  In this paper, we want to check whether or not the carbon abundance can be
affected by initial metallicity. We calculated a series of stellar evolution.
We found that when $Z\leq0.02$, the carbon abundance is almost independent of
metallicity if it is plotted against the initial WD mass. However, when
$Z>0.02$, the carbon abundance is not only a function of the initial WD mass,
but also metallicity, i.e. for a given initial WD mass, the higher the
metallicity, the lower the carbon abundance. Based on some previous studies,
i.e. both a high metallicity and a low carbon abundance lead to a lower
production of $^{\rm 56}$Ni formed during SN Ia explosion, the effects of the
carbon abundance and the metallicity on the amount of $^{\rm 56}$Ni are
enhanced by each other, which may account for the variation of maximum
luminosity of SNe Ia, at least qualitatively. Considering that the central
density of WD before supernova explosion may also play a role on the production
of $^{\rm 56}$Ni and the carbon abundance, the metallicity and the central
density are all determined by the initial parameters of progenitor system, i.e.
the initial WD mass, metallicity, orbital period and secondary mass, the amount
of $^{\rm 56}$Ni might be a function of the initial parameters. Then, our
results might construct a bridge linking the progenitor model and the explosion
model of SNe Ia.
