A support theorem for the geodesic ray transform of symmetric tensor fields

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