  In this paper we consider the inverse problem of determining on a compact
Riemannian manifold the metric tensor in the wave equation with Dirichlet data
from measured Neumann boundary observations. This information is enclosed in
the dynamical Dirichlet-to-Neumann map associated to the wave equation. We
prove in dimension $n\geq 2$ that the knowledge of the Dirichlet-to-Neumann map
for the wave equation uniquely determines the metric tensor and we establish
logarithm-type stability.
