The Linearized Calderon Problem In Transversally Anisotropic Geometries

In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an Fourier-Bros-Iagolnitzer-type transform at certain points in the transversal manifold. This leads to proving a uniqueness result for transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic X-ray transform, which has limited earlier results on this problem.