An Integral Geometry Problem in a Nonconvex Domain

Abstract

We consider the problem of recovering the solenoidal part of a symmetric tensor field f on a compact Riemannian manifold (M,g) with boundary from the integrals of f over all geodesics joining boundary points. All previous results on the problem are obtained under the assumption that the boundary ∂M is convex. This assumption is related to the fact that the family of maximal geodesics has the structure of a smooth manifold if ∂M is convex and there is no geodesic of infinite length in M. This implies that the ray transform of a smooth field is a smooth function and so we may use analytic techniques. Instead of convexity of ∂M we assume that ∂M is a smooth domain in a larger Riemannian manifold with convex boundary and the problem under consideration admits a stability estimate. We then prove uniqueness of a solution to the problem for