Reconstructing a metric from local data

Question

Suppose you have a Riemannian manifold $\mathcal{M}$ where the metric $g$ is unknown. Let $U$ be an open subset of the manifold with a coordinate chart $x^\mu$. Now imagine that for each fixed coordinate value $x_0^\mu$ you can compute the curvature tensor $R(0)$ in Riemann normal coordinates adapted to the anchor point $x_0$. With only this data, is it possible to reconstruct the metric on $U$ in the original coordinates $x^\mu$, or on any open subset of $U$? If not, is there any simple additional data which can be specified to make the problem well posed?