Reconstruction along a geodesic from sphere data in Finsler geometry and anisotropic elasticity

Abstract

Dix formulated the inverse problem of recovering an elastic body from the measurements of wave fronts of point scatterers. We geometrize this problem in the framework of linear elasticity, leading to the geometrical inverse problem of recovering a Finsler manifold from certain sphere data in a given open subset of the manifold. We solve this problem locally along any geodesic through the measurement set.

Stable reconstruction of simple Riemannian manifolds from unknown interior sources

“…The sphere data described in connection to the Riemannian results of [11] above has also been studied on Finsler manifolds [12]. Knowledge of the spheres uniquely determine the fundamental tensor and the curvature operator along any geodesic passing through the known domain, but in contrast to Riemannian geometry this information is insufficient for a full reconstruction of the universal cover.…”

“…As these functions differ from the boundary distance functions r s : ∂M → R, r s (x) = d(x, π(s)), only by a constant, we thus know the differential and the Hessian of each r s on all of ∂M . Therefore the data determines the critical points of these functions and the function E(p, y), for p = π(s) and y ∈ c(p), of (12). From this one can easily compute E(x) and E from ( 13) and ( 14).…”