with smooth boundary. Both inputs and observations are restricted to the same patch. The metric and connection are determined up to a diffeomorphism and a unitary gauge that restrict to the identity on the measured patch.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Automatically collected notes (unreviewed)

work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Comments and discussion

News and blogs

  • Mikko Salo, Mathematics and AI (FAME Flagship, 2026-10-15). Identifies the release’s scalar and system versions of anisotropic Calderón recovery and its three-dimensional measurable-conductivity nonuniqueness claim as results of direct interest to inverse problems. [family-level]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026,
  author = {{OpenAI}},
  title = {{Determination of a metric and a unitary connection from one boundary patch}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026/paper.pdf}{OAI:Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/365.md. The Lean statements have not been checked by people against the paper's claims.

Family 365: Joint metric and connection recovery from one boundary patch

Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data.

Other papers in this family: