Inverse problems for semilinear wave equations on Lorentzian manifolds

We consider inverse problems in space-time $(M, g)$, a $4$-dimensional Lorentzian manifold. For semilinear wave equations $\square_g u + H(x, u) = f$, where $\square_g$ denotes the usual Laplace-Beltrami operator, we prove that the source-to-solution map $L: f \rightarrow u|_V$, where $V$ is a neighborhood of a time-like geodesic $\mu$, determines the topological, differentiable structure and the conformal class of the metric of the space-time in the maximal set where waves can propagate from $\mu$ and return back. Moreover, on a given space-time $(M, g)$, the source-to-solution map determines some coefficients of the Taylor expansion of $H$ in $u$.