Boundary distance rigidity for general simple Riemannian metrics

Prove that every simple Riemannian metric is determined, up to the natural boundary-preserving isometry gauge, by its boundary distance function in all dimensions.

Background

The paper studies boundary rigidity for simple Finsler metrics within a fixed conformal class, proving uniqueness and stability for metrics of the form F=λF0F=\lambda F_0. In the Riemannian setting, the broader boundary rigidity problem asks whether a general simple metric is determined by the distances between boundary points, modulo boundary-preserving isometries.

The authors note that this general Riemannian rigidity statement is established in dimension two and for generic simple metrics in dimensions at least three, but remains unproved for arbitrary simple Riemannian metrics. This unresolved conjecture provides broader context for the conformal Finsler rigidity result proved in the paper.

References

It is a conjecture that general simple Riemannian metrics are boundary distance rigid .

— Conformal boundary rigidity for simple Finsler metrics  (2609.09527 - Lam et al., 8 Sep 2026) in Introduction, paragraph beginning “When the metric is a Riemannian metric”

Whether all convex two dimensional manifolds that have no trapped geodesics are lens rigid is among the most important open geometric inverse problems.

— Travel time tomography for elliptically anisotropic elastic planar media  (2610.13747 - Ilmavirta et al., 9 Oct 2026) in Section 1, Introduction

Third, boundary rigidity is not known for general simple manifolds in three or more dimensions.

— Travel time tomography for elliptically anisotropic elastic planar media  (2610.13747 - Ilmavirta et al., 9 Oct 2026) in Section 1, Section 1.4 (Our scope)