The geodesic ray transform on two-dimensional Cartan-Hadamard manifolds

TL;DR

This paper proves two injectivity theorems for the geodesic ray transform on two-dimensional Cartan-Hadamard manifolds with non-positive curvature, advancing understanding of integral geometry in these geometries.

Contribution

It establishes injectivity results for the geodesic ray transform on 2D Cartan-Hadamard manifolds with bounded and decaying non-positive curvature, filling a gap in geometric analysis.

Findings

Proved injectivity for bounded non-positive curvature.

Proved injectivity for decaying non-positive curvature.

Extended the class of manifolds where the geodesic ray transform is injective.

Abstract

We prove two injectivity theorems for the geodesic ray transform on two-dimensional, complete, simply connected Riemannian manifolds with non-positive Gaussian curvature, also known as Cartan-Hadamard manifolds. The first theorem is concerned with bounded non-positive curvature and the second with decaying non-positive curvature.