Support theorems for the transverse ray transform of tensor fields of rank m

Abstract

Let m and n be integers satisfying m≥2 and n≥m+2. Let (M,g) be a simple, real analytic, Riemannian manifold of dimension n with boundary and f be a rank m-tensor field defined over it. In this work, we prove a support theorem for the transverse ray transform of such tensor fields. More specifically, we prove that for a tensor field f of rank m, if the transverse ray transform of f vanishes over an appropriate open set of maximal geodesics of M, then the support of f vanishes on the points of M that lie on the union of the aforementioned open set of geodesics. We also show that if the tensor field is assumed to be symmetric, then one has a similar support theorem for the transverse ray transform of symmetric tensor fields of rank up to n−1.