  The progenitor systems accounting for explosions of type Ia supernovae (SNe
Ia) is still under debate. Symbiotic channel is one of the possible progenitor
scenarios, in which the WDs in these systems increase in mass through wind
accretion from their red giant companions. The mass-loss processes of the
giants in the symbiotic systems could produce amount of circumstellar medium
(CSM), and the detections of the interaction signals between SN ejecta and CSM
can be used as an ideal way to distinguish the different progenitor models.
However, the density distribution and geometric structure of the CSM around the
symbiotic systems remain highly uncertain. By assuming that the tidal torque
from binary interaction can increase the mass-loss rate of the red giant and
cause the stellar wind concentrate towards the equatorial plane, we provide a
simplified method to estimate the density and the degree of deviation from
spherical symmetry of the CSM. Based on the calculations of the binary
evolutions of symbiotic systems using stellar evolution code MESA, we obtained
the parameter space for producing SNe Ia. We found that SNe Ia could originate
from symbiotic systems with massive carbon-oxygen white dwarfs (CO WDs), while
the half-opening angle of the stellar wind from red giant towards the WD varies
with the binary evolution, resulting in the formation of surrounding CSM with
different geometric structures. The corresponding properties of ejecta-CSM
interactions may be examined by the spectropolarimetry observations in the
future, from which one may find additional relationship between circumstellar
environment of SNe Ia and their progenitor systems.
