  We consider a strongly damped wave equation on compact manifolds, both with
and without boundaries, and formulate the corresponding inverse problems. For
closed manifolds, we prove that the metric can be uniquely determined, up to an
isometry, from the knowledge of the source-to-solution map. Similarly, for
manifolds with boundaries, we prove that the metric can be uniquely determined,
up to an isometry, from partial knowledge of the Dirichlet-to-Neumann map. The
key point is to retrieve the spectral information of the Laplace-Beltrami
operator, from the Laplace transform of the measurements. Further we show that
the metric can be determined up to an isometry, using a single measurement in
both scenarios.
