Semiglobal boundary rigidity for Riemannian metrics

Abstract

Let (M, g) be a Riemannian manifold with boundary. In this paper we consider the problem of determining the metric g from its associated boundary distance function dg(x, y) = dist (x, y), x, y ∈ ∂M, that is the geodesic distance between boundary points. This problem arose in rigidity questions in Riemannian geometry [12], [6], [8]. For the case in which M is a bounded domain of Euclidean space and the metric is conformal to the Euclidean one, this problem is called the inverse kinematic problem which arose in Geophysics and has a long history starting at least in the early part of the 20th century with Herglotz [10]. He considered the case where M is a ball {x ∈ R | r = |x| ≤ R} equipped with a spherically symmetric metric ds2 = dx2/c2(r) where c(r) is a positive function depending only on the radius r = |x|. Herglotz found a formula to determine c(r) from the boundary distance function. Physically this corresponds to the case of a spherically symmetric Earth model with an index of refraction depending only on the radius. The boundary distance function corresponds to the travel times of e.g. acoustic waves going through the Earth and measured at the surface. The general problem for the case that the sound speed depends on all variables has been extensively studied (see for instance [17] and the references given there). Also, this problem has a close connection for other inverse problems related to determining the sound speed from boundary measurements, see [20].