In this talk we will discuss the differences in the methodology of determining time-dependent and time-independent coefficients appearing in a hyperbolic equation in a Riemannian manifold. The talk is based on two recent research projects: 1) We will prove that a local source-to-solution map of a hyperbolic partial differential operator on a complete Riemannian manifold (no boundary, and possible non-compact) determines a) the topology and the geometry of the manifold uniquely and b) the lower order time-independent coefficients up to a natural obstruction. 2) We will prove that under certain geometric assumptions the knowledge of a partial Cauchy data set uniquely determines time-dependent lower order coefficients appearing in a hyperbolic initial / boundary value problem.

This talk is based on joint works with Boya Liu (NC State University), Andrew Shedlock (NC State University) and Lili Yan (University of Minnesota).