Finiteness theorem for the ray transform on a Riemannian manifold

Abstract

The ray transform I on a compact Riemannian manifold M with boundary is the operator sending a symmetric tensor field f to the set of integrals of f over all geodesics joining boundary points. A field f is called potential if it can be represented as the symmetric part of the covariant derivative of another tensor field vanishing on the boundary: The main result asserts that the space of potential tensor fields is a subspace of a finite codimension in Ker I if M is simple. A Riemannian manifold is called simple if every two points are joined by a unique geodesic.