Umbilic points on Gaussian random surfaces

An umbilic point U on a surface Sigma is a place where the two principal curvatures of Sigma are equal. U is a singularity of Sigma in three different senses: (i) it is the source of elliptic (E) or hyperbolic (H) umbilic catastrophes in the envelope of normals ('focal surface') of Sigma ; (ii) it has index +or- 1 / 2 depending on whether the principal curvature directions of Sigma (defining the lines of curvature) rotate by +or- pi during a circuit of U; (iii) it has a pattern of the 'star' (S), 'lemon' (L) or 'monstar' (M) type depending on the configuration of lines of curvature near U. We calculate the average density of umbilic points, and the fractions alpha of umbilics of the different sorts, for the case where Sigma is a surface whose (small) deviation from a plane is specified by a Gaussian random function. It is always the case that alpha S = alpha +or-1 2/= 1 / 2 . The other fractions depend on the degree of isotropy of the statistics of Sigma . In the isotropic case the elliptic umbilic fraction is alpha E =1- alpha H =0.268, and the monstar fraction is alpha M = 1 / 2 - alpha L =0.053.