  This paper studies Gaussian random fields with Mat\'ern covariance functions
with smooth parameter $\nu>2$. Two cases of parameter spaces, the Euclidean
space and $N$-dimensional sphere, are considered. For such smooth Gaussian
fields, we have derived the explicit formulae for the expected Euler
characteristic of the excursion set, the expected number and height
distribution of critical points. The results are valuable for approximating the
excursion probability in family-wise error control and for computing p-values
in peak inference.
