In a homonymous paper, with F. Pistolato, we obtain a new formula to compute the expected intrinsic volumes (i.e., L.-K. curvatures, or Minkowski functionals) of the excursion set of any non-isotropic Gaussian fields on any three-dimensional compact Riemannian manifold. This work is motivated by the study of the Cosmic Microwave Background (CMB), a radiation that serves as the primary source of information about the nature of the universe. Its fluctuations are modeled as random fields. So far, most studies (and space missions) have focused on the CMB temperature, modeled as an isotropic random field on the two-sphere, where the geometric functionals (area, boundary length, Euler characteristic) of the excursion set have proven to be powerful statistical tools. Currently, the main focus is shifting to the CMB polarization, which can be modeled as a spin-2 random field on SO(3), but is not isotropic. Besides presenting the formula and the model, I will explain the challenges involved in this more general framework.