Joint metric and connection recovery from one boundary patch

2 · Going further

The main problem is to recover a smooth Riemannian metric and a smooth unitary connection on a trivial Hermitian rank-two bundle over a compact, connected, smooth manifold of dimension $n\ge 3$ with smooth boundary. For any nonempty open boundary patch $\Gamma$, the available zero-frequency measurements have both inputs and observations restricted to $\Gamma$. The main manuscript claims that these data determine the pair up to a diffeomorphism and a unitary gauge, each restricting to the identity on $\Gamma$.

The metric-only companion addresses what it calls the smooth anisotropic Calderón uniqueness problem with arbitrary same-patch measurements. It claims that the zero-frequency Dirichlet-to-Neumann energy form on $\Gamma$ determines a smooth metric up to a diffeomorphism fixing $\Gamma$. Thus, the two recovery manuscripts concern the same boundary-access restriction, with the main manuscript additionally recovering the unitary connection. The abstracts state these conclusions but do not describe the argument or its intermediate steps, so no proof mechanism can be extracted from the supplied text.

The third manuscript supplies a contrasting nonuniqueness construction. It gives two distinct uniformly positive bounded measurable scalar conductivities on a ball in ${\mathbb{R}}^3$ with the same full Dirichlet-to-Neumann operator; both conductivities equal one near the boundary. This is the result explicitly identified as Lean-formalized. It concerns bounded measurable scalar conductivities rather than the smooth metric-and-connection setting of the recovery statements.

Useful notions

• Boundary inputs and responses
• Smoothness versus bounded measurable regularity
• Riemannian metrics
• Connections on complex vector bundles
• Recovery up to changes of description

What was known before / what is new

Limits and assumptions

Joint recovery assumes a compact connected smooth manifold of dimension at least three with smooth boundary, a smooth metric, and a smooth unitary connection on a trivial Hermitian rank-two bundle. Uniqueness is only up to a diffeomorphism and unitary gauge that are the identity on the measured patch. The formalization explicitly identified in the supplied source covers the bounded measurable scalar-conductivity nonuniqueness result; the smooth recovery statements are presented as manuscript claims, not as separately identified Lean-verified results. The nonuniqueness example is on a three-dimensional ball and does not contradict the stated smooth recovery results. The abstracts supply neither argument details nor an explicit list of remaining open questions.

Mini glossary

Riemannian metric

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A mathematical description of the geometry whose recovery is sought in the smooth uniqueness statements.

Unitary connection

—

A rule for transporting complex vectors that respects their Hermitian inner product.

Trivial Hermitian rank-two bundle

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A globally product-shaped collection of two-dimensional complex vector spaces equipped with Hermitian inner products over the manifold.

Dirichlet-to-Neumann operator

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The boundary measurement map relating imposed boundary values to the corresponding boundary flux responses.

Diffeomorphism

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A smooth invertible change of coordinates with a smooth inverse, allowed here only when it fixes the measured patch.

Unitary gauge

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An inner-product-preserving change of description in the bundle, allowed here only when it is the identity on the measured patch.

Keywords : Inverse boundary problems · Same-patch recovery · Conductivity nonuniqueness

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family