  Random fields in nature often have, to a good approximation, Gaussian
characteristics. For such fields, the relative densities of umbilical points --
topological defects which can be classified into three types -- have certain
fixed values. Phenomena described by nonlinear laws can however give rise to a
non-Gaussian contribution, causing a deviation from these universal values. We
consider a Gaussian field with a perturbation added to it, given by a nonlinear
function of that field, and calculate the change in the relative density of
umbilical points. This allows us not only to detect a perturbation, but to
determine its size as well. This geometric approach offers an independent way
of detecting non-Gaussianity, which even works in cases where the field itself
cannot be probed directly.
