  We review boundary rigidity theorems assessing that, under appropriate
conditions, Riemannian manifolds with the same spectrum of boundary geodesics
are isometric. We show how to apply these theorems to the problem of
reconstructing a $d+1$ dimensional, negative curvature space-time from boundary
data associated to two-point functions of high-dimension local operators in a
conformal field theory. We also show simple, physically relevant examples of
negative-curvature spaces that fail to satisfy in a subtle way some of the
assumptions of rigidity theorems. In those examples, we explicitly show that
the spectrum of boundary geodesics is not sufficient to reconstruct the metric
in the bulk. We also survey other reconstruction procedures and comment on
their possible implementation in the context of the holographic AdS/CFT
duality.
