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 being zero.

Comparison of curves 2 and 3 with the results of [10] 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 [13], 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].