  The isotope coefficient $\beta$ of the magnetic penetration depth in the
superconducting state is studied at T=0 for a $d$-CDW and a nodal metal model.
Disregarding superconductivity the Fermi surface of the first model possesses
arcs whereas the second model has no arcs. We show that a large increase of
$\beta$ in the pseudogap region is generically incompatible with Fermi arcs in
the pseudogap state. Thus only the second model shows a large increase of
$\beta$ with decreasing doping. The required electron-phonon coupling is small
and compatible with first-principles calculations based on the local density
approximation (LDA).
