  In this paper we report on the bulk features of the hole carved by the
companion star in the material ejected during a Type Ia supernova explosion. In
particular we are interested in the long term evolution of the hole as well as
in its fingerprint in the geometry of the supernova remnant after several
centuries of evolution, which is a hot topic in current Type Iasupernovae
studies. We use an axisymmetric smoothed particle hydrodynamics code to
characterize the geometric properties of the supernova remnant resulting from
the interaction of this ejected material with the ambient medium. Our aim is to
use supernova remnant observations to constrain the single degenerate scenario
for Type Ia supernova progenitors. Our simulations show that the hole will
remain open during centuries, although its partial or total closure at later
times due to hydrodynamic instabilities is not excluded. Close to the edge of
the hole, the Rayleigh-Taylor instability grows faster, leading to plumes that
approach the edge of the forward shock. We also discuss other geometrical
properties of the simulations, like the evolution of the contact discontinuity.
