Principal curvatures

Generalization to hypersurfaces: Gauss' Equation#For hypersurfaces.
It is the normal curvature of an immersed surface along a principal direction, i.e. along a direction in which it assumes an extremal value. They are usually denoted by ${\kappa }_1$ and ${\kappa }_2$. Their product is the Gaussian curvature.

They can be computed as the eigenvalues of the shape operator, that is, the second fundamental form with raised index.

In matrix notation if

then ${\kappa }_1$ and ${\kappa }_2$ are the eigenvalues of ${\mathcal{F}}_1^{-1}{\mathcal{F}}_2$.