  Modern proposed atmospheric neutrino oscillation experiments, such as PINGU
in the Antarctic ice or ORCA in Mediterranean sea water, aim for precision
measurements of the oscillation parameters including the ordering of the
neutrino masses. They can, however, go far beyond that: Since neutrino
oscillations are affected by the coherent forward scattering with matter,
neutrinos can provide a new view on the interior of the earth. We show that the
proposed atmospheric oscillation experiments can measure the lower mantle
density of the earth with a precision at the level of a few percent, including
the uncertainties of the oscillation parameters and correlations among
different density layers. While the earth's core is, in principle, accessible
by the angular resolution, new technology would be required to extract
degeneracy-free information.
