A Support Theorem for the Geodesic Ray Transform of Functions

TL;DR

This paper proves a support theorem for the geodesic ray transform on real-analytic simple Riemannian manifolds, showing that vanishing on certain geodesics implies the function is zero on their union.

Contribution

It establishes a new support theorem for the geodesic ray transform using microlocal unique continuation techniques in the real-analytic setting.

Findings

Support theorem for geodesic ray transform established

Vanishing on open set of geodesics implies function vanishes on their union

Uses microlocal analysis and unique continuation methods

Abstract

Let $(M,g)$ be a simple Riemannian manifold. Under the assumption that the metric $g$ is real-analytic, it is shown that if the geodesic ray transform of a function $f\in L^{2}(M)$ vanishes on an appropriate open set of geodesics, then $f=0$ on the set of points lying on these geodesics. The approach is based on a microlocal version of unique continuation of analytic functions.