  The electronic structure of NdTe$_3$ in the charge density wave phase (CDW)
is investigated by angle-resolved photoemission spectroscopy. The combination
of high-quality crystals and careful surface preparation reveals subtle and
previously unobserved details in the Fermi surface topology, allowing an
interpretation of the rich and unexplained quantum oscillations in the rare
earth tritellurides RTe$_3$. In particular, several closed Fermi surface
elements can be observed that are related to CDW-induced replicas of the
original bands, leading to the curious situation in which a CDW does not only
remove Fermi surface elements but creates new ones that are observable in
transport experiments. Moreover, a large residual Fermi surface is found in the
CDW gap, very close to the position of the gapped normal-state Fermi surface.
Its area agrees very well with high-frequency quantum oscillations in NdTe$_3$
and its presence is explained by strong electron-phonon coupling combined with
the quasi one-dimensional character of the CDW. Finally, we identify the origin
of the low-frequency $\alpha$ quantum oscillations ubiquitous for the lighter R
elements in the RTe$_3$ family and responsible for the high mobility in these
compounds.
