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 (SN Ia) 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 (SNR) after several centuries of evolution, which is a hot topic in current SN Ia studies. We use an axisymmetric smoothed particle hydrodynamics code to characterize the geometric properties of the SNR resulting from the interaction of this ejected material with the ambient medium.Our aim is to use SNR observations to constrain the single degenerate scenario for SN Ia 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.