A Metric Property of Umbilic Points

In the space U of cubic forms of surfaces, regarded as a G-space and endowed with a natural invariant metric, the ratio of the volumes of those representing umbilic points with negative to those with positive indexes is evaluated in terms of the asymmetry of the metric, defined here. A connection of this ratio with that reported by Berry and Hannay (1977) in the domain of Statistical Physics, is discussed. 1. Umbilic Points, Invariant Metrics and Volume Ratios At an umbilic point p of an oriented C3 surface S embedded in an oriented Euclidean 3-space R3 the principal curvatures coincide. In a neighborhood of such point, S can be written in a Monge chart as the graph z = h(x, y) of a function of the form h(x, y) = k 2 (x + y) + 1 6 (ax + 3bxy + 3bxy + ay) + o((x + y)). (1) The frame (x, y; z) is positive and adapted to S at p. This means that the plane orthonormal frame (x, y) is attached to the tangent plane, positively oriented, and the z-axis is along the unit positive normal to S at p. Any other such presentation as the graph Z = H(X,Y ) of a function H(X,Y ) = K 2 (X + Y ) + 1 6 (AX + 3BXY + 3BXY 2 +AY ) + o((X + Y ) differs by a rotation x = cos θX − sin θY, y = sin θX + cos θY, z = Z, linking the positively oriented frames (X,Y ;Z) and (x, y; z), adapted to the surface at p.