  Type Ia supernovae (SNe Ia) can be used as the standard candlelight to
determine the cosmological distances because they are thought to have a uniform
fuel amount. Recent observations of several overluminous SNe Ia suggest that
the white dwarf masses at supernova explosion may significantly exceed the
canonical Chandrasekhar mass limit. These massive white dwarfs may be supported
by rapid differential rotation. Based on single degenerate model and the
assumption that the white dwarf would differentially rotate when the accretion
rate $\dot{M}>3\times10^{-7}M_{\odot}\rm yr^{-1}$, we have calculated the
evolutions of close binaries consisting of a white dwarf and a normal
companion. To include the effect of rotation, we introduce an effective mass
$M_{\rm eff}$ for white dwarfs. For the donor stars with two different
metallicities $Z=0.02$ and 0.001, we present the distribution of the initial
donor star masses and the orbital periods of the progenitors of
super-Chandrasekhar mass SNe Ia. The calculation results indicate that, for an
initial massive white dwarf of $1.2M_{\odot}$, a considerable fraction of SNe
Ia may result from super-Chandresekhar mass white dwarfs, but very massive
($>1.7 M_{\sun}$) white dwarfs are difficult to form, and none of them could be
found in old populations. However, super-Chandrasekhar mass SNe Ia are very
rare when the initial mass of white dwarfs is $1.0M_{\odot}$. Additionally, SNe
Ia in low metallicity environment are more likely to be homogeneous.
