X-ray Transforms in Pseudo-Riemannian Geometry
Ilmavirta, J. (2018). X-ray Transforms in Pseudo-Riemannian Geometry. Journal of Geometric Analysis, 28(1), 606-626. https://doi.org/10.1007/s12220-017-9834-z
Published inJournal of Geometric Analysis
We study the problem of recovering a function on a pseudo-Riemannian manifold from its integrals over all null geodesics in three geometries: pseudo-Riemannian products of Riemannian manifolds, Minkowski spaces, and tori. We give proofs of uniqueness and characterize non-uniqueness in different settings. Reconstruction is sometimes possible if the signature (n1,n2) satisfies n1≥1 and n2≥2 or vice versa and always when n1,n2≥2 . The proofs are based on a Pestov identity adapted to null geodesics (product manifolds) and Fourier analysis (other geometries). The problem in a Minkowski space of any signature is a special case of recovering a function in a Euclidean space from its integrals over all lines with any given set of admissible directions, and we describe sets of lines for which this is possible. Characterizing the kernel of the null geodesic ray transform on tori reduces to solvability of certain Diophantine systems.
ISSN Search the Publication Forum1050-6926
Publication in research information system
MetadataShow full item record
Showing items with similar title or keywords.
The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds Ilmavirta, Joonas; Mönkkönen, Keijo (Springer Science and Business Media LLC, 2023)We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce ...
Linearized Calderón problem and exponentially accurate quasimodes for analytic manifolds Krupchyk, Katya; Liimatainen, Tony; Salo, Mikko (Elsevier Inc., 2022)In this article we study the linearized anisotropic Calderón problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a ...
Geodesic ray transform with matrix weights for piecewise constant functions Ilmavirta, Joonas; Railo, Jesse (Suomalainen tiedeakatemia, 2020)We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and ...
The Light Ray Transform in Stationary and Static Lorentzian Geometries Feizmohammadi, Ali; Ilmavirta, Joonas; Oksanen, Lauri (Springer, 2021)Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural ...
Fixed angle inverse scattering in the presence of a Riemannian metric Ma, Shiqi; Salo, Mikko (Walter de Gruyter GmbH, 2022)We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a ...