A SUPPORT THEOREM FOR THE GEODESIC RAY TRANSFORM OF SYMMETRIC TENSOR FIELDS

Let (M, g) be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of such a field f along maximal geodesics vanish on an appropriate open subset of the space of geodesics in M. Under the assumption that the metric g is real-analytic, it is shown that there exists a vector field v satisfying f = dv on the set of points lying on these geodesics and v = 0 on the intersection of this set with the boundary partial derivative M of the manifold M. Using this result, a Helgason's type of a support theorem for the geodesic ray transform is proven. The approach is based on analytic microlocal techniques.