A Metric property of Umbilic points

Abstract

In the space of cubic forms of surfaces, regarded as a G-space and endowed with a natural invariant metric, the ratio of umbilic points with a negative index to those with a positive index is evaluated in terms of the asymmetry of the metric, defined here. This work contains also a section with references pertinent to umbilic points and one with a discussion of a connection of the above defined ratio with that reported in 1977 by Berry and Hannay in the domain of Statistical Physics.