A Support Theorem for the Geodesic Ray Transform on Functions

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∈L2(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 analytic microlocal analysis.