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.
n]] denoting the principal curvatures
of E, at the point p [member of] [SIGMA].
If the curvature diagram degenerates to exactly one point then the surface has two constant principal curvatures
which is possible only for a piece of a plane, a sphere or a circular cylinder.
Firstly, the definition of principal curvatures
on an object surface is given.
However, the types of singularities occurring in these sets are essentially the same as in the convex case: where a convex set may be characterized as one for which all principal curvatures
are, in a generalized sense, positive, sets with positive reach admit a parallel description as those for which all principal curvatures
are bounded below.
Niebergall  and Cecil and Jensen  studied proper Dupin hypersurfaces with four distinct principal curvatures
and constant Lie curvature.
Assume that the hypothesis as above and in addition that at each point exactly two principal curvatures
are distinct and they have multiplicities > 1.
2n+1] has at most two distinct constant principal curvatures
, then it is congruent to one of the following: