Geodesic Tomography Problems on Riemannian Manifolds

Abstract

This dissertation is concerned with integral geometric inverse problems. The geodesic ray transform is an operator that encodes the line integrals of a function along geodesics. The dissertation establishes many conditions when such information determines a function uniquely and stably. A new numerical model for computed tomography imaging is created as a part of the dissertation. The introduction of the dissertation contains an introduction to inverse problems and mathematical models associcated to computed tomography. The main focus is in definitions of integral geometry problems, survey of the related literature, and introducing the main results of the dissertation. A list of important open problems in integral geometry is given. In the first article of the dissertation, it is shown that a symmetric solenoidal tensor field can be determined uniquely from its geodesic ray transform on Cartan-Hadamard manifolds, when certain geometric decay conditions are satisfied. The studied integral transforms appear in inverse scattering theory in quantum physics and general relativity. In the second article of the dissertation, it is shown that a piecewise constant vector-valued function can be determined uniquely from its geodesic ray transform with a continuous and non-singular matrix weight on Riemannian manifolds that admit a strictly convex function and have a strictly convex boundary. These integral transforms can be used to model attenuated ray transforms and inverse problems for connections and Higgs fields. The third and fourth articles of the dissertation study the geodesic ray transform over closed geodesics on flat tori when the functions have low regularity assumptions. The fourth article studies a generalization of the geodesic ray transform when the integrals of a function are known over lower dimensional isometrically embedded flat tori. New inversion formulas, regularization strategies and stability estimates are proved in the articles. The new results have applications in different computational tomography methods.