As the shell becomes thicker, the contribution of Gauss curvature tensor in terms of energy is no more negligible as compared to that of the first two tensors used in classical thin shells.
Finally as previous, the Gauss curvature tensor is expressed as
The mathematical root of our paper is the pioneering work of mathematician Richard Hamilton on the Gauss curvature
The Gauss curvature of the affine translation surface given by (3.
The Gauss curvature K of the generalized Scherk surface (3.
3) and the Gauss equation that the Gauss curvature K of M is given by
In this case, M is a Lorentz surface of constant Gauss curvature and [alpha][delta] = [beta][gamma] = 0 from (3.
When n = 2, the Gauss-Kronecker curvature is simply called the Gauss curvature
, which is intrinsic due to well-known Gauss' theorema egregium.
On the other hand, the Gauss curvature
K, the mean curvature H are given by
Next, looking at the local isometric embedding of surfaces in R3, they discuss metrics with Gauss curvature
that is everywhere positive, negative, nonnegative, nonpositive, as well as the case of the mixed sign.
3] gives us that the Gauss curvature
is the product of the principal curvatures.
For this purpose, we restrict to the family of rotation surfaces because the constant Gauss curvature
equation reduces into an ordinary differential equation.