An Inverse Problem for symmetric hyperbolic Partial Differential Operators on Complete Riemannian Manifolds

We show that a complete Riemannian manifold, as well as time independent smooth lower order terms appearing in a first order symmetric perturbation of a Riemannian wave operator can be uniquely recovered, up to the natural obstructions, from a local source to solution map of the respective hyperbolic initial value problem. Our proofs are based on an adaptation of the classical Boundary Control method (BC-method) originally developed by Belishev and Kurylev. The BC-method reduces the PDE-based problem to a purely geometric problem involving the so-called travel time data. For each point in the manifold the travel time data contains the distance function from this point to any point in a fixed \textit{a priori} known compact observation set. It is well known that this geometric problem is solvable. The main novelty of this paper lies in our strategy to recover the lower order terms via a further adaptation of the BC-method.