Context : Type Ia Supernovae ( SNe Ia ) have been widely used in cosmology as distance indicators .
However , to fully exploit their potential in cosmology , a better control over systematic uncertainties is required .
Some of the uncertainties are related to the unknown nature of the SN Ia progenitors .

Aims : We aim to test the use of integral field unit ( IFU ) spectroscopy for correlating the properties of nearby SNe Ia with the properties of their host galaxies at the location of the SNe .
The results are to be compared with those obtained from an analysis of the total host spectrum .
The goal is to explore this path of constraining the nature of the SN Ia progenitors and further improve the use of SNe Ia in cosmology .

Methods : We used the wide-field IFU spectrograph PMAS/PPAK at the 3.5m telescope of Calar Alto Observatory to observe six nearby spiral galaxies that hosted SNe Ia .
Spatially resolved 2D maps of the properties of the ionized gas and the stellar populations were derived .

Results : Five of the observed galaxies have an ongoing star formation rate of 1-5 M _ { \odot } yr ^ { -1 } and mean stellar population ages \sim 5 Gyr .
The sixth galaxy shows no star formation and has an about 12 Gyr old stellar population .
All galaxies have stellar masses larger than 2 \times 10 ^ { 10 } M _ { \odot } and metallicities above solar .
Four galaxies show negative radial metallicity gradients of the ionized gas up to -0.058 dex kpc ^ { -1 } and one has nearly uniform metallicity with a possible shallow positive slope .
The stellar components show shallower negative metallicity gradients up to -0.03 dex kpc ^ { -1 } .
We find no clear correlation between the properties of the galaxy and those of the supernovae , which may be because of the small ranges spanned by the galaxy parameters .
However , we note that the Hubble residuals are on average positive while negative Hubble residuals are expected for SNe Ia in massive hosts such as the galaxies in our sample .

Conclusions : The IFU spectroscopy on 4-m telescopes is a viable technique for studying host galaxies of nearby SNe Ia .
It allows one to correlate the supernova properties with the properties of their host galaxies at the projected positions of the supernovae .
Our current sample of six galaxies is too small to draw conclusions about the SN Ia progenitors or correlations with the galaxy properties , but the ongoing CALIFA IFU survey will provide a solid basis to exploit this technique more and improve our understanding of SNe Ia as cosmological standard candles .