  There is still considerable debate over the progenitors of type Ia supernovae
(SNe Ia). Likewise, it is not agreed how single white dwarfs with masses less
than ~0.5 Msun can be formed in the field, even though they are known to exist.
We consider whether single low-mass white dwarfs (LMWDs) could have been formed
in binary systems where their companions have exploded as a SN Ia. In this
model, the observed single LMWDs are the remnants of giant-branch donor stars
whose envelopes have been stripped off by the supernova explosion. We
investigate the likely remnants of SNe Ia, including the effects of the
explosion on the envelope of the donor star. We also use evolutionary arguments
to examine alternative formation channels for single LMWDs. In addition, we
calculate the expected kinematics of the potential remnants of SNe Ia. SN Ia in
systems with giant-branch donor stars can naturally explain the production of
single LMWDs. It seems difficult for any other formation mechanism to account
for the observations, especially for those single LMWDs with masses less than
~0.4 Msun. Independent of those results, we find that the kinematics of one
potentially useful population containing single LMWDs is consistent with our
model. Studying remnant white-dwarf kinematics seems to be a promising way to
investigate SN Ia progenitors. The existence of single LMWDs appears to
constitute evidence for the production of SNe Ia in binary systems with a
red-giant donor star. Other single white dwarfs with higher space velocities
support a second, probably dominant, population of SN Ia progenitors which
contained main-sequence or subgiant donor stars at the time of explosion. The
runaway stars LP400-22 and US 708 suggest the possibility of a third formation
channnel for some SNe Ia in systems where the donor stars are hot subdwarfs.
