The rulings are principal curvature
lines with vanishing normal curvature and the Gaussian curvature vanishes at all surface points.
Thus the Gauss-Kronecker curvature G(p) is nothing but the product of the principal curvature
of the sampling) is given by the maximal principal curvature
In their framework, ridge and valley vertices are detected as zero-crossings, and then connected along the direction of principal curvature
The eigenvalues and eigenvectors of the quadratic form correspond to the principal curvature
values and to the directions of principal curvature
jik] = 0, in terms of the principal curvature
functions and three vector valued functions of one variable.
E] denote a principal curvature
with the induced hyperbolic and Euclidean metric on S respectively, then [kappa] and [[kappa].
Thus, the principal curvatures
can be obtained as the roots of the quadratic equation [k.
Different normal sections correspond to different curvatures at the evaluated point, and the maximum and minimum values of these curvatures are called the principal curvatures
Comparison of curves 2 and 3 with the results of  shows, that the rigidity of shallow spherical shells with the inclusion, for equal values of the dimensionless principal curvatures
and other parameters, significantly higher than the rigidity of corresponding cylindrical shells with inclusion.
In recent years, one of the principal research subjects already current in this theory is to characterize complete spacelike hypersurfaces with constant mean curvature (or constant scalar curvature) and two distinct principal curvatures
one of which is simple.
Finally, Chapter 7 is dedicated to various aspects of extrinsic geometry, such as Frenet formulas for curves, ruled and developable surfaces, the shape operator, principal curvatures
and curvature lines, etc.