Two dimensional compact simple Riemannian manifolds are boundary distance rigid

Abstract

We prove that knowing the lengths of geodesics joining points of the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

“…Unfortunately, notations of [13] do not agree with [14]. For example, the manifold ∂ + ΩM of [14] is denoted by ∂ − Ω(M ) in [13] and vice versa, the Hilbert transform is denoted by H in [13] while H stands for the differentiation with respect to the geodesic flow in [14], and so on. All notations of the current paper are agreed with [14] as far as possible.…”

“…To explain the similarity and difference between the proofs of Theorems 1.2 and 1.3, let us cite the crucial paragraph of [13]:…”

“…In our opinion, the main achievement of [13] is discovering a relationship between the boundary distance function and Dirichlet-to-Neumann (DN) map of a simple 2- The next step is the following…”

“…One of the main tools of [13] is the commutator formula for the Hilbert transform and the operator H of differentiation with respect to the geodesic flow. We present an alternative proof of the formula in Section 8.…”

“…In the recent paper [13] by Pestov-Uhlmann, boundary rigidity of a simple two-dimensional Riemannian manifold is proved. The present article uses the same approach for solving the corresponding linear problem.…”