gpt 6 astra ultra attempt on Math-279

Michel Boundary Rigidity Conjecture for Simple Manifolds

Definitions and mathematical statement

A simple Riemannian metric on a compact manifold with boundary has strictly convex boundary, no conjugate points, and unique geodesic connections between points, in the cited convention. Its boundary distance function is $d_g:\partial M\times\partial M\to{\mathbb{R}}_{\ge0}$, computed using curves in the whole manifold. For smooth simple metrics $g,g'$ on the same $n$-manifold, $n\ge3$, the conjecture asserts \[ d_g|_{\partial M\times\partial M}=d_{g'}|_{\partial M\times\partial M} \quad\Longrightarrow\quad g'=\phi^*g, \qquad\phi|_{\partial M}=\mathrm{id}. \] The coordinate ambiguity is unavoidable. Boundary-to-boundary distances are the given data, not all interior distances or a full Dirichlet-to-Neumann operator.

Formulation and concept references: [S1].

Short English statement

Do all boundary-to-boundary distances determine a simple interior Riemannian metric, up to a diffeomorphism fixing the boundary?

Research attempt

Scope and source audit.

This unresolved attempt was prepared by GPT-6 Astra Ultra on 27 September 2026 using first-variation calculations, an integral-geometric argument, and inspection of Stefanov--Uhlmann--Vasy [S1]. Their local and global results have specified convexity or foliation hypotheses; simplicity is not silently replaced by the existence of such a foliation. The data here remain boundary distances. The route proves ordered-metric and conformal subcases directly, identifies the linearized gauge kernel, and gives an exact sufficient stability package for local uniqueness. None of these supplies the full global result for arbitrary simple metrics in dimension at least three.

The boundary-fixing ambiguity is real.

If $\phi:M\to M$ is a diffeomorphism restricting to the identity on the boundary, the $\phi^*g$-length of a curve equals the $g$-length of its image. Curves with prescribed boundary endpoints correspond bijectively, so their length infima agree: \[ d_{\phi^*g}(x,y)=d_g(x,y),\qquad x,y\in\partial M. \] This proves the invariance exactly, without linearization. A claimed reconstruction of metric coefficients in a fixed interior coordinate system would therefore demand more than the data determine. The correct target is the indicated quotient by such diffeomorphisms.

There is also a useful rigidity result when a pointwise comparison is available. Suppose $g'\ge g$ as quadratic forms and both metrics are simple, with equal boundary distances. If $g'_p(v,v)>g_p(v,v)$ for some nonzero interior vector, follow the $g'$ geodesic through $(p,v)$ to its two boundary endpoints. Simplicity makes this whole segment minimizing and ensures it exits in both directions. Along it the $g$ speed is at most the $g'$ speed and strictly smaller on an open subsegment. Hence \[ d_g(x,y)\le L_g(\gamma)<L_{g'}(\gamma)=d_{g'}(x,y), \] a contradiction. Strict inequality at a boundary point would persist at nearby interior points by continuity. Thus $g'=g$. The same argument works with the order reversed. In general two metrics are not pointwise ordered, so this does not settle the conjecture.

The full conformal subcase from length inequalities.

Assume $g'=c^2g$ with a smooth positive function $c$, and both metrics simple. For each maximal unit-speed $g$ geodesic with boundary endpoints, equality of the boundary distances implies \[ \int_\gamma(c-1)\,dt\ge0. \] Indeed its $g$ length is the boundary distance, whereas its $g'$ length is at least the same boundary distance. Integrate this inequality over the incoming unit vectors at the boundary with the natural flux weight. The Santaló formula gives \begin{equation}\label{wsref-0-61-32-37-35-3a-63-6f-6e-66-6f-72-6d-61-6c-31} \int_M(c-1)\,dV_g\ge0. \end{equation} The formula used here parametrizes almost every unit tangent vector by its entrance point, entrance direction and travel time. Geodesic flow preserves Liouville measure, and the boundary Jacobian is the absolute inward normal component. For a function depending only on the base point, integration over each unit sphere supplies the same positive sphere-volume factor, which cancels. Simplicity gives nontrapping and makes these maximal segments the minimizing ones needed above. This standard integral-geometric identity is the external input to the conformal argument.

Apply the same reasoning to $g'$ geodesics, now measuring their length in $g$. Since the speed ratio is $c^{-1}$ and $dV_{g'}=c^n dV_g$, \[ \int_M(c^{n-1}-c^n)\,dV_g\ge0. \] Normalize $dV_g$ to a probability measure and write $\mathbb E$ for its integral. We have $\mathbb E c\ge1$ and $\mathbb E c^n\le\mathbb E c^{n-1}$. Hölder's inequality gives \[ \mathbb E c^{n-1}\le(\mathbb E c^n)^{(n-1)/n}. \] Since $c>0$, these inequalities force $\mathbb E c^n\le1$. On the other hand, Jensen gives $(\mathbb E c)^n\le\mathbb E c^n$. Thus all these inequalities are equalities and strict convexity of $x\mapsto x^n$, for $n>1$, forces $c$ constant almost everywhere. Smoothness and the mean value make $c\equiv1$. This proves rigidity within one fixed conformal class.

The two Santaló integrations use each metric's own flux measure; there is no unjustified identification of their boundary sphere bundles. Only their separately obtained interior inequalities are compared. For general anisotropic metrics a single scalar $c$ does not encode the directional length ratio and volume change simultaneously. The moment argument therefore cannot be transferred by replacing a tensor with its determinant.

Linearization produces the tensor X-ray transform.

Let $g_s=g+s h+O(s^2)$ be a smooth family of simple metrics and let $\gamma_s$ be the minimizing geodesic from fixed distinct boundary points $x$ to $y$. Differentiating length at $s=0$ has two terms: the metric derivative and the curve variation. The curve term vanishes because $\gamma_0$ is a geodesic with fixed endpoints. With unit-speed $g$ parametrization, \[ \left.\frac d{ds}\right|_{s=0}d_{g_s}(x,y) =\frac12\int_{\gamma_0}h(\dot\gamma_0,\dot\gamma_0)\,dt =\tfrac12 I_g h(x,y). \] The factor one half comes from differentiating a square root. It is not an assertion that a finite change in boundary distance equals this fixed-geodesic integral.

The gauge ambiguity survives linearization. If $X$ is a vector field vanishing on the boundary and $h=\mathcal L_Xg$, then along a unit geodesic \[ h(\dot\gamma,\dot\gamma) =2\langle\nabla_{\dot\gamma}X,\dot\gamma\rangle =2\frac d{dt}\langle X,\dot\gamma\rangle. \] The endpoint values vanish, so $I_gh=0$. Thus injectivity on all symmetric tensors is false even at the infinitesimal level. One must remove these potential tensors, for example by a solenoidal gauge, before asking for injectivity or a stability estimate.

A complete flat, compact-support linearized calculation.

Take a smooth compactly supported symmetric tensor $h$ on $\mathbb R^n$, satisfying the solenoidal equations $\sum_i\partial_i h_{ij}=0$. Suppose its integrals $\int_{\mathbb R}h_{ij}(x+tv)v_iv_j\,dt$ vanish on every line. Fourier transformation in the line's perpendicular displacement gives \[ \widehat h(\xi)(v,v)=0 \quad\text{whenever }\xi\cdot v=0. \] The Fourier-transformed solenoidal equations say $\widehat h(\xi)\xi=0$. For $\xi\ne0$, use an orthonormal basis with last vector parallel to $\xi$. The last row and column of the symmetric matrix vanish. Its restriction to $\xi^\perp$ has zero quadratic form on every real vector, so polarization makes that whole block zero as well. Hence $\widehat h(\xi)=0$ for $\xi\ne0$; continuity at zero finishes $h=0$.

This proves injectivity in that flat solenoidal class. Compact support allows ordinary Fourier transformation and line integration without boundary terms. A general tensor on a manifold cannot be extended by zero and assumed smooth and solenoidal across the boundary. Nor do curved geodesics obey this Fourier-slice identity. The computation therefore supplies a valid model calculation, not an unstated solution of the curved tensor-transform problem.

A precise conditional route from linear to nonlinear uniqueness.

Suppose a gauge has been chosen near a fixed metric so that a nearby competitor can be written $g'=g+h$ in a Banach space $X$, and suppose there are a data norm $Y$ and constants $C,C'$ such that \[ \|h\|_X\le C\|I_gh\|_Y, \qquad \|d_{g+h}-d_g-\tfrac12 I_gh\|_Y\le C'\|h\|_X^2. \] If the distances agree, then $\|h\|_X\le2CC'\|h\|_X^2$. For $\|h\|_X<(2CC')^{-1}$, this forces $h=0$. This is a rigorous local uniqueness deduction once the two estimates and the gauge construction hold in compatible spaces.

The assumptions are substantial. A mere injectivity statement with no quantitative estimate does not control the nonlinear error. If the stability estimate loses derivatives, the quadratic remainder must be estimated in a matching scale of norms; one cannot substitute an unrelated small norm into the absorption inequality. The normal-gauge and convex-foliation analysis in [S1] addresses precisely such analytic issues in its stated settings. This paragraph specifies a sufficient package and does not assert it for every simple metric.

Why local recovery does not automatically globalize.

Along a path $g_s$ joining two metrics, integrating the first-variation formula involves geodesics $\gamma_s$ that change with $s$. Zero total boundary-distance difference therefore does not imply $I_g(g'-g)=0$ for the initial metric. The linearized transform cannot be applied to a finite difference while discarding this path dependence. Likewise, recovering a collar near the boundary only permits repeated layer stripping if one can produce the next suitable strictly convex interior boundary and retain the required data. Simplicity alone has not been shown here to supply that global foliation.

Verification and outcome.

The ordered-metric argument checks the direction of both length inequalities. The conformal proof checks the two different volume measures and all powers of $c$. The linearization retains its factor one half, and the gauge kernel is an exact endpoint derivative. The strongest conclusions are the ordered and conformal subcases, the flat solenoidal transform calculation, and the stated conditional local uniqueness criterion. The first general gap is a global rigidity mechanism for arbitrary simple anisotropic metrics, beyond the verified source hypotheses. Michel's full conjecture remains UNRESOLVED.

Sources

• S1 Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge, arXiv1702.03638v3,12May2021; internal revision13May2021. https://arxiv.org/pdf/1702.03638v3. Formulation source.