Shape operator ($S_p$)

For an oriented surface with Gauss map $N$, the shape operator at $p$ is the self-adjoint linear map

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.

For a proper Euclidean motion of Euclidean three-space $E(x)=Ax+b$, transport the unit normal by $\tilde{N}(E(p))=AN(p)$. The corresponding shape operators are orthogonally conjugate:Their determinant and trace, hence Gaussian curvature and mean curvature, are unchanged.