In this case, the principal curvatures
[[lambda].sup.[delta].sub.1], ..., [[lambda].sup.[delta].sub.n] of [[lambda].sup.[delta]] are given by [alpha] [not equal to] a, AS = 0 (j = 2, ...,n).
where k = [square root of [k.sup.2.sub.1] + [k.sup.2.sub.2] + [k.sup.2.sub.3]] is defined as curvature and [k.sub.1] [k.sub.2] and [k.sub.3] denote principal curvatures
of the curve a according to parallel frame (Erdogdu, 2015).
The Gaussian curvature is calculated as the product of the principal curvatures
; thus, at points where any of the principal curvatures
is zero, GC is also zero, in contrast to the MC, which averages the principal curvatures
and is not necessarily zero at these points.
the principal curvatures
[k.sub.1] and [k.sub.2], the shape index S and the curvedness index C.
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
[k.sub.1] and [k.sub.2], respectively.
Also, the Gauss-curvature bending energy vanishes everywhere since the imposed geometry has a 1D periodicity, which results in one of the two 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.
For example, the average principal curvatures
(mean curvature) can describe the local folding of the surface.
where [kappa] and [bar.[kappa]] are bending constants, [C.sub.1] and [C.sub.2] are principal curvatures
of a vesicle surface, and [C.sub.0] is the spontaneous curvature.
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.
By the differential geometry , the principal curvatures
at the vertex O of the revolutionary paraboloid are [c.sub.1] = [c.sub.2] = c with c > 0 in the illustrated coordinate system.
We give a relation between one of the principal curvatures
of the invariant surface and hyperbolic curvature of profile curve of the invariant surface in [H.sup.3].