  In this paper, we investigate the anisotropic Calder{\'o}n problem on
cylindrical Riemannian manifolds with boundary having two ends and equipped
with singular metrics of (simple or double) warped product type, that is whose
warping factors only depend on the horizontal direction of the cylinder. By
singular, we mean that these factors are only assumed to be positive almost
everywhere and to belong to some L p space with 0 < p $\le$ $\infty$. Using
recent developments on the Weyl-Titchmarsh theory for singular Sturm-Liouville
operators, we prove that the local Dirichlet-to-Neumann maps at each end are
well defined and determine the metric uniquely if: 1. (Doubly warped product
case) the coefficients of the metric are L $\infty$ and bounded from below by a
positive constant. 2. (Warped product case) the coefficients of the metrics
belong to a critical L p space where p < $\infty$ depends on the dimension of
the compact fibers of the cylinder. Finally, we show (in the warped product
case and for zero frequency) that these uniqueness results are sharp by giving
simple counterexamples for a class of singular metrics whose coefficients do
not belong to the critical L p space. All these counterexamples lead to a
region of space that is invisible to boundary measurements.
