The distribution of the polarization of the Cosmic Microwave Background (CMB)
in the sky is determined by the hypothesis of random Gaussian distribution of
the primordial density perturbations. This hypotheses is well motivated by the
inflationary cosmology. Therefore, the test of consistency of the statistical
properties of the CMB polarization field with the Gaussianity of primordial
density fluctuations is a realistic way to study the nature of primordial
inhomogeneities in the Universe. This paper contains the theoretical
predictions of the general statistical properties of the CMB polarization
field. All results obtained under assumption of the Gaussian nature of the
signal. We pay the special attention to the following two problems. First, the
classification and statistics of the singular points of the polarization field
where polarization is equal to zero. Second, the topology of contours of the
value of the degree of polarization. We have investigated the percolation
properties for the zones of ``strong'' and ``weak'' polarization. We also have
calculated Minkowski functionals for the CMB polarization field. All results
are analytical.Comment: Latex, 22 pages, including 5 figure