On problem of polarization tomography, I

Abstract

The polarization tomography problem consists of recovering a matrix function f from the fundamental matrix of the equation $D\eta /dt={\pi }_{\dot{\gamma }}f\eta$ known for every geodesic $\gamma$ of a given Riemannian metric. Here ${\pi }_{\dot{\gamma }}$ is the orthogonal projection onto the hyperplan ${\dot{\gamma }}^{\perp }$. The problem arises in optical tomography of slightly anisotropic media. The local uniqueness theorem is proved: a $C^1$- small function f can be recovered from the data uniquely up to a natural obstruction. A partial global result is obtained in the case of the Euclidean metric on $R^3$.