2024

Kykkänen, Antti

Is a function uniquely determined by its integrals over geodesics of a Riemannian manifold? This question — known as geodesic X-ray tomography — is a geometric generalization of the classical problem of recovering a function from its integrals along lines encountered in medical applications of X-ray tomography. The geometric question naturally arises from a various geometric inverse problems such as boundary rigidity and spectral rigidity. This thesis studies geodesic X-tomography problems in non-smooth Riemannian geometries. The central objects of interest — known as geodesic X-ray transforms — are various integral transforms encoding the integrals of a function or a tensor field over the geodesics. We encounter two different types of non-smooth geometries: globally non-smooth Riemannian metrics and Riemannian metrics singular at the boundary of the manifold. The thesis contains four articles recording results on X-ray transforms and the geometries themselves. We prove that the geodesic X-ray transform of Lipschitz scalar functions is injective on simple Riemannian manifolds with C1,1 regular metrics. We prove that the X-ray transforms of C1,1 smooth 1-forms and tensor fields of higher rank are solenoidally injective on simple Riemannian manifolds of non-positive sectional curvature with C1,1 regular metrics. These results are based on energy methods and the use of the so called Pestov identity. In addition to injectivity results, we produce a redefinition of simplicity that is compatible with non-smooth geometry, and prove that the redefinition is equivalent to any standard definition of simplicity for C∞ smooth Riemannian metrics. We supplement the injectivity results by considering the normal operator of the X-ray transform in non-smooth geometry. Based on non-smooth microlocal analysis of the normal operator we prove that the geodesic X-ray transform is injective on L2 when the Riemannian metric is simple but only finitely differentiable. The number of derivatives needed depends explicitly on the dimension of the manifold. Riemannian metrics that are C∞ smooth in the interior of a manifold with boundary but have a conformal blow up of a specific strength at the boundary are called gas giant metrics. Such Riemannian metrics are different from but relatives of asymptotically hyperbolic metrics, and arise naturally in the study of wave propagation in gas giant planets. The specific type of singularity is related to the fact that unlike on terrestrial planets the density of a gas giant planet goes to zero at the surface. The specific blow up rate comes from a polytropic model. We prove and apply Pestov identities in gas giant geometry to show that the X-ray transform on a gas giant is injective. We develop the differential geometry of gas giant metrics with an emphasis on the geometry of geodesics, and study the basic analytic properties of the Laplace–Beltrami operator associated to a gas giant metric. The introduction part of the thesis contains an overview of the X-ray tomography in Riemannian geometry and the geometric preliminaries behind it. An overview of the included articles is also provided.