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Math-2849 · mathematics · Pde

Recovery of both metric and U(2) connection from one boundary patch

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a compact connected smooth manifold M of dimension $n\geq 3$ with smooth boundary and one nonempty open patch Gamma, do equal zero-energy connection-Laplacian Dirichlet-to-Neumann pairings on compactly supported Gamma data determine smooth g and smooth u(2)-valued connection A up to a diffeomorphism Phi and a smooth unitary gauge U fixing Gamma? Explicitly ask $g_2=Phi^*g_1$ and $A_2=U^{-1}(Phi^*A_1)U+U^{-1}dU$, with $Phi|\Gamma =id$ and $U|\Gamma =I$.

Short English statement

For a compact connected smooth manifold M of dimension $n\geq 3$ with smooth boundary and one nonempty open patch Gamma, do equal zero-energy connection-Laplacian Dirichlet-to-Neumann pairings on compactly supported Gamma data determine smooth g and smooth u(2)-valued connection A up to a diffeomorphism Phi and a smooth unitary gauge U fixing Gamma? Explicitly ask $g_2=Phi^*g_1$ and $A_2=U^{-1}(Phi^*A_1)U+U^{-1}dU$, with $Phi|\Gamma =id$ and $U|\Gamma =I$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2849-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2849

Recovery of a metric and connection: The boundary data and observation are confined to the same arbitrary nonempty patch. The conclusion retains precisely the metric diffeomorphism and unitary gauge ambiguities, both equal to the identity on that patch; it recovers the rank-two connection together with the metric.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Recovery of both metric and U$2$ connection from one boundary patch

OpenAI's claim

Recovery of a metric and connection: Zero-frequency measurements with input and observation on one arbitrary open boundary patch are claimed to recover both a smooth metric and a unitary rank-two connection. Recovery is up to a diffeomorphism and unitary gauge that fix the measured patch.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix $n\ge3$ and a smooth connected manifold $M$ of that dimension, assumed compact with $\partial M$ smooth and nonempty. Choose a relatively open $\Gamma\subset\partial M$ which is nonempty and proper. On the trivial Hermitian rank-two bundle, set $d_A=d+A$, $A\in C^\infty(T^*M\otimes\mathfrak u(2))$, and define the local Dirichlet energy form using solutions of $d_A^{*g}d_Au=0$ with boundary data supported in $\Gamma$. If two smooth metrics and connections give the same form for every $f,h\in C_c^\infty(\Gamma;\mathbb C^2)$, the claim supplies smooth $\Phi:M\to M$ and $U:M\to U(2)$ with \[\Phi|_\Gamma=\operatorname{id},\quad U|_\Gamma=I,\quad g_2=\Phi^*g_1,\quad A_2=U^{-1}(\Phi^*A_1)U+U^{-1}dU.\] Thus the ambiguity in recovering the two geometric unknowns consists of the displayed diffeomorphism and gauge, each trivial on the observation patch.

Outline of the argument

Recovery of a metric and connection: Boundary factorization identifies normal-gauge jets, and a common exterior cap starts equality of Green matrices. Source solutions provide separating points, harmonic jets, and bundle frames. Mixed-kernel continuation in the product, followed by moving-sphere transfer and angular coercivity, extends the maximal local metric-and-bundle identification throughout the manifold.

Formal verification

Family 365 describes Lean coverage for the measurable scalar-conductivity obstruction; smooth anisotropic and coupled connection recovery are separate.

Recovery of a metric and connection: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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