  The detection of the magnetic type $B$-mode polarization is the main goal of
future cosmic microwave background (CMB) experiments. In the standard model,
the $B$-mode map is a strongly non-gaussian field due to the lensed component.
Besides the two-point correlation function, the other statistics are also very
important to dig the information of the polarization map. In this paper, we
employ the Minkowski functionals to study the morphological properties of the
lensed $B$-mode maps. We find that the deviations from Gaussianity are very
significant for both full and partial-sky surveys. As an application of the
analysis, we investigate the morphological imprints of the foreground residuals
in the $B$-mode map. We find that even for very tiny foreground residuals, the
effects on the map can be detected by the Minkowski functional analysis.
Therefore, it provides a complementary way to investigate the foreground
contaminations in the CMB studies.
