  Spherical spin random fields are used to model the Cosmic Microwave
Background polarization, the study of which is at the heart of modern Cosmology
and will be the subject of the LITEBIRD mission, in the 2030s. Its scope is to
collect datas to test the theoretical predictions of the Cosmic Inflation
model. In particular, the Minkowski functionals, or the Lipschitz-Killing
curvatures, of excursion sets can be used to detect deviations from Gaussianity
and anisotropies of random fields, being fine descriptors of their geometry and
topology.
  In this paper we give an explicit, non-asymptotic, formula for the
expectation of the Lipshitz-Killing curvatures of the excursion set of the real
part of an arbitrary left-invariant Gaussian spin spherical random field, seen
as a field on $SO(3)$. Our findings are coherent with the asymptotic ones
presented in Carr\'on Duque, J. et al. "Minkowski Functionals in $SO(3)$ for
the spin-2 CMB polarisation field", Journal of Cosmology and Astroparticle
Physics (2024). We also give explicit expressions for the Adler-Taylor metric,
and its curvature. We obtain such result as an application of a general formula
that applies to any nondegenerate Gaussian random field defined on an arbitrary
three dimensional compact Riemannian manifold. The novelty is that the
Lipschitz-Killing curvatures are computed with respect to an arbitrary metric,
possibly different than the Adler-Taylor metric of the field.
