Principal curvature ($k_1,k_2$)

The principal curvatures are the two eigenvalues $k_1,k_2$ of the shape operator. Thus $K=k_1{k}_2$ and $H=(k_1+k_2)/2$.

A principal direction is an eigenvector of the shape operator. The normal curvature in a unit principal direction is the corresponding principal curvature.

An umbilical point of a surface is a point where the two principal curvatures are equal, equivalently where the shape operator is a scalar multiple of the identity.