  We obtain formulae for the expected number and height distribution of
critical points of smooth isotropic Gaussian random fields parameterized on
Euclidean space or spheres of arbitrary dimension. The results hold in general
in the sense that there are no restrictions on the covariance function of the
field except for smoothness and isotropy. The results are based on a
characterization of the distribution of the Hessian of the Gaussian field by
means of the family of Gaussian orthogonally invariant (GOI) matrices, of which
the Gaussian orthogonal ensemble (GOE) is a special case. The obtained formulae
depend on the covariance function only through a single parameter (Euclidean
space) or two parameters (sphere), and include the special boundary case of
random Laplacian eigenfunctions.
