Extrema of principal curvature and symmetry

Source abstract

In this paper we show that away from umbilic points certain measures of the local reflectional symmetry of a surface in Euclidean 3-space are detected by the extrema of the sectional curvatures along lines of curvature. There are two types of reflectional symmetry, with one detected by the contact between the surface and spheres, and in this case the result is due to Porteous and is 20 years old. We show that an analogous result remains true for the second type of symmetry.